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geodesic flow, Arnol'd diffusion, Melnikov, normal hyperbolicity, ionization
---------------9809081917531
Content-Type: application/x-tex; name="paper.tex"
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% This TeX file was generated by ./texpsinclude
% from the original TeX file paper.tex
% and the PostScript files psfig.sty tori.eps cross.eps.
% TeX writes out the included PostScript to files.
%
% Here is part of paper.tex:
\documentstyle[12pt,psfig,amssymb,draft]{article}

% Here are the
% TeX macros for dumping included Postscript to files.
% Adapted from Knuth's \answer macro in the TeXbook.
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94

\def\endofps{EndOfTheIncludedPostscriptMagicCookie}
\chardef\other=12
\newwrite\psdumphandle 
\outer\def\psdump#1{\par\medbreak
  \immediate\openout\psdumphandle=#1
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copypsline}
\def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials
  \catcode`\\=\other \obeylines}
{\obeylines \gdef\copypsline#1
  {\def\next{#1}%
  \ifx\next\endofps\let\next=\endgroup %
  \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}}
\outer\def\closepsdump{
  \immediate\closeout\psdumphandle}

% Here is the PostScript for psfig.sty:
\message{Writing file psfig.sty}
\psdump{psfig.sty}% Psfig/TeX 
\def\PsfigVersion{1.10}
\def\setDriver{\DvipsDriver} % \DvipsDriver or \OzTeXDriver
%
% All software, documentation, and related files in this distribution of
% psfig/tex are Copyright 1993 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex 
% providing that this notice is clearly maintained. The right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author(s) of that portion.
%
% To use with LaTeX, use \documentstyle[psfig,...]{...}
% To use with TeX, use \input psfig.sty
%
% Bugs and improvements to trevor@media.mit.edu.
%
% Thanks to Ned Batchelder, Greg Hager (GDH), J. Daniel Smith (JDS),
% Tom Rokicki (TR), Robert Russell (RR), George V. Reilly (GVR),
% Ken McGlothlen (KHC), Baron Grey (BG), Gerhard Tobermann (GT).
% and all others who have contributed code and comments to this project!
%
% ======================================================================
% Modification History:
%
%  9 Oct 1990   JDS	used more robust bbox reading code from Tom Rokicki
% 29 Mar 1991   JDS	implemented rotation= option
% 25 Jun 1991   RR	if bb specified on cmd line don't check
%			for .ps file.
%  3 Jul 1991	JDS	check if file already read in once
%  4 Sep 1991	JDS	fixed incorrect computation of rotated
%			bounding box
% 25 Sep 1991	GVR	expanded synopsis of \psfig
% 14 Oct 1991	JDS	\fbox code from LaTeX so \psdraft works with TeX
%			changed \typeout to \ps@typeout
% 17 Oct 1991	JDS	added \psscalefirst and \psrotatefirst
% 23 Jun 1993   KHC     ``doclip'' must appear before ``rotate''
% 27 Oct 1993   TJD	removed printing of filename to avoid 
%			underscore problems. changed \frame to \fbox.
%			Added OzTeX support from BG. Added new
%			figure search path code from GT.
%
% ======================================================================
%
% Command synopsis:
%
% \psdraft	draws an outline box, but doesn't include the figure
%		in the DVI file.  Useful for previewing.
%
% \psfull	includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
% 		before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
% 		 figure after rotation.  Asymetric figures will
% 		 appear to shrink.
%
% \psfigurepath{dir:dir:...}  sets the path to search for the figure
%
% \psfig
% usage: \psfig{file=, figure=, height=, width=,
%			bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=, angle=, silent=}
%
%	"file" is the filename.  If no path name is specified and the
%		file is not found in the current directory,
%		it will be looked for in directory \psfigurepath.
%	"figure" is a synonym for "file".
%	By default, the width and height of the figure are taken from
%		the BoundingBox of the figure.
%	If "width" is specified, the figure is scaled so that it has
%		the specified width.  Its height changes proportionately.
%	If "height" is specified, the figure is scaled so that it has
%		the specified height.  Its width changes proportionately.
%	If both "width" and "height" are specified, the figure is scaled
%		anamorphically.
%	"bbllx", "bblly", "bburx", and "bbury" control the PostScript
%		BoundingBox.  If these four values are specified
%               *before* the "file" option, the PSFIG will not try to
%               open the PostScript file.
%	"rheight" and "rwidth" are the reserved height and width
%		of the figure, i.e., how big TeX actually thinks
%		the figure is.  They default to "width" and "height".
%	The "clip" option ensures that no portion of the figure will
%		appear outside its BoundingBox.  "clip=" is a switch and
%		takes no value, but the `=' must be present.
%	The "angle" option specifies the angle of rotation (degrees, ccw).
%	The "silent" option makes \psfig work silently.
%
% ======================================================================
% check to see if macros already loaded in (maybe some other file says
% "\input psfig") ...
\ifx\undefined\psfig\else\endinput\fi
%
% from a suggestion by eijkhout@csrd.uiuc.edu to allow
% loading as a style file. Changed to avoid problems
% with amstex per suggestion by jbence@math.ucla.edu

\let\LaTeXAtSign=\@
\let\@=\relax
\edef\psfigRestoreAt{\catcode`\@=\number\catcode`@\relax}
%\edef\psfigRestoreAt{\catcode`@=\number\catcode`@\relax}
\catcode`\@=11\relax
\newwrite\@unused
\def\ps@typeout#1{{\let\protect\string\immediate\write\@unused{#1}}}

\def\DvipsDriver{
	\ps@typeout{psfig/tex \PsfigVersion -dvips}
\def\PsfigSpecials{\DvipsSpecials} 	\def\ps@dir{/}
\def\ps@predir{} }
\def\OzTeXDriver{
	\ps@typeout{psfig/tex \PsfigVersion -oztex}
	\def\PsfigSpecials{\OzTeXSpecials}
	\def\ps@dir{:}
	\def\ps@predir{:}
	\catcode`\^^J=5
}

%% Here's how you define your figure path.  Should be set up with null
%% default and a user useable definition.

\def\figurepath{./:}
\def\psfigurepath#1{\edef\figurepath{#1:}}

%%% inserted for Searching Unixpaths
%%% (the path must end with :)
%%% (call: \DoPaths\figurepath )
%%%------------------------------------------------------
\def\DoPaths#1{\expandafter\EachPath#1\stoplist}
%
\def\leer{}
\def\EachPath#1:#2\stoplist{% #1 part of the list (delimiter :)
  \ExistsFile{#1}{\SearchedFile}
  \ifx#2\leer
  \else
    \expandafter\EachPath#2\stoplist
  \fi}
%
% exists the file (does not work for directories!)
%
\def\ps@dir{/}
\def\ExistsFile#1#2{%
   \openin1=\ps@predir#1\ps@dir#2
   \ifeof1
       \closein1
       %\ps@typeout{...not: \ps@predir#1\ps@dir#2}
   \else
       \closein1
       %\ps@typeout{...in:  \ps@predir#1\ps@dir#2}
        \ifx\ps@founddir\leer
          %\ps@typeout{set founddir #1}
           \edef\ps@founddir{#1}
        \fi
   \fi}
%------------------------------------------------------
%
% Get dir in path or error
%
\def\get@dir#1{%
  \def\ps@founddir{}
  \def\SearchedFile{#1}
  \DoPaths\figurepath
%  \fi
}
%------------------------------------------------------
%%% END of Searching Unixpaths


%
% @psdo control structure -- similar to Latex @for.
% I redefined these with different names so that psfig can
% be used with TeX as well as LaTeX, and so that it will not 
% be vunerable to future changes in LaTeX's internal
% control structure,
%
\def\@nnil{\@nil}
\def\@empty{}
\def\@psdonoop#1\@@#2#3{}
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}
% 
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 3pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip 
                 \fboxsep\vrule width \fboxrule}
                 \hrule height \fboxrule}}}}
\fi
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% file reading stuff from epsf.tex
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%   Portions modified/removed for use in PSFIG package by
%      J. Daniel Smith, 9 October 1990.
%
\newread\ps@stream
\newif\ifnot@eof       % continue looking for the bounding box?
\newif\if@noisy        % report what you're making?
\newif\if@atend        % %%BoundingBox: has (at end) specification
\newif\if@psfile       % does this look like a PostScript file?
%
% PostScript files should start with `%!'
%
{\catcode`\%=12\global\gdef\epsf@start{%!}}
\def\epsf@PS{PS}
%
\def\epsf@getbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\ps@stream=\ps@predir#1
\ifeof\ps@stream\ps@typeout{Error, File #1 not found}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\not@eoftrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
    \loop
       \if@psfile
	  \read\ps@stream to \epsf@fileline
       \else{
	  \obeyspaces
          \read\ps@stream to \epsf@tmp\global\let\epsf@fileline\epsf@tmp}
       \fi
       \ifeof\ps@stream\not@eoffalse\else
%
%   Check the first line for `%!'.  Issue a warning message if its not
%   there, since the file might not be a PostScript file.
%
       \if@psfile\else
       \expandafter\epsf@test\epsf@fileline:. \\%
       \fi
%
%   We check to see if the first character is a % sign;
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:' and the `(atend)' specification was not found.
%   That is, the only way to stop is when the end of file is reached,
%   or a `%%BoundingBox: llx lly urx ury' line is found.
%
          \expandafter\epsf@aux\epsf@fileline:. \\%
       \fi
   \ifnot@eof\repeat
   }\closein\ps@stream\fi}%
%
% This tests if the file we are reading looks like a PostScript file.
%
\long\def\epsf@test#1#2#3:#4\\{\def\epsf@testit{#1#2}
			\ifx\epsf@testit\epsf@start\else
\ps@typeout{Warning! File does not start with `\epsf@start'.  It may not be a PostScript file.}
			\fi
			\@psfiletrue} % don't test after 1st line
%
%   We still need to define the tricky \epsf@aux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12\global\let\epsf@percent=%\global\def\epsf@bblit{%BoundingBox}}
%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.  We continue searching if `(at end)'
%   was found after the `%BoundingBox:'.
%
\long\def\epsf@aux#1#2:#3\\{\ifx#1\epsf@percent
   \def\epsf@testit{#2}\ifx\epsf@testit\epsf@bblit
	\@atendfalse
        \epsf@atend #3 . \\%
	\if@atend	
	   \if@verbose{
		\ps@typeout{psfig: found `(atend)'; continuing search}
	   }\fi
        \else
        \epsf@grab #3 . . . \\%
        \not@eoffalse
        \global\no@bbfalse
        \fi
   \fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsf@grab #1 #2 #3 #4 #5\\{%
   \global\def\epsf@llx{#1}\ifx\epsf@llx\empty
      \epsf@grab #2 #3 #4 #5 .\\\else
   \global\def\epsf@lly{#2}%
   \global\def\epsf@urx{#3}\global\def\epsf@ury{#4}\fi}%
%
% Determine if the stuff following the %%BoundingBox is `(atend)'
% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)} 
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}


% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% trigonometry stuff from "trig.tex"
\chardef\psletter = 11 % won't conflict with \begin{letter} now...
\chardef\other = 12

\newif \ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\c@mputetrue % but assume that we do

\let\then = \relax
\def\r@dian{pt }
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}

{ %%% Things that need abnormal catcodes %%%
	\catcode `\@ = \psletter
	\gdef \nodimen {\expandafter \n@dimen \the \dimen}
	\gdef \term #1 #2 #3%
	       {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
		\edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
				   %%% freeze parameter 2 (dimen, by value)
		\t@rm {\t@} {\t@@} {#3}%
	       }
	\gdef \t@rm #1 #2 #3%
	       {{%
		\count 0 = 0
		\dimen 0 = 1 \dimensionless@nit
		\dimen 2 = #2\relax
		\Mess@ge {Calculating term #1 of \nodimen 2}%
		\loop
		\ifnum	\count 0 < #1
		\then	\advance \count 0 by 1
			\Mess@ge {Iteration \the \count 0 \space}%
			\Multiply \dimen 0 by {\dimen 2}%
			\Mess@ge {After multiplication, term = \nodimen 0}%
			\Divide \dimen 0 by {\count 0}%
			\Mess@ge {After division, term = \nodimen 0}%
		\repeat
		\Mess@ge {Final value for term #1 of 
				\nodimen 2 \space is \nodimen 0}%
		\xdef \Term {#3 = \nodimen 0 \r@dians}%
		\aftergroup \Term
	       }}
	\catcode `\p = \other
	\catcode `\t = \other
	\gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}

\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym

\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
	\count 0 = #1\relax
	\count 2 = #2\relax
	\count 4 = 65536
	\Mess@ge {Before scaling, count 0 = \the \count 0 \space and
			count 2 = \the \count 2}%
	\ifnum	\count 0 > 32767 %%% do our best to avoid overflow
	\then	\divide \count 0 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 0 < -32767
		\then	\divide \count 0 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\ifnum	\count 2 > 32767 %%% while retaining reasonable accuracy
	\then	\divide \count 2 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 2 < -32767
		\then	\divide \count 2 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\multiply \count 0 by \count 2
	\divide \count 0 by \count 4
	\xdef \product {#1 = \the \count 0 \internal@nits}%
	\aftergroup \product
       }}

\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x+90) = sin(180-x)
		\multiply\dimen0 by -1
		\advance\dimen0 by 180\r@dian
		\r@duce
	    \else \ifdim\dimen0 < -90\r@dian \then  % sin(-x) = sin(360+x)
		\advance\dimen0 by 360\r@dian
		\r@duce
		\fi
	    \fi}

\def\Sine#1%
       {{%
	\dimen 0 = #1 \r@dian
	\r@duce
	\ifdim\dimen0 = -90\r@dian \then
	   \dimen4 = -1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 90\r@dian \then
	   \dimen4 = 1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 0\r@dian \then
	   \dimen4 = 0\r@dian
	   \c@mputefalse
	\fi
%
	\ifc@mpute \then
        	% convert degrees to radians
		\divide\dimen0 by 180
		\dimen0=3.141592654\dimen0
%
		\dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
		\divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
		\Mess@ge {Sin: calculating Sin of \nodimen 0}%
		\count 0 = 1 %%% see power-series expansion for sine
		\dimen 2 = 1 \r@dian %%% ditto
		\dimen 4 = 0 \r@dian %%% ditto
		\loop
			\ifnum	\dimen 2 = 0 %%% then we've done
			\then	\stillc@nvergingfalse 
			\else	\stillc@nvergingtrue
			\fi
			\ifstillc@nverging %%% then calculate next term
			\then	\term {\count 0} {\dimen 0} {\dimen 2}%
				\advance \count 0 by 2
				\count 2 = \count 0
				\divide \count 2 by 2
				\ifodd	\count 2 %%% signs alternate
				\then	\advance \dimen 4 by \dimen 2
				\else	\advance \dimen 4 by -\dimen 2
				\fi
		\repeat
	\fi		
			\xdef \sine {\nodimen 4}%
       }}

% Now the Cosine can be calculated easily by calling \Sine
\def\Cosine#1{\ifx\sine\UnDefined\edef\Savesine{\relax}\else
		             \edef\Savesine{\sine}\fi
	{\dimen0=#1\r@dian\advance\dimen0 by 90\r@dian
	 \Sine{\nodimen 0}
	 \xdef\cosine{\sine}
	 \xdef\sine{\Savesine}}}	      
% end of trig stuff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\psdraft{
	\def\@psdraft{0}
	%\ps@typeout{draft level now is \@psdraft \space . }
}
\def\psfull{
	\def\@psdraft{100}
	%\ps@typeout{draft level now is \@psdraft \space . }
}

\psfull

\newif\if@scalefirst
\def\psscalefirst{\@scalefirsttrue}
\def\psrotatefirst{\@scalefirstfalse}
\psrotatefirst

\newif\if@draftbox
\def\psnodraftbox{
	\@draftboxfalse
}
\def\psdraftbox{
	\@draftboxtrue
}
\@draftboxtrue

\newif\if@prologfile
\newif\if@postlogfile
\def\pssilent{
	\@noisyfalse
}
\def\psnoisy{
	\@noisytrue
}
\psnoisy
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@angle
\newif\if@clip
\newif\if@verbose
\def\@p@@sclip#1{\@cliptrue}
%
%
\newif\if@decmpr
%
\def\@p@@sfigure#1{\def\@p@sfile{null}\def\@p@sbbfile{null}\@decmprfalse
   % look directly for file (e.g. absolute path)
   \openin1=\ps@predir#1
   \ifeof1
	\closein1
	% failed, search directories for file
	\get@dir{#1}
	\ifx\ps@founddir\leer
		% failed, search directly for file.bb
		\openin1=\ps@predir#1.bb
		\ifeof1
			\closein1
			% failed, search directories for file.bb
			\get@dir{#1.bb}
			\ifx\ps@founddir\leer
				% failed, lose.
				\ps@typeout{Can't find #1 in \figurepath}
			\else
				% found file.bb in search dir
				\@decmprtrue
				\def\@p@sfile{\ps@founddir\ps@dir#1}
				\def\@p@sbbfile{\ps@founddir\ps@dir#1.bb}
			\fi
		\else
			\closein1
			%found file.bb directly
			\@decmprtrue
			\def\@p@sfile{#1}
			\def\@p@sbbfile{#1.bb}
		\fi
	\else
		% found file in search dir
		\def\@p@sfile{\ps@founddir\ps@dir#1}
		\def\@p@sbbfile{\ps@founddir\ps@dir#1}
	\fi
   \else
	% found file directly
	\closein1
	\def\@p@sfile{#1}
	\def\@p@sbbfile{#1}
   \fi
}
%
%
%
\def\@p@@sfile#1{\@p@@sfigure{#1}}
%
\def\@p@@sbbllx#1{
		%\ps@typeout{bbllx is #1}
		\@bbllxtrue
		\dimen100=#1
		\edef\@p@sbbllx{\number\dimen100}
}
\def\@p@@sbblly#1{
		%\ps@typeout{bblly is #1}
		\@bbllytrue
		\dimen100=#1
		\edef\@p@sbblly{\number\dimen100}
}
\def\@p@@sbburx#1{
		%\ps@typeout{bburx is #1}
		\@bburxtrue
		\dimen100=#1
		\edef\@p@sbburx{\number\dimen100}
}
\def\@p@@sbbury#1{
		%\ps@typeout{bbury is #1}
		\@bburytrue
		\dimen100=#1
		\edef\@p@sbbury{\number\dimen100}
}
\def\@p@@sheight#1{
		\@heighttrue
		\dimen100=#1
   		\edef\@p@sheight{\number\dimen100}
		%\ps@typeout{Height is \@p@sheight}
}
\def\@p@@swidth#1{
		%\ps@typeout{Width is #1}
		\@widthtrue
		\dimen100=#1
		\edef\@p@swidth{\number\dimen100}
}
\def\@p@@srheight#1{
		%\ps@typeout{Reserved height is #1}
		\@rheighttrue
		\dimen100=#1
		\edef\@p@srheight{\number\dimen100}
}
\def\@p@@srwidth#1{
		%\ps@typeout{Reserved width is #1}
		\@rwidthtrue
		\dimen100=#1
		\edef\@p@srwidth{\number\dimen100}
}
\def\@p@@sangle#1{
		%\ps@typeout{Rotation is #1}
		\@angletrue
%		\dimen100=#1
		\edef\@p@sangle{#1} %\number\dimen100}
}
\def\@p@@ssilent#1{ 
		\@verbosefalse
}
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}
\def\@p@@spostlog#1{\@postlogfiletrue\def\@postlogfileval{#1}}
\def\@cs@name#1{\csname #1\endcsname}
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}
%
% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{
		\@bbllxfalse \@bbllyfalse
		\@bburxfalse \@bburyfalse
		\@heightfalse \@widthfalse
		\@rheightfalse \@rwidthfalse
		\def\@p@sbbllx{}\def\@p@sbblly{}
		\def\@p@sbburx{}\def\@p@sbbury{}
		\def\@p@sheight{}\def\@p@swidth{}
		\def\@p@srheight{}\def\@p@srwidth{}
		\def\@p@sangle{0}
		\def\@p@sfile{} \def\@p@sbbfile{}
		\def\@p@scost{10}
		\def\@sc{}
		\@prologfilefalse
		\@postlogfilefalse
		\@clipfalse
		\if@noisy
			\@verbosetrue
		\else
			\@verbosefalse
		\fi
}
%
% Go through the options setting things up.
%
\def\parse@ps@parms#1{
	 	\@psdo\@psfiga:=#1\do
		   {\expandafter\@setparms\@psfiga,}}
%
% Compute bb height and width
%
\newif\ifno@bb
\def\bb@missing{
	\if@verbose{
		\ps@typeout{psfig: searching \@p@sbbfile \space  for bounding box}
	}\fi
	\no@bbtrue
	\epsf@getbb{\@p@sbbfile}
        \ifno@bb \else \bb@cull\epsf@llx\epsf@lly\epsf@urx\epsf@ury\fi
}	
\def\bb@cull#1#2#3#4{
	\dimen100=#1 bp\edef\@p@sbbllx{\number\dimen100}
	\dimen100=#2 bp\edef\@p@sbblly{\number\dimen100}
	\dimen100=#3 bp\edef\@p@sbburx{\number\dimen100}
	\dimen100=#4 bp\edef\@p@sbbury{\number\dimen100}
	\no@bbfalse
}
% rotate point (#1,#2) about (0,0).
% The sine and cosine of the angle are already stored in \sine and
% \cosine.  The result is placed in (\p@intvaluex, \p@intvaluey).
\newdimen\p@intvaluex
\newdimen\p@intvaluey
\def\rotate@#1#2{{\dimen0=#1 sp\dimen1=#2 sp
%            	calculate x' = x \cos\theta - y \sin\theta
		  \global\p@intvaluex=\cosine\dimen0
		  \dimen3=\sine\dimen1
		  \global\advance\p@intvaluex by -\dimen3
% 		calculate y' = x \sin\theta + y \cos\theta
		  \global\p@intvaluey=\sine\dimen0
		  \dimen3=\cosine\dimen1
		  \global\advance\p@intvaluey by \dimen3
		  }}
\def\compute@bb{
		\no@bbfalse
		\if@bbllx \else \no@bbtrue \fi
		\if@bblly \else \no@bbtrue \fi
		\if@bburx \else \no@bbtrue \fi
		\if@bbury \else \no@bbtrue \fi
		\ifno@bb \bb@missing \fi
		\ifno@bb \ps@typeout{FATAL ERROR: no bb supplied or found}
			\no-bb-error
		\fi
		%
%\ps@typeout{BB: \@p@sbbllx, \@p@sbblly, \@p@sbburx, \@p@sbbury} 
%
% store height/width of original (unrotated) bounding box
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\ps@bbw{\number\count203}
		\edef\ps@bbh{\number\count204}
		%\ps@typeout{ psbbh = \ps@bbh, psbbw = \ps@bbw }
		\if@angle 
			\Sine{\@p@sangle}\Cosine{\@p@sangle}
	        	{\dimen100=\maxdimen\xdef\r@p@sbbllx{\number\dimen100}
					    \xdef\r@p@sbblly{\number\dimen100}
			                    \xdef\r@p@sbburx{-\number\dimen100}
					    \xdef\r@p@sbbury{-\number\dimen100}}
%
% Need to rotate all four points and take the X-Y extremes of the new
% points as the new bounding box.
                        \def\minmaxtest{
			   \ifnum\number\p@intvaluex<\r@p@sbbllx
			      \xdef\r@p@sbbllx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluex>\r@p@sbburx
			      \xdef\r@p@sbburx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluey<\r@p@sbblly
			      \xdef\r@p@sbblly{\number\p@intvaluey}\fi
			   \ifnum\number\p@intvaluey>\r@p@sbbury
			      \xdef\r@p@sbbury{\number\p@intvaluey}\fi
			   }
%			lower left
			\rotate@{\@p@sbbllx}{\@p@sbblly}
			\minmaxtest
%			upper left
			\rotate@{\@p@sbbllx}{\@p@sbbury}
			\minmaxtest
%			lower right
			\rotate@{\@p@sbburx}{\@p@sbblly}
			\minmaxtest
%			upper right
			\rotate@{\@p@sbburx}{\@p@sbbury}
			\minmaxtest
			\edef\@p@sbbllx{\r@p@sbbllx}\edef\@p@sbblly{\r@p@sbblly}
			\edef\@p@sbburx{\r@p@sbburx}\edef\@p@sbbury{\r@p@sbbury}
%\ps@typeout{rotated BB: \r@p@sbbllx, \r@p@sbblly, \r@p@sbburx, \r@p@sbbury}
		\fi
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\@bbw{\number\count203}
		\edef\@bbh{\number\count204}
		%\ps@typeout{ bbh = \@bbh, bbw = \@bbw }
}
%
% \in@hundreds performs #1 * (#2 / #3) correct to the hundreds,
%	then leaves the result in @result
%
\def\in@hundreds#1#2#3{\count240=#2 \count241=#3
		     \count100=\count240	% 100 is first digit #2/#3
		     \divide\count100 by \count241
		     \count101=\count100
		     \multiply\count101 by \count241
		     \advance\count240 by -\count101
		     \multiply\count240 by 10
		     \count101=\count240	%101 is second digit of #2/#3
		     \divide\count101 by \count241
		     \count102=\count101
		     \multiply\count102 by \count241
		     \advance\count240 by -\count102
		     \multiply\count240 by 10
		     \count102=\count240	% 102 is the third digit
		     \divide\count102 by \count241
		     \count200=#1\count205=0
		     \count201=\count200
			\multiply\count201 by \count100
		 	\advance\count205 by \count201
		     \count201=\count200
			\divide\count201 by 10
			\multiply\count201 by \count101
			\advance\count205 by \count201
			%
		     \count201=\count200
			\divide\count201 by 100
			\multiply\count201 by \count102
			\advance\count205 by \count201
			%
		     \edef\@result{\number\count205}
}
\def\compute@wfromh{
		% computing : width = height * (bbw / bbh)
		\in@hundreds{\@p@sheight}{\@bbw}{\@bbh}
		%\ps@typeout{ \@p@sheight * \@bbw / \@bbh, = \@result }
		\edef\@p@swidth{\@result}
		%\ps@typeout{w from h: width is \@p@swidth}
}
\def\compute@hfromw{
		% computing : height = width * (bbh / bbw)
	        \in@hundreds{\@p@swidth}{\@bbh}{\@bbw}
		%\ps@typeout{ \@p@swidth * \@bbh / \@bbw = \@result }
		\edef\@p@sheight{\@result}
		%\ps@typeout{h from w : height is \@p@sheight}
}
\def\compute@handw{
		\if@height 
			\if@width
			\else
				\compute@wfromh
			\fi
		\else 
			\if@width
				\compute@hfromw
			\else
				\edef\@p@sheight{\@bbh}
				\edef\@p@swidth{\@bbw}
			\fi
		\fi
}
\def\compute@resv{
		\if@rheight \else \edef\@p@srheight{\@p@sheight} \fi
		\if@rwidth \else \edef\@p@srwidth{\@p@swidth} \fi
		%\ps@typeout{rheight = \@p@srheight, rwidth = \@p@srwidth}
}
%		
% Compute any missing values
\def\compute@sizes{
	\compute@bb
	\if@scalefirst\if@angle
% at this point the bounding box has been adjsuted correctly for
% rotation.  PSFIG does all of its scaling using \@bbh and \@bbw.  If
% a width= or height= was specified along with \psscalefirst, then the
% width=/height= value needs to be adjusted to match the new (rotated)
% bounding box size (specifed in \@bbw and \@bbh).
%    \ps@bbw       width=
%    -------  =  ---------- 
%    \@bbw       new width=
% so `new width=' = (width= * \@bbw) / \ps@bbw; where \ps@bbw is the
% width of the original (unrotated) bounding box.
	\if@width
	   \in@hundreds{\@p@swidth}{\@bbw}{\ps@bbw}
	   \edef\@p@swidth{\@result}
	\fi
	\if@height
	   \in@hundreds{\@p@sheight}{\@bbh}{\ps@bbh}
	   \edef\@p@sheight{\@result}
	\fi
	\fi\fi
	\compute@handw
	\compute@resv}
%
%
%
\def\OzTeXSpecials{
	\special{empty.ps /@isp {true} def}
	\special{empty.ps \@p@swidth \space \@p@sheight \space
			\@p@sbbllx \space \@p@sbblly \space
			\@p@sbburx \space \@p@sbbury \space
			startTexFig \space }
	\if@clip{
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special{empty.ps doclip \space }
	}\fi
	\if@angle{
		\if@verbose{
			\ps@typeout{(rotate)}
		}\fi
		\special {empty.ps \@p@sangle \space rotate \space} 
	}\fi
	\if@prologfile
	    \special{\@prologfileval \space } \fi
	\if@decmpr{
		\if@verbose{
			\ps@typeout{psfig: Compression not available
			in OzTeX version \space }
		}\fi
	}\else{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile \space }
		}\fi
		\special{epsf=\@p@sfile \space }
	}\fi
	\if@postlogfile
	    \special{\@postlogfileval \space } \fi
	\special{empty.ps /@isp {false} def}
}
\def\DvipsSpecials{
	%
	\special{ps::[begin] 	\@p@swidth \space \@p@sheight \space
			\@p@sbbllx \space \@p@sbblly \space
			\@p@sbburx \space \@p@sbbury \space
			startTexFig \space }
	\if@clip{
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special{ps:: doclip \space }
	}\fi
	\if@angle
		\if@verbose{
			\ps@typeout{(clip)}
		}\fi
		\special {ps:: \@p@sangle \space rotate \space} 
	\fi
	\if@prologfile
	    \special{ps: plotfile \@prologfileval \space } \fi
	\if@decmpr{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile.Z \space }
		}\fi
		\special{ps: plotfile "`zcat \@p@sfile.Z" \space }
	}\else{
		\if@verbose{
			\ps@typeout{psfig: including \@p@sfile \space }
		}\fi
		\special{ps: plotfile \@p@sfile \space }
	}\fi
	\if@postlogfile
	    \special{ps: plotfile \@postlogfileval \space } \fi
	\special{ps::[end] endTexFig \space }
}
%
% \psfig
% usage : \psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=}
%
% "clip=" is a switch and takes no value, but the `=' must be present.
\def\psfig#1{\vbox {
	% do a zero width hard space so that a single
	% \psfig in a centering enviornment will behave nicely
	%{\setbox0=\hbox{\ }\ \hskip-\wd0}
	%
	\ps@init@parms
	\parse@ps@parms{#1}
	\compute@sizes
	%
	\ifnum\@p@scost<\@psdraft{
		\PsfigSpecials 
		% Create the vbox to reserve the space for the figure.
		\vbox to \@p@srheight sp{
		% 1/92 TJD Changed from "true sp" to "sp" for magnification.
			\hbox to \@p@srwidth sp{
				\hss
			}
		\vss
		}
	}\else{
		% draft figure, just reserve the space and print the
		% path name.
		\if@draftbox{		
			% Verbose draft: print file name in box
			% 10/93 TJD changed to fbox from frame
			\hbox{\fbox{\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{ \hss 
			        % 10/93 TJD deleted to avoid ``_'' problems
				% \@p@sfile
			 \hss }
			\vss
			}}}
		}\else{
			% Non-verbose draft
			\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{\hss}
			\vss
			}
		}\fi	



	}\fi
}}
\psfigRestoreAt
\setDriver
\let\@=\LaTeXAtSign





EndOfTheIncludedPostscriptMagicCookie

\closepsdump

% Here is the PostScript for tori.eps:
\message{Writing file tori.eps}
\psdump{tori.eps}%!PS-Adobe-2.0
%%Title: tori.fig
%%Creator: fig2dev Version 3.1 Patchlevel 1
%%CreationDate: Sat Jun 13 18:33:24 1998
%%For: llave@tere (Rafael de la Llave,,,)
%%Orientation: Landscape
%%BoundingBox: 62 167 532 675
%%Pages: 1
%%BeginSetup
%%IncludeFeature: *PageSize A4
%%EndSetup
%%EndComments
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {} def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
45.5 97.0 translate
 90 rotate
1 -1 scale
.9 .9 scale % to make patterns same scale as in xfig

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup				% cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
    false charpath		% cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint			% cx cy cchar rx ry char x y
    newpath
    moveto			% cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {			% cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto		% cx cy cchar rx ry
  } forall
  pop pop pop pop pop		% -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index show		% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proe char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  userdict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat					% tilingtype
    matrix currentmatrix exch			% cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
	% Distort the pattern so that it occupies
	% an integral number of device pixels
	dup 4 get exch dup 5 get exch		% tx ty cmtx
	XStep 0 dtransform
	round exch round exch			% tx ty cmtx dx.x dx.y
	XStep div exch XStep div exch		% tx ty cmtx a b
	0 YStep dtransform
	round exch round exch			% tx ty cmtx a b dy.x dy.y
	YStep div exch YStep div exch		% tx ty cmtx a b c d
	7 -3 roll astore			% { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% right-shingles
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 48 48 true [ 48 0 0 -48 0 48 ]
	{<000000000080000000000080000000000040000000000040
	000000000020000000000020000000000010000000000010
	000000000008000000000008000000000004000000000004
	000000000002000000000002000000000001ffffffffffff
	008000000000008000000000004000000000004000000000
	002000000000002000000000001000000000001000000000
	000800000000000800000000000400000000000400000000
	000200000000000200000000000100000000ffffffffffff
	000000800000000000800000000000400000000000400000
	000000200000000000200000000000100000000000100000
	000000080000000000080000000000040000000000040000
	000000020000000000020000000000010000ffffffffffff>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P13 exch def
1.1111 1.1111 scale %restore scale

/clp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/l {lineto} bind def
/m {moveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

$F2psBegin
10 setmiterlimit
 0.06000 0.06000 sc
7.500 slw
% Ellipse
n 5850 3450 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P13 [24 0 0 -24 382.93 222.93]  PATmp PATsp ef gr PATusp gs col0 s gr

% Ellipse
n 5806 6300 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P13 [24 0 0 -24 380.00 412.93]  PATmp PATsp ef gr PATusp gs col0 s gr

% Ellipse
n 5594 6675 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P13 [24 0 0 -24 365.87 437.93]  PATmp PATsp ef gr PATusp gs col0 s gr

% Polyline
n 1200 2100 m 1200 8100 l  7200 8100 l  7200 2100 l  1200 2100 l  gs col-1 s gr 
% Polyline
n 1200 2100 m 3600 300 l  9600 300 l  9600 6300 l  7200 8100 l  gs col-1 s gr 
% Polyline
n 7200 2100 m 9600 300 l  gs col-1 s gr 
% Polyline
	[66.7] 0 setdash
n 3600 300 m 3600 6300 l  9600 6300 l  gs col-1 s gr 	[] 0 setdash
% Polyline
	[66.7] 0 setdash
n 1200 8100 m 3600 6300 l  gs col-1 s gr 	[] 0 setdash
% Polyline
	[1 50.0] 50.000000 setdash
n 5850 3450 m 5775 6300 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P13 [24 0 0 -24 385.00 230.00]  PATmp PATsp ef gr PATusp  gs col0 s gr 	[] 0 setdash
30.000 slw
% Interp Spline
n 1200 3675 m
	1501.83 3657.30 1633.08 3657.30 1725 3675 curveto
	1819.29 3693.16 2016.66 3784.76 2100 3825 curveto
	2224.77 3885.24 2503.44 4059.94 2625 4125 curveto
	2759.86 4197.17 3072.94 4380.10 3225 4425 curveto
	3378.95 4470.46 3744.95 4491.41 3900 4500 curveto
	4122.59 4512.33 4647.98 4541.17 4875 4500 curveto
	5024.20 4472.94 5340.74 4333.54 5475 4275 curveto
	5615.54 4213.71 5943.74 4056.13 6075 3975 curveto
	6148.38 3929.65 6298.97 3792.73 6375 3750 curveto
	6456.88 3703.98 6655.71 3618.16 6750 3600 curveto
	6828.79 3584.83 6941.29 3584.83 7200 3600 curveto
 gs col-1 s gr

% Interp Spline
n 7200 3600 m
	7543.78 3625.07 7693.78 3625.07 7800 3600 curveto
	7958.34 3562.62 8264.75 3371.34 8400 3300 curveto
	8555.68 3217.89 8937.47 3054.57 9075 2925 curveto
	9205.63 2801.94 9336.88 2595.69 9600 2100 curveto
 gs col-1 s gr

% Interp Spline
	[133.3] 0 setdash
n 3600 2175 m
	3859.24 2162.30 3971.74 2162.30 4050 2175 curveto
	4159.99 2192.85 4400.19 2280.38 4500 2325 curveto
	4627.23 2381.88 4903.44 2559.94 5025 2625 curveto
	5159.86 2697.17 5472.94 2880.10 5625 2925 curveto
	5778.95 2970.46 6144.95 2991.41 6300 3000 curveto
	6522.59 3012.33 7047.98 3041.17 7275 3000 curveto
	7424.20 2972.94 7740.74 2833.54 7875 2775 curveto
	8015.54 2713.71 8343.74 2556.13 8475 2475 curveto
	8548.38 2429.65 8698.97 2292.73 8775 2250 curveto
	8856.88 2203.98 9055.71 2118.16 9150 2100 curveto
	9228.79 2084.83 9341.29 2084.83 9600 2100 curveto
 gs col-1 s gr
	[] 0 setdash
% Interp Spline
n 7200 6675 m
	7543.78 6700.07 7693.78 6700.07 7800 6675 curveto
	7958.34 6637.62 8264.75 6446.34 8400 6375 curveto
	8555.68 6292.88 8937.47 6129.57 9075 6000 curveto
	9205.63 5876.94 9336.88 5670.69 9600 5175 curveto
 gs col-1 s gr

% Interp Spline
	[133.3] 0 setdash
n 1218 6707 m
	1561.78 6732.07 1711.78 6732.07 1818 6707 curveto
	1976.34 6669.62 2282.75 6478.34 2418 6407 curveto
	2573.68 6324.89 2955.47 6161.57 3093 6032 curveto
	3223.63 5908.94 3354.88 5702.69 3618 5207 curveto
 gs col-1 s gr
	[] 0 setdash
7.500 slw
% Ellipse
n 2850 6750 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P13 [24 0 0 -24 182.93 442.93]  PATmp PATsp ef gr PATusp gs col0 s gr

30.000 slw
% Interp Spline
	[133.3] 0 setdash
n 3588 5275 m
	3889.83 5257.30 4021.08 5257.30 4113 5275 curveto
	4207.29 5293.16 4400.66 5395.14 4488 5425 curveto
	4606.07 5465.36 4892.81 5542.82 5013 5575 curveto
	5149.14 5611.45 5473.31 5700.69 5613 5725 curveto
	5783.24 5754.63 6190.99 5791.42 6363 5800 curveto
	6568.39 5810.24 7057.53 5811.38 7263 5800 curveto
	7418.05 5791.41 7784.91 5755.15 7938 5725 curveto
	8061.08 5700.76 8346.53 5625.53 8463 5575 curveto
	8556.29 5534.53 8743.54 5389.13 8838 5350 curveto
	8904.79 5322.33 9068.90 5289.37 9138 5275 curveto
	9214.46 5259.10 9326.96 5240.35 9588 5200 curveto
 gs col-1 s gr
	[] 0 setdash
% Interp Spline
	[133.3] 0 setdash
n 1215 3705 m
	1507.35 3593.57 1634.85 3548.57 1725 3525 curveto
	1880.66 3484.30 2259.49 3459.95 2415 3405 curveto
	2586.46 3344.42 2952.47 3159.57 3090 3030 curveto
	3220.63 2906.94 3351.88 2700.69 3615 2205 curveto
 gs col0 s gr
% Interp Spline
	[1 50.0] 50.000000 setdash
n 2925 6825 m
	3229.05 6868.44 3360.30 6887.19 3450 6900 curveto
	3569.60 6917.08 3853.76 6966.38 3975 6975 curveto
	4060.73 6981.09 4264.57 6975.00 4350 6975 curveto
	4435.43 6975.00 4638.99 6983.52 4725 6975 curveto
	4812.72 6966.31 5015.83 6928.88 5100 6900 curveto
	5136.91 6887.34 5215.83 6842.09 5250 6825 curveto
	5301.26 6799.37 5376.26 6761.87 5550 6675 curveto
 gs col0 s gr
	[] 0 setdash
30.000 slw
% Interp Spline
n 1200 6750 m
	1501.83 6732.30 1633.08 6732.30 1725 6750 curveto
	1819.29 6768.16 2012.66 6870.14 2100 6900 curveto
	2218.07 6940.36 2504.81 7017.82 2625 7050 curveto
	2761.14 7086.45 3085.31 7175.69 3225 7200 curveto
	3395.24 7229.63 3802.99 7266.42 3975 7275 curveto
	4180.39 7285.24 4669.53 7286.38 4875 7275 curveto
	5030.05 7266.41 5396.91 7230.15 5550 7200 curveto
	5673.08 7175.76 5958.53 7100.53 6075 7050 curveto
	6168.29 7009.53 6355.54 6864.13 6450 6825 curveto
	6516.80 6797.33 6680.90 6764.37 6750 6750 curveto
	6826.46 6734.10 6938.96 6715.35 7200 6675 curveto
 gs col-1 s gr

/Times-Bold findfont 450.00 scalefont setfont
6000 6375 m
gs 1 -1 sc (y) col0 show gr
/Times-Bold findfont 375.00 scalefont setfont
7525 3550 m
gs 1 -1 sc (-) col0 show gr
/Times-Bold findfont 375.00 scalefont setfont
6275 3550 m
gs 1 -1 sc (+) col0 show gr
/Times-Bold findfont 450.00 scalefont setfont
3000 6675 m
gs 1 -1 sc (x) col0 show gr
/Times-Bold findfont 375.00 scalefont setfont
3225 6775 m
gs 1 -1 sc (-) col0 show gr
/Times-Bold findfont 450.00 scalefont setfont
5700 6750 m
gs 1 -1 sc (u) col0 show gr
/Times-Bold findfont 450.00 scalefont setfont
6075 3450 m
gs 1 -1 sc (x  = S\(x \)) col0 show gr
showpage
%%Page: 1 1
$F2psEnd
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/MLshow { currentpoint stroke M
  0 exch R MFshow } bind def
/MRshow { currentpoint stroke M
  exch dup MFwidth neg 3 -1 roll R MFshow } def
/MCshow { currentpoint stroke M
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%%EndProlog
gnudict begin
gsave
50 50 translate
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0 setgray
newpath
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1.000 UL
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1209 4839 M
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[(Symbol) 210.0 0.0 true true (e)]
[(Helvetica) 168.0 95.0 true true (m/2)]
[(Helvetica) 210.0 0.0 true true (\))]
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4529 5003 M
[ [(Helvetica) 210.0 0.0 true true (O\()]
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[(Helvetica) 168.0 95.0 true true (3)]
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2618 3549 M
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2879 4651 M
[ [(Helvetica) 210.0 0.0 true true (S\()]
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1.000 UL
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1275 4775 M
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4487 4395 M
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1395 3534 M
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1395 3534 M
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8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
currentpoint stroke M
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
currentpoint stroke M
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -35 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -23 V
20 -23 V
21 -21 V
22 -21 V
23 -20 V
25 -19 V
26 -19 V
28 -19 V
28 -20 V
30 -19 V
31 -21 V
32 -20 V
33 -22 V
34 -22 V
35 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -25 V
40 -26 V
40 -26 V
41 -25 V
42 -25 V
43 -25 V
43 -23 V
44 -23 V
44 -20 V
45 -20 V
45 -17 V
45 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
47 0 V
47 2 V
46 5 V
47 7 V
47 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
45 18 V
45 19 V
45 19 V
44 19 V
44 19 V
43 19 V
43 19 V
42 18 V
41 17 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
35 15 V
34 14 V
33 15 V
32 15 V
31 17 V
30 17 V
28 18 V
28 19 V
26 22 V
25 23 V
23 24 V
22 27 V
21 29 V
20 30 V
18 33 V
17 35 V
15 36 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 43 V
5 44 V
4 44 V
2 44 V
1 43 V
1395 3534 M
90 -386 R
258 -229 R
395 -258 R
485 -217 R
514 48 R
485 200 R
395 169 R
258 277 R
90 458 R
LTb
LT0
1372 6072 M
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
currentpoint stroke M
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
currentpoint stroke M
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
3 -85 V
9 -79 V
16 -74 V
21 -67 V
27 -59 V
32 -53 V
39 -47 V
44 -43 V
49 -40 V
54 -39 V
59 -40 V
64 -42 V
68 -44 V
72 -48 V
77 -50 V
79 -51 V
83 -52 V
86 -50 V
88 -46 V
91 -41 V
92 -33 V
94 -23 V
94 -14 V
96 -3 V
94 7 V
96 17 V
95 25 V
94 31 V
93 36 V
92 38 V
89 39 V
87 38 V
85 36 V
81 33 V
78 32 V
74 30 V
70 29 V
67 31 V
61 34 V
57 39 V
52 45 V
46 53 V
41 60 V
36 68 V
30 76 V
24 81 V
18 86 V
12 89 V
6 88 V
0 87 V
-6 82 V
-12 77 V
-18 70 V
-24 63 V
-30 56 V
-36 50 V
-41 45 V
-46 41 V
-52 39 V
-57 40 V
-61 40 V
-67 43 V
-70 46 V
-74 49 V
-78 51 V
-81 52 V
-85 51 V
-87 48 V
-89 44 V
-92 37 V
-93 28 V
-94 19 V
-95 8 V
-96 -2 V
-94 -12 V
-96 -21 V
-94 -28 V
-94 -34 V
-92 -37 V
-91 -39 V
-88 -38 V
-86 -37 V
-83 -35 V
-79 -33 V
-77 -30 V
-72 -30 V
-68 -30 V
-64 -32 V
-59 -36 V
-54 -42 V
-49 -49 V
-44 -56 V
-39 -64 V
-32 -72 V
-27 -79 V
-21 -84 V
-16 -88 V
-9 -88 V
-3 -88 V
353 -626 R
894 -480 R
1015 250 R
663 454 R
0 855 R
-663 496 R
2619 7193 M
1725 6818 M
1372 6072 M
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
currentpoint stroke M
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
currentpoint stroke M
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
1362 5078 M
17 130 V
14 114 V
9 82 V
7 39 V
7 -12 V
8 -66 V
8 -116 V
5 -159 V
0 -190 V
-1 -206 V
2 -202 V
11 -181 V
17 -143 V
22 -95 V
22 -42 V
22 14 V
24 66 V
28 109 V
36 144 V
40 160 V
42 157 V
38 135 V
34 96 V
33 48 V
33 -8 V
34 -64 V
33 -116 V
32 -161 V
29 -195 V
29 -209 V
32 -201 V
38 -177 V
42 -137 V
43 -88 V
44 -33 V
43 20 V
45 73 V
47 117 V
50 150 V
51 166 V
50 159 V
48 135 V
47 96 V
46 47 V
46 -6 V
47 -57 V
47 -106 V
46 -147 V
48 -173 V
47 -182 V
49 -167 V
49 -134 V
49 -90 V
48 -37 V
48 18 V
49 72 V
47 124 V
46 166 V
43 195 V
41 204 V
40 193 V
41 162 V
42 117 V
42 65 V
42 10 V
42 -46 V
42 -95 V
45 -137 V
49 -162 V
48 -167 V
45 -152 V
39 -121 V
36 -76 V
35 -25 V
35 29 V
34 84 V
31 131 V
26 171 V
18 195 V
15 200 V
15 187 V
19 155 V
22 113 V
22 63 V
21 9 V
20 -42 V
23 -87 V
28 -121 V
31 -138 V
29 -137 V
21 -119 V
14 -84 V
9 -39 V
8 14 V
8 69 V
6 123 V
2 168 V
-6 200 V
-14 216 V
currentpoint stroke M
1362 5078 M
76 -586 R
302 324 R
2109 3692 M
514 842 R
3147 3453 M
475 1262 R
439 -691 R
229 843 R
108 273 R
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
currentpoint stroke M
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
currentpoint stroke M
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
1 -43 V
2 -43 V
4 -41 V
5 -40 V
7 -38 V
9 -37 V
10 -35 V
11 -33 V
13 -31 V
15 -30 V
16 -28 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -21 V
27 -20 V
29 -20 V
29 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -23 V
36 -23 V
36 -23 V
38 -25 V
38 -25 V
39 -26 V
41 -26 V
41 -26 V
42 -26 V
42 -27 V
44 -25 V
44 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
47 -18 V
47 -15 V
48 -14 V
48 -10 V
48 -9 V
49 -5 V
48 -3 V
49 0 V
49 2 V
48 5 V
49 7 V
49 10 V
48 11 V
49 14 V
48 15 V
48 16 V
48 18 V
47 19 V
47 19 V
46 19 V
46 20 V
45 19 V
45 20 V
44 19 V
44 18 V
42 18 V
42 18 V
41 17 V
41 16 V
39 16 V
38 16 V
38 15 V
36 15 V
36 15 V
34 16 V
33 17 V
32 17 V
31 18 V
29 19 V
29 21 V
27 22 V
26 23 V
24 26 V
23 27 V
22 30 V
20 32 V
19 34 V
17 36 V
16 37 V
15 40 V
13 40 V
11 43 V
10 43 V
9 44 V
7 45 V
5 45 V
4 45 V
2 45 V
1 44 V
1341 4802 M
93 -399 R
267 -240 R
410 -266 R
502 -222 R
534 47 R
502 205 R
410 178 R
267 288 R
93 471 R
LT1
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
currentpoint stroke M
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
currentpoint stroke M
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
1 -43 V
2 -43 V
4 -41 V
5 -39 V
7 -38 V
8 -37 V
10 -34 V
12 -33 V
13 -31 V
14 -30 V
16 -27 V
17 -26 V
19 -25 V
20 -23 V
22 -23 V
23 -21 V
24 -21 V
26 -20 V
26 -20 V
29 -20 V
29 -20 V
30 -21 V
32 -21 V
33 -21 V
34 -22 V
35 -23 V
36 -24 V
38 -23 V
38 -25 V
39 -26 V
40 -26 V
41 -26 V
41 -26 V
43 -26 V
43 -25 V
43 -25 V
45 -24 V
45 -23 V
46 -21 V
46 -20 V
46 -17 V
47 -16 V
47 -13 V
48 -11 V
48 -8 V
48 -6 V
48 -2 V
49 -1 V
48 3 V
48 5 V
48 7 V
49 9 V
48 12 V
48 13 V
48 16 V
48 16 V
47 18 V
47 18 V
46 19 V
46 19 V
46 20 V
45 19 V
45 20 V
43 19 V
43 18 V
43 18 V
41 17 V
41 17 V
40 16 V
39 16 V
38 15 V
38 16 V
36 14 V
35 15 V
34 15 V
33 16 V
32 17 V
30 18 V
29 19 V
29 21 V
26 22 V
26 24 V
24 25 V
23 28 V
22 29 V
20 32 V
19 33 V
17 36 V
16 37 V
14 39 V
13 41 V
12 42 V
10 43 V
8 44 V
7 44 V
5 45 V
4 45 V
2 45 V
1 44 V
1352 4968 M
92 -396 R
265 -240 R
407 -263 R
499 -221 R
530 48 R
499 204 R
407 174 R
265 287 R
92 469 R
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
currentpoint stroke M
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
currentpoint stroke M
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
1 -42 V
2 -42 V
4 -40 V
5 -39 V
7 -38 V
8 -35 V
10 -34 V
11 -33 V
13 -30 V
14 -29 V
15 -27 V
17 -25 V
19 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -20 V
26 -19 V
28 -20 V
29 -19 V
30 -20 V
31 -21 V
32 -21 V
33 -21 V
35 -23 V
35 -23 V
37 -24 V
37 -24 V
38 -26 V
40 -25 V
40 -26 V
40 -26 V
42 -25 V
42 -26 V
43 -24 V
44 -24 V
44 -22 V
44 -21 V
46 -20 V
45 -17 V
46 -15 V
47 -13 V
46 -11 V
47 -7 V
47 -5 V
48 -3 V
47 0 V
48 2 V
46 5 V
48 7 V
47 9 V
48 11 V
47 13 V
47 15 V
46 17 V
47 17 V
46 18 V
45 19 V
46 19 V
44 20 V
44 19 V
44 19 V
43 19 V
42 18 V
42 17 V
40 17 V
40 17 V
40 15 V
38 16 V
37 15 V
37 14 V
35 15 V
35 15 V
33 15 V
32 15 V
31 17 V
30 17 V
29 19 V
28 19 V
26 22 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
19 33 V
17 35 V
15 36 V
currentpoint stroke M
14 39 V
13 40 V
11 41 V
10 42 V
8 44 V
7 43 V
5 45 V
4 44 V
2 44 V
1 43 V
1382 5299 M
90 -389 R
261 -232 R
398 -260 R
489 -218 R
520 47 R
489 202 R
398 171 R
261 280 R
90 461 R
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
currentpoint stroke M
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
currentpoint stroke M
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
1 -43 V
2 -41 V
4 -40 V
5 -39 V
7 -37 V
8 -36 V
10 -34 V
11 -32 V
13 -30 V
14 -28 V
15 -27 V
17 -25 V
18 -24 V
20 -23 V
21 -21 V
22 -21 V
24 -20 V
25 -19 V
26 -20 V
27 -19 V
29 -19 V
30 -20 V
31 -20 V
32 -21 V
33 -22 V
34 -22 V
36 -23 V
36 -24 V
37 -24 V
38 -25 V
39 -26 V
40 -25 V
40 -26 V
42 -26 V
42 -25 V
42 -24 V
44 -24 V
44 -22 V
44 -21 V
45 -20 V
45 -17 V
46 -15 V
46 -13 V
47 -11 V
46 -8 V
47 -5 V
47 -3 V
48 0 V
47 2 V
46 5 V
47 7 V
48 10 V
47 11 V
47 14 V
46 15 V
47 16 V
46 17 V
46 18 V
45 19 V
45 19 V
44 20 V
44 19 V
44 19 V
42 18 V
42 18 V
42 18 V
40 17 V
40 16 V
39 16 V
38 15 V
37 15 V
36 14 V
36 15 V
34 14 V
33 15 V
32 16 V
31 16 V
30 17 V
29 18 V
27 20 V
26 21 V
25 23 V
24 25 V
22 27 V
21 29 V
20 31 V
18 32 V
17 35 V
15 37 V
14 38 V
13 40 V
11 41 V
10 42 V
8 43 V
7 44 V
5 44 V
4 43 V
2 44 V
1 43 V
1390 5410 M
90 -387 R
259 -231 R
396 -258 R
487 -218 R
516 48 R
487 201 R
396 169 R
259 278 R
90 459 R
LT0
1395 3534 M
-2 282 R
-22 282 R
-25 281 R
-9 282 R
13 282 R
26 283 R
19 282 R
-1 282 R
-22 282 R
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
currentpoint stroke M
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
currentpoint stroke M
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
23 -2538 R
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
-1 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-3 25 V
-2 25 V
-2 26 V
-3 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-1 25 V
-1 26 V
0 26 V
-1 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
1 25 V
2 26 V
1 26 V
2 25 V
2 26 V
2 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
3 26 V
2 25 V
2 26 V
2 26 V
2 25 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
1 26 V
1 25 V
1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
-1 26 V
0 26 V
-1 25 V
-1 26 V
-2 25 V
-1 26 V
-2 26 V
-2 25 V
-1 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
4365 3596 M
2 282 R
22 282 R
25 281 R
9 282 R
-13 282 R
-26 282 R
-19 283 R
1 282 R
22 282 R
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
currentpoint stroke M
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
currentpoint stroke M
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 25 V
-3 26 V
-2 26 V
-3 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-2 26 V
-2 25 V
-2 26 V
-1 26 V
-2 25 V
-1 26 V
-1 26 V
-1 25 V
-1 26 V
0 25 V
0 26 V
0 26 V
0 25 V
0 26 V
1 26 V
0 25 V
1 26 V
1 26 V
2 25 V
1 26 V
2 26 V
2 25 V
1 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
4365 3596 M
-1 25 V
-1 26 V
0 26 V
0 25 V
0 26 V
0 26 V
0 25 V
1 26 V
1 25 V
1 26 V
1 26 V
1 25 V
2 26 V
2 26 V
1 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
3 26 V
2 26 V
3 24 V
2 26 V
3 25 V
2 26 V
2 26 V
3 25 V
2 26 V
2 26 V
2 25 V
2 26 V
1 26 V
2 25 V
1 26 V
2 26 V
1 25 V
1 26 V
0 26 V
1 25 V
0 26 V
0 25 V
0 26 V
0 26 V
-1 25 V
0 26 V
-1 26 V
-1 25 V
-1 26 V
-2 26 V
-1 25 V
-2 26 V
-2 26 V
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\title{ A geometric approach  to the existence of orbits with
unbounded energy in  generic periodic perturbations 
by a potential of
generic geodesic flows of  $\torus ^{2}$}
\author{Amadeu Delshams$^{1}$\thanks{\tt
amadeu@ma1.upc.es}\and
Rafael de la Llave$^{2}$\thanks{\tt
llave@math.utexas.edu}\and
Tere M.~Seara$^1$\thanks{\tt tere@ma1.upc.es}}
\date{August 31, 1998}


\begin{document}
%\bibliographystyle{alpha}%BiBTeX
\maketitle
\medskip
\begin{center}$^1$
Departament de Matem\`atica Aplicada I\\
Universitat Polit\`ecnica de Catalunya\\
Diagonal 647, 08028 Barcelona, Spain
\end{center}
\smallskip
\begin{center}$^2$
Department of Mathematics\\
University of Texas at Austin\\
Austin, TX, 78712, USA\\
\smallskip
\end{center}


\begin{abstract}
We give a proof based in geometric perturbation theory of a result
proved by J.N.~Mather using variational methods.
Namely, the existence of orbits with unbounded energy in 
perturbations of a generic geodesic flows in $\torus^2$ 
by a generic periodic potential. 
\end{abstract}
\section{Introduction}

The goal of this paper is to give a proof, using geometric perturbation
methods, of a result proved by J.N.~Mather using variational methods. We
will prove:
\begin{thm}\label{unboundedorbit}
Let  $g$ be a  ${\cal C}^{r}$ generic metric on  $\torus^{2}$,
$U:\torus^{2}\times \torus \to \real$ a generic ${\cal C}^{r}$
function
(on $\torus^{2}$, and periodic in time),  $r$ sufficiently large.

Consider the time dependent Lagrangian
\[
L(q,\dot q,t) = \frac{1}{2} g_{q}(\dot q,\dot q) - U (q,t).
\]
Then, the Euler-Lagrange equation of $L$ has a solution $q(t)$  whose
energy
\[
E(t) = H(\dot q(t),q(t),t) = \frac{1}{2} g_{q}(\dot q(t),\dot q(t)) + U
(q(t),t),
\]
goes to infinity as  $t\to \infty $.
\end{thm}

\begin{rem}
Note that, in fact, the only unbounded part in  $H(\dot q(t), q(t),t)$
is  $\dot q(t)$, so that the theorem could be expressed as unbounded
growth in the velocity.
\end{rem}

\begin{rem}
As it is usually the case in these problems of diffusion,
one not only constructs orbits whose energy grows unbounded,
but also orbits whose energy makes more or less arbitrary
excursions. We formulate this precisely in 
Theorem~\ref{thm:main}, and deduce
Theorem~\ref{unboundedorbit} from it. 
\end{rem}

\begin{rem}
The argument presented here shows that $r \ge 15 $ is 
large enough for Theorem~\ref{unboundedorbit}. 
(See the proof of Lemma~\ref{lem:potmather}.)
We do not claim that this is optimal for  the 
geometric method to go through.
\end{rem}

Actually, the results of Mather contain this as a
particular case,
as well as ours. This theorem as stated seems to be just a common ground
that allows some comparison of the methods.
Notably, Mather can deal with situations involving much less
regularity.

\begin{rem}
The assumptions of genericity will be made quite explicit in
Theorem~\ref{thm:main}, a more general result than
Theorem~\ref{unboundedorbit}. 
They amount to the existence of a closed hyperbolic geodesic
with a homoclinic connecting orbit for the metric  $g$, and that a
certain functional, called Melnikov potential or Poincar\'e function,
computed from the potential on the homoclinic
orbit, is not constant.
\end{rem}

The work of Mather \cite{Mather96} also requires similar assumptions.
As far as we can see, the main difference in the
hypothesis of
\cite{Mather96} and this paper
is that \cite{Mather96} also uses that the periodic orbits
and the connecting ones are minimizing and class A. On the other
hand, the differentiability hypotheses  of
this note are much more restrictive than
those in \cite{Mather96}.
The orbits with growing energy produced in this work and those produced
in~\cite{Mather96} are not necessarily the same: the orbits we produce here
shadow smooth invariant curves, whereas those in~\cite{Mather96}
shadow minimizing Aubry-Mather sets (which could be Cantor sets). 
We think that it is remarkable that the
functional  that needs not to be constant is
the same in both approaches. We hope that this could 
lead to a more geometric understanding of 
\cite{Mather96}, which could perhaps lead 
to some new results.


We note that it is possible to choose  $g$ and  $U$ as arbitrarily close
to the flat metric and zero as desired in an analytic topology. Hence,
this could be considered as an analogue of Arnol'd diffusion.
Depending on what one defines precisely as Arnol'd diffusion
it may not be appropriate to call the phenomenon  described
in \cite{Mather96} and here by this name.
Since  a universally accepted precise definition
of Arnol'd diffusion seems to be lacking we just point out that
the phenomena described here has a similar flavor and
indeed the methods that we use here are very similar to
the methods traditionally used in the field.


The analogy with the traditional approaches of
Arnol'd diffusion is much closer when we consider
what happens for a bounded range of  (rather high) energies.
We note that  in this case, there are two smallness
parameters. One is  the distance of the metric to the
flat metric and another one is the size of
the potential. For high energy, the
potential is a very small perturbation of
the geodesic flow (we will make all this more precise later).
If we chose $g$ close to flat, 
for the theorem to go through we need to choose the energy
for which the potential can be considered as a sufficiently 
small perturbation.
The same  feature of two smallness parameters was present in the original
example \cite{Arnold64}.

Note that the geodesic flow, which in our case plays the role of the
unperturbed system, is assumed to have some hyperbolicity properties.
In the notation of \cite{ChierchiaG94} this is an {\em a priori}
unstable system.

An important feature of this problem is
that, besides two smallness parameters, it has two
time scales.  For high energy, the
frequency of the unperturbed problem is
high while the frequency of
the perturbation is small.
Hence, one can bring to bear methods of
adiabatic theory to obtain small gaps between KAM tori.
(This phenomenon also happens in  the models
considered in \cite{ChierchiaG94}, who emphasized  
the important role played by this fact in the conclusions 
and also identified several physical models where this is a natural 
assumption.)


The main difference of the
methods presented here with more traditional
approaches to Arnold diffusion is the reliance in hyperbolic
perturbation theory and center manifold reduction rather
than the exclusive reliance in KAM perturbation theories.
(An sketch of the method proposed,  was known in 
\cite{Llave96}.)

We think that the locally invariant normally hyperbolic manifold is an
interesting structure since one can study the dynamics on it using the
powerful methods of two dimensional dynamics, notably Aubry-Mather
theory.  We hope to come back 
to these issues exploiting the  
many structures present in the  invariant center manifold in a near future.
Similar ideas were used in \cite{LlaveW89}.
We note that the use of methods based in  normal 
hyperbolicity to deal with systems with two 
scales of time in a geometric way has been 
successfully used
for a long time  (See e.g. \cite{Fenichel79}). 


We want to draw attention to  \cite{BolotinT98}, which presents 
another method to obtain similar results without using 
the theory of normally hyperbolic manifolds.  They construct the 
transition chain relying in standard KAM theory and 
Poincar\'e-Melnikov theory.



Of course, the methods used  in \cite{Mather96}
are completely different from all the
methods based on perturbation theory.

We have hopes that a  blending of
the traditional methods, with
hyperbolic perturbation theory, a more
geometric understanding and variational
methods could lead to progress in the
problem of Arnol'd diffusion.

\subsection{Summary of the method}

The proof we present here can be conveniently divided into
different stages.

In a first stage, we use classical Riemannian geometry to establish the
existence of a family of periodic orbits. The whole family is a  two
dimensional  normally hyperbolic manifold which carries an exact
symplectic
form (restriction of the symplectic form in the phase space). Its stable
and unstable
manifolds intersect transversally and the motion on it is a twist map
with an unbounded frequency. This step is due to Morse and Mather, and
is covered is sections~\ref{sec:classical} and \ref{subs:hyperbolic}.

In a second stage (section \ref{subs:scaled}), we show that, for high
enough energy,
the perturbation introduced by the potential can be considered small. 
This is just an elementary scaling argument. We give full details 
mainly to set the notation.

In a third stage, we use perturbation theory of normally hyperbolic
manifolds to show that this normally hyperbolic
manifold persists into a locally invariant normally hyperbolic manifold,
and its stable and unstable manifolds keep on intersecting
transversally. Also, we note that the 
perturbed invariant manifold inherits a symplectic 
structure from the ambient space and that, therefore, 
the  rich methods of Hamiltonian perturbation theory 
can be brought to bear on the motion restricted to it.
A brief summary of hyperbolic perturbation theory is presented in
appendix \ref{ap:hyperbolic}, and the application to our problem is
presented in sections \ref{subs:extended} and \ref{subs:perturbed}. The
motion on this invariant manifold is faster than the perturbation
introduced by the potential.


In a fourth stage (section~\ref{subs:averaging}), we use averaging
theory to eliminate the fast
angles from the Hamiltonian to obtain that the motion on the normally
hyperbolic invariant manifold can be reduced to integrable up to an
error which is of very high order in the perturbation parameter.
Hence, the error decreases as an inverse power of the energy.
 
In a fifth stage (section~\ref{subs:KAM}), we use quantitative versions
of KAM theory to show
that the smallness of the perturbation in the 
invariant manifold leads to the fact that this 
invariant manifold is filled very densely with KAM 
tori, and we obtain explicit expressions for the main terms of these
tori.

In a sixth stage we use Poincar\'e-Melnikov theory to compute the change
of energy in a homoclinic excursion and show that, under appropriate
non-degeneracy assumptions, the---very close---KAM tori have to have
homoclinic and heteroclinic intersections.
 
These calculations are not completely standard due to the presence of
two time scales. We also note that the literature about Melnikov
functions for quasiperiodic objects is somewhat confusing.
Notably, some of the terms that make the naive Melnikov integrals
not absolutely converging
are incorrectly omitted in many papers. Hence, we decided
to present rather full details in section~\ref{subs:poincare}.

In a seventh stage (section~\ref{subs:transition}) we use the results
which
show that given transition chains, one can find orbits that shadow them.

We emphasize that all these stages use only readily available theorems
and techniques.
(Perhaps the less standard part is 
the part on the calculation of Poincar\'e-Melnikov functions, so
it appears fully expanded.)
Moreover, these stages are significantly independent, so
that if we assume---or arrive by other methods---to the conclusions of
one, all the subsequent results apply.

In particular, the above strategy was designed to be compatible with
variational methods. The invariant manifolds produced using  the theory of
normally hyperbolic manifolds carry Aubry-Mather sets,
as pointed by J.N.~Mather. Moreover, variational methods can be used
to provide powerful shadowing lemmas that can be used in the last stage.



\section{Classical geometry of the geodesic flow}\label{sec:classical}


The following geometric facts were proved by Morse and Mather and their
relevance
for the problem we are considering was discovered and emphasized in
\cite{Mather96}.
\begin{thm}\label{thm:Morse}
For a  ${\cal C}^{r}$ open and dense set of metrics in  $\torus^{2}$,
$r=1,\ldots , \infty ,\omega $, there exists a closed geodesic
$``\Lambda " $ which is hyperbolic in the dynamical systems sense as
a periodic orbit of the unit geodesic flow.

Moreover, there exists another geodesic  $``\gamma "$ and  real numbers
$a_{+}$, $a_{-}$, such that
\begin{equation}
\dist (``\Lambda "(t+a_{\pm}),``\gamma "(t)) \to 0 \quad \mbox{as}
\quad t \to \pm \infty
\end{equation}
\end{thm}


Here we will take the standard
definition that a  geodesic  $``\Lambda "$ is a curve  $``\Lambda
":\real \to \torus^{2}$, parameterized by arc length
which is a critical point for the length among any two
of its points.
Later, we will consider curves in the cotangent
bundle that are orbits of the geodesic flow.
Clearly, these orbits are closely related to
the geometric geodesics in the manifold.
We will use for the orbits in the cotangent bundle the
same letter as for the geodesics but suppress the quotation marks.
When we want to speak about the orbits of the geodesic 
flow as manifolds in phase space (more properly, the 
range of the mapping $\Lambda$), we will use a $ \hat {}$
(i.e. $ \hat \Lambda = {\rm Range }  \Lambda$).
Note that by changing the origin of time, we obtain another geodesic, so
that the geodesics satisfying geometric properties are always
one parameter families.
This consideration will be important when we consider time
dependent perturbations.

Actually Morse showed much more. He showed that there exists
one  $``\Lambda "$ in each  free homotopy class. Moreover, he showed that
$``\Lambda" $ can be taken to be minimizing and  $``\gamma "$ satisfies
other minimizing properties (class A). Such properties play an important
role in the work \cite{Mather96}, but we will not need them here. For the
approach presented here,  the hyperbolicity and the connection property
are what is important.
In particular, we note that dynamical systems theory shows that the
homoclinic connections $\gamma $ imply the existence of other periodic
orbits and other homoclinic connections. Even if these are not minimizing,
the proof developed here works without any change.


\section{ The unperturbed problem}


\subsection{Hamiltonian formalism and notation}\label{subs:Hamiltonian}


The present problem admits natural Lagrangian and Hamiltonian formulations.
 From our point of view neither of them plays a large role, but it seems
that the Hamiltonian point of view is somewhat more convenient. Hence,
this is the formalism that we will consider. 
%One interesting remark is
%that the Hamiltonian formalism allows a very easy incorporation of
%magnetic fields (a change in the symplectic structure), which seems hard
%to study  in the Lagrangian formalism.

The phase space of the geodesic flow is   ${\bf T}^{*}\torus^{2}=
\torus^{2}\times\real^{2} $. We will denote the coordinates in  $\torus^{2}$ by
$q$ and the cotangent directions by  $p$. Note that we are taking some
advantage---but mainly in the notation---of
the fact that the cotangent bundle of
$\torus^{2}$ is trivial.

We point  out that, as it is
well known,  the phase space, being a cotangent
bundle admits a canonical symplectic form,
which moreover is exact.

It is well known that for a
cotangent manifold ${\bf T}^* \torus ^{2}$
there is a unique
1-form $\theta$ such that
$\alpha^* \theta = \alpha$
for any one form $\alpha$ on  $\torus ^{2}$  .
(Here we think of forms as maps
from $\torus ^{2}$ to  ${\bf T}^* \torus ^{2}$.)

Then, $\Omega = d \theta$ is a symplectic form.
In local coordinates, $\theta = \sum_i p_i d q_i$,
$\Omega = \sum_i d p_i \wedge d q_i$.

With respect to the  form $\Omega$, the geodesic flow
is  Hamiltonian and the Hamiltonian function is
$H_0(p,q) = {1 \over 2} g_q(p,p)$. We will denote by  $\Phi_t $
this geodesic flow.

For each  $E$, we will denote  $\Sigma _{E}= \{ (p,q) \ | \
H_{0}(p,q)=E\}$, and observe that, for any  $E_{0}>0$ (later we will
use this for large  $E_{0}$),
 $\tilde \Sigma _{E_{0}}=  \cup _{E\ge E_{0}}\Sigma _{E} \simeq
[E_{0},\infty )\times \torus^{1}\times \torus^{2}$,
that is, we can take the energy as a part of a coordinate system.
Note that the energy is  one half the square of $|p|$
so that the energy  can be used as a radial coordinate in $p$. 
This is quite convenient.
We will also need an angle coordinate, to complete the polar
coordinate system.

We also note that  $\Sigma _{E}$---a three dimensional manifold
diffeomorphic to $\torus^{1} \times \torus^{2}  $---is invariant under the
geodesic flow.

Given an arbitrary geodesic  $``\lambda ":\real \to \torus^{2}$ we will
denote by $\lambda _{E}(t) =(\lambda ^{q}_{E}(t),\lambda ^{p}_{E}(t))$
the orbit of the geodesic flow that lies in the energy surface
$\Sigma _{E}$, and whose projection over  $q$ turns along the range of
$``\lambda "$. Moreover, we fix the origin of time in  $\lambda _{E}$ so
that it corresponds to the origin of the parameterization in  $``\lambda
"$. (Formally  $H_{0}(\lambda _{E}(t))=E$, and
$\mbox{Range}(``\lambda")= \mbox{Range}(\lambda^{q}_{E})
$,  $``\lambda "(0)=\lambda _{E}^{q}(0)$.)

It is easy to check that the
above conditions determine uniquely the orbit of the geodesic flow, in
particular determine  $\lambda ^{p}_{E}(t)$.
Note that
\begin{equation}
\label{rescaling}
\left( \lambda ^{p}_{E}(t),\lambda ^{q}_{E}(t) \right) =
\left( \sqrt{\! E}\lambda ^{p}_{1}(\sqrt{\! E}t),\lambda
^{q}_{1}(\sqrt{\! E}t) \right),
\end{equation}
so that, for the geodesic, the role of  $E$ is just a rescaling of
time.



\subsection{Hyperbolicity properties} \label{subs:hyperbolic}


Extending the methods of Morse for Theorem~\ref{thm:Morse}, 
J.N.~Mather
showed:

\begin{thm}\label{thm:morseflow}
For a  ${\cal C}^{r}$ generic metric, $r = 2,\ldots, \infty,\omega$,
and for any value of the
Hamiltonian $H_{0}(p,q)=E>0$, there exists a
periodic orbit  $\las(t)$, as in~(\ref{rescaling}), of the geodesic flow
whose range $\lao$ is a normally
hyperbolic invariant manifold in the energy surface.  Its stable and
unstable
manifolds $W^{s,u}_{\lao}$ are two dimensional, and there exists a
homoclinic orbit  $\gas (t)$, that is, its range  $\gao$ satisfies
\[
\label{transversal-gamma}
\gao  \subset  \left( W^{\st}_{\lao} \setminus \lao  \right) \cap \left(
W^{\un}_{\lao} \setminus \lao \right).
\]
Moreover, this intersection is transversal as intersection of
invariant manifolds in the energy surface along  $\gao$.

For  $E=1$, we have that
\begin{equation}\label{eq:asymp}
\dist {(\Lambda _{1}(t+a_{\pm}), \gamma _{1}(t))}\to 0 \quad
\mbox{as}\quad t \to \pm\infty .
\end{equation}
\end{thm}

We also note that since $\gao$ is 
one dimensional, $W^{\st}_{\lao}$,
$W^{\un}_{\lao}$ are two dimensional, and the ambient 
manifold $\Sigma_E$ is three dimensional, 
we have 
$T_x \gao = T_x W^{\st}_{\lao} \cap  T_x W^{\un}_{\lao}$
for all the points $x \in \gao$. Hence, by the implicit 
function theorem, $\gao$ is the locally unique intersection.
Since we are considering manifolds invariant under flows, 
their intersection has to contain orbits and $\gamma_E$ is
locally the only
possible---up to change in the origin of the parameter---orbit 
in the intersection of 
$ W^{\st}_{\lao} $ and $W^{\un}_{\lao}$.


In this situation the energy is preserved and then the dynamics can
be analyzed on the energy surface. This, however, will not be useful
when
we consider the external periodic potential which changes the energy.
Hence,
it will be useful to discuss what happens for all energy surfaces. The
following is a description of the behavior of  $\Lambda =
\bigcup_{E \ge E_0} \lao $ for all values of the energy.

\begin{lem}
\label{hyperbolic}
Define
$\Lambda   = \bigcup_{E \ge E_0} \lao$.
This is a
manifold  with boundary which is diffeomorphic to
$[E_{0},\infty )\times \torus^{1}$, and the canonical symplectic form
$\Omega$  on
${\bf T}^{*}\torus^{2}$ $\Omega $ restricted to  $\Lambda $ is
non-degenerate.  The form 
$\Omega |_\Lambda $ is invariant under the geodesic flow
$\Phi_t$.

We have for
some $C, \alpha > 0 $ and  for all  $x \in \lao$,
$$
T_x \Sigma_E = E^s_x \oplus E^u_x \oplus  T_x\lao
\label{eqn:hyperbolicdef}
$$
with $|| D \Phi_t(x) |_{ E^s_x} || \le C e^{-\alpha t} $ for
$ t \ge 0 $,
$|| D \Phi_t(x) |_{ E^u_x} || \le C e^{\alpha t} $ for
$ t \le 0 $ and
$|| D \Phi_t(x) |_{ T_{x}\lao} || \le C $ for
all $t \in \real$.



The stable and unstable manifolds to  $\Lambda $:
$
W^{\st}_{\Lambda }$,  $W^{\un}_{\Lambda }$,    are three
dimensional manifolds diffeomorphic to
$[E_{0},\infty )\times \torus^{1}\times \real $,
and
\[
\gamma = \bigcup_{E \ge E_0} \gao \subset
\left( W^{\st}_{\Lambda } \setminus \Lambda \right) \cap
\left( W^{\un}_{\Lambda } \setminus \Lambda \right)
\]
is diffeomorphic to  $[E_{0},\infty )\times \real$.
\end{lem}

We also note  that, since the definition of transversal intersection 
of manifolds only requires that the tangent spaces 
span the ambient space, when we add an extra dimension (in this case
the energy, but later we will consider 
other parameters) the intersection of the extended manifolds 
is still transversal. The intersection of the extended
manifolds will not be just one orbit 
but we will have
$$
T_x \gamma = T_x W^{\st}_\Lambda \cap  T_x W^{\un}_\Lambda .
\nonumber
$$
Hence, $\gamma$ will still be a locally unique intersection.

 We note that the only properties of the geodesic
flow
that we will use are the conclusions of Theorem~\ref{thm:morseflow} and
Lemma~\ref{hyperbolic}.



\subsection{Extended phase spaces}\label{subs:extended}


Since we are going to consider periodic
perturbations, it will be convenient to
introduce an extra angle variable, which 
we will denote by $s$, which moves at a
constant rate $1$.
Then, the phase space will be
$ {\bf T}^* \torus^2 \times \torus ^{1}$.

We will introduce the notation
$\tilde \Lambda = \Lambda \times \torus^1 $,
and analogously for $\gamma$, etc. to denote the
corresponding objects  in the extended phase
space.

In the case that we do not have any external potential,
the dynamics in this extended phase space is
just the product of the unperturbed dynamics in
$ {\bf T}^* \torus^2 $  and
the motion with constant speed in the circle.

In this extended phase space the results of section \ref{subs:hyperbolic}
simply translate into:
\begin{itemize}
\item
$\lao \times \torus ^{1}$   is a two dimensional invariant manifold. Its
(un)stable manifold is a three dimensional manifold.
They intersect  transversally in $\Sigma_E \times \torus ^{1}$.
(Of course, they are not transversal in the whole
extended space since they lie on the energy surface.)
\item
When we consider the results for all the energies, we
obtain normal hyperbolicity.

$\Lambda \times \torus^1 $ is a  $3$-dimensional manifold, and it is
normally hyperbolic for the extended flow $\tilde \Phi_t$ (see
Definition~\ref{def:hyperbolic} in appendix~\ref{ap:hyperbolic}). The
(un)stable manifolds of $\Lambda \times
\torus^1 $ are $ W^{\un,\st}_{\Lambda }\times \torus ^{1}$, and are
$4$-dimensional.
\item
Moreover, $\gamma \times \torus^{1}$ lies  
in the intersection of
$W^s_{\Lambda} \times \torus^ {1} =
W ^s _{\Lambda \times \torus^{1}}$
and of
$W^u_{\Lambda} \times \torus^ {1} =
W ^u _{\Lambda \times \torus^{1}}$,
and the intersection is transversal.

\item
The extended flow  $\tilde \Phi_ t$   restricted to the invariant
manifold  $\tilde \Lambda $  is neither contracting nor expanding:
\begin{equation}
|| D  \tilde \Phi_t (x) |_{T_x \tilde \Lambda} || \le C \ \
\forall t \in \real, \ \ x \in \tilde \Lambda.
\label{neutral}
\end{equation}


\end{itemize}

These observations will be important because they will allow us to use
the rich theory of hyperbolic invariant manifolds summarized in
appendix~\ref{ap:hyperbolic} when we consider the problem with the
external potential.

This extended phase space is obviously not symplectic
(it has odd dimension). In order to perform some
other calculations, we will find it convenient to
perform a symplectic extension.
This is accomplished by adding another real  variable $a$
symplectically conjugate to $s$, which does not change
with time.

Then, the symplectically extended phase space is
$ {\bf T}^* \torus^2 \times \real \times \torus^1 $.
The symplectic form in this space is
$\tilde \Omega = \Omega +d a \wedge d s  $.
The flow is Hamiltonian and its
Hamiltonian function is
$h(a, s, p,q) = a + H_0(p,q)$.

Since $a$ is conserved, the restriction of the flow of  $h$
to each of the manifolds  $a=\cte$ is identical to the
flow of  $H_{0}$  in the extended phase space.  In this case, the
neutral direction given by $a$ spoils all the hyperbolicity properties.
This situation is very common in Hamiltonian systems since
the  neutrality along a manifold as in (\ref{neutral}) implies 
similar bounds for the symplectic conjugate space.





\subsection{The inner map}\label{subs:inner}


We will consider  $F$ the time 1 map of the geodesic flow restricted to
$\Lambda $ i.~e. $F = \Phi_1 |_\Lambda$.   (This will make it easier
to analyze the time periodic external forcing.) As we are dealing with
the autonomous case, we note:
\begin{enumerate}
\item
It is still true that  $\Lambda $ is a normally hyperbolic surface
for  $\Phi_1$.
\item
The stable and the unstable manifolds  for $\Phi_1$ are the same as for the
flow $\Phi_t$. In particular, they are still transversal.
\item
$\Omega |_{\Lambda }$ is a symplectic form on  $\Lambda $.
\item
${\Phi_1}^{*}\Omega =\Omega $. Hence $F^{*}\Omega |_{\Lambda } =\Omega
|_{\Lambda }$.
\item
We have the canonical 1-form  $\theta $,
called the symplectic potential,  such that  $d\theta
=\Omega $. We note that  $\Omega |_{\Lambda }=d\theta |_{\Lambda }$.
\item
${\Phi_1}^{*}\theta  =
\theta +dS$. Hence, $F^{*}\theta|_{\Lambda } =
\theta|_{\Lambda } +dS|_{\Lambda }$. Therefore, the map  $F$ restricted to
$\Lambda $ is an exact symplectic map.
\end{enumerate}

\begin{rem}\label{rem:scaling}
Note that the rescaling properties~(\ref{rescaling})
of the geodesic flow  imply scaling properties
for the variational equations.
As a consequence of them, the angle
 $\langle T_{\lao}W^{\st}_{\lao},
T_{\lao} W^{\un}_{\lao} \rangle $
between the stable and
the unstable bundles in  $\lao$, remains bounded independently
of $E$. On the other hand,  the Lyapunov
exponents scale with $\sqrt{\! E}$.
Therefore,
\begin{eqnarray*}
\norm{D{\Phi_1}|_{T_{\lao} W^{\st}_{\lao}}}
&\le& \alpha ^{\sqrt{\! E}}\\
\norm{D{\Phi_{-1}}|_{T_{\lao}W^{\un}_{\lao}}}
&\le& \alpha ^{\sqrt{\! E}}
\end{eqnarray*}
where  $\alpha <1$ is independent of  $E$, even if
it  depends on the metric.
\end{rem}

\subsection{ A coordinate system on  $\Lambda $}

Now we want to describe a coordinate system in  $\Lambda $ that can be
used to compute the motions on it as well as their perturbations.
We want coordinate functions that are not only defined on $\Lambda$
but also on a neighborhood of it.
This will be particularly important for us mainly in the 
calculation of the Poincar\'e function.
Since the manifolds we are going to  consider
are cylinders, we will take one real coordinate and one
angle coordinate.

The real coordinate will be the unperturbed energy  $H_{0}$, which is
natural, given the fundamental role played by it in the problem.
For the conjugate angle, we will choose an angle coordinate (in
the $q$-space) of the geodesic $``\Lambda " $
 and extend it to  $\torus ^{2}$, and then, lift it to
${\bf T}^{*}\torus ^{2}$.
We can identify a point in  $\Lambda $ with the values
$(E,\varphi )$ taken by these coordinates. The coordinate representation
of the periodic orbit  $\las (t)$ in these coordinates
(see~(\ref{rescaling}))
is given by
$(E, \sqrt{\! E}t)$. Note that  
for any $\varphi_0 \in \real$, $\las (t+\varphi_{0} /\sqrt{\! E})$ is
another periodic orbit that in these coordinates is given by
$(E,\varphi_{0} +\sqrt{\! E}t)$.


For emphasis, the unperturbed map will be denoted also by  $F_{0}$.
Then:
\begin{equation}
F_{0}(E,\varphi ) = (E,\varphi +\sqrt{\! E})
\label{F0}
\end{equation}
We call attention to the fact that  $F_{0}$ is a twist map. The
restriction of the symplectic form to  $\Lambda $ can be expressed in
these coordinates as
\[
\Omega |_{\Lambda } = \frac{1}{\sqrt{\! E}} dE\wedge
d\varphi = d(2\sqrt{\! E})\wedge d\varphi
\]
and  is clearly  preserved by  $F_{0}$. Moreover, we have that
\begin{eqnarray*}
\theta |_{\Lambda } &=& 2\sqrt{\! E}d\varphi \\
F_{0}^{*}\theta |_{\Lambda } &=& \theta |_{\Lambda }+dE.
\end{eqnarray*}




\subsection{The outer map}


Another  important ingredient in our approach  is the map  $S:\Lambda
\to \Lambda $ that we will call the ``scattering map" or the ``outer map"
associated to  $\gamma $. This map  $S$ will transform the
asymptotic point at  $-\infty $ of a homoclinic orbit to  $\Lambda $
into the asymptotic point at  $+\infty$. We define the scattering
map as  $x_{+}=S(x_{-})$ if  $W^{s}(x_{+}) \cap W^{u}(x_{-})$ lies in a
curve which is in a neighborhood of  $\gamma $.

We note that this map is well defined locally due to the smooth
dependence on the base point of the (un)stable manifold and to the
fact that the intersection
is locally unique.

The following  construction  will make 
this definition more precise and will make it possible
to discuss perturbations.

If we  fix an small section 
${\cal S}$ of the flow near a portion of 
the homoclinic orbit  $\gamma$,  then
$ {\cal V} =  W^s_\Lambda \cap 
W^u_\Lambda \cap  {\cal S}$  
is a differentiable 
manifold because of the explicit 
description in section 
\ref{subs:hyperbolic}.
(Later, for perturbed mappings,
we will use another argument 
from appendix \ref{ap:hyperbolic}.)




We can define two mappings  $x_\pm: {\cal V}  \rightarrow \Lambda$
by $p \in W^s_{x_+(p)}$,
$p \in W^u_{x_-(p)}$. That is, we associate to 
a point $p\in{\cal V}$, the point in $\Lambda$ to whose
strong (un)stable manifold the point 
belongs. 

If we consider $  x_+({\cal V}) \times x_-({\cal V}) \subset \Lambda \times \Lambda$,
this manifold is  the graph of the mapping $S$ in an small neighborhood
in $\Lambda$.

Clearly, the mapping $S$ associates to an asymptotic 
orbit in $-\infty$ another asymptotic orbit at 
$+\infty$.

We note that this definition of the mapping $S$ is independent of
the choice of the transversal section $\cal S$ because the 
points $x_\pm$ are a property of the orbits in 
$\cal V$ and choosing another transversal section does 
not change the pairs $(x_-, x_+)$ that appear. 
(More geometrically, we only use the transversal 
section
to be able to parameterize the space of orbits.)

We call attention  
to the fact that all the elements in the above 
construction of
the map $S$ are based in  
intersecting manifolds constructed 
using hyperbolicity theory and 
that, therefore, using the results 
of appendix \ref{ap:hyperbolic}, 
it will be easy to show that they 
persist under perturbation and that they 
depend somewhat smoothly on 
parameters. See 
section
\ref{sec:hyperbolic-theoretical}.


We could worry that, as  $x_{-}$ moves around in a
loop,  $S(x_{-})$ does not come back to the same point. This, however
cannot happen because in such a case we would have two points in the
manifold  $\Lambda $,  $x_{+}^{1}$,  $x_{+}^{2}$  such that their stable
manifolds intersect.
Hence, we conclude that  $S$ is a well defined map, in the universal
cover of  $\Lambda $ and that a deck transformation in the domain
induces a deck transformation in the range:  $S\circ g = g'\circ S$.

A direct calculation  shows that in our
case  $S_{\#}:\pi (\Lambda )\to \pi (\Lambda )$ verifies  $S_{\#}={\rm
Id}$.
In any case, we note for future references that
even if $S$ gets deformed in a continuous way, $S_{\#}$ cannot depend
on parameters since it is an element of a discrete group.

\begin{rem}\label{rem:locally}
We note that even if the map $S$ is well defined as a map on  the 
manifold $\Lambda$, the mapping that to each point $x_{-}$ associates 
the orbit (with parameterization) effecting the connection, 
even if defined locally, is not well defined globally. If we move 
$x_{-}$ along a non-trivial loop in $\Lambda$, 
the connecting orbit gets transformed into another orbit 
with a different parameterization. (This amounts to 
incrementing or decrementing $a_\pm$ by a unit).
This remark will play a role when we discuss the Poincar\'e
function since it is expressed as an integral over the 
connecting orbit.
\end{rem}

We now proceed to compute the outer map for the unperturbed
problem, which will be denoted by $S_0:\Lambda \to \Lambda$.
For $x_{+}$, $x_{-}$ being two points of $\Lambda $,
 $x_{+}= S_0(x_{-})$ means that  $\exists z\in \gamma \subset {\bf
T}^{*}\torus^{2}$,  such that
\[
\dist {\left( \Phi_t (x_{\pm}),\Phi_t (z)\right) }\to  0 \ , \quad
\mbox{as}\quad t\to \pm \infty
\]

 From the results of Morse of Theorem~\ref{thm:morseflow}:
\begin{equation}
\dist (\Lambda _{1}(t+a_{\pm}), \gamma _{1}(t))\to 0,
\ \ {\rm as }  \  \
t\to \pm \infty
\end{equation}
and the rescaling properties give
\begin{equation}
\dist (\Lambda _{E}(t/\sqrt{\! E}+a_{\pm}/\sqrt{\! E}), \gamma
_{E}(t/\sqrt{\! E}))\to 0
\ \ {\rm as }  \  \
t\to \pm \infty
\label{limits1}
\end{equation}
and for any  $\varphi _{0}\in \real$,
\begin{equation}
\dist (\Lambda _{E}(t/\sqrt{\! E}+\varphi_{0} /\sqrt{\! E}
+a_{\pm}/\sqrt{\! E}),
\gamma _{E}(t/\sqrt{\! E}+\varphi_{0} /\sqrt{\! E}))\to 0
\ \ {\rm as }  \  \
t\to \pm \infty
\label{limits2}
\end{equation}


Hence, expressing the map  $S_0$ in the coordinates  $(E,\varphi )$,
we have
$S_{0}(\Lambda_E) = \Lambda_E$, and
 \[
S_{0}(E,a_{-}+\varphi ) = (E,a_{+}+\varphi )
\]
or more simply, calling  $\Delta =a_{+}-a_{-}$  the phase shift:
\begin{equation}\label{unperturbedouter}
S_{0}(E, \varphi ) = \left(E,\varphi +\Delta  \right).
\end{equation}

\section{The problem with external potential}

\subsection{Summary}

The main idea is that, for high energy, the external potential is an
small (and slow) perturbation of the geodesic flow.

Therefore, all the geometric structures that we constructed based on
normal hyperbolicity and transversality persist for high energy. In
particular, the manifold  $\Lambda $ will persist as well as the
transversality
of the intersection of its stable and unstable manifolds. This will
allow us to define $F,S$ analogues   of the maps  $F_{0},S_{0}$, and to
compute them perturbatively.

Using the information that we have of these maps, we will construct a
sequence  $n_{1}, \ldots, n_{k}, \ldots $, such that there is some
point $x$
\begin{equation}\label{transitionchain}
p_{k} = F^{n_{k}}\circ S \circ \cdots \circ F^{n_{1}}\circ S (x) \to
\infty. \end{equation}
This sequence of points  $p_{k}$  will be used as the skeleton for
orbits of the perturbed geodesic flow whose energy grows to infinity.
Heuristically, these orbits can be described as follows:
the orbits make  excursions roughly along the 
homoclinic orbit when the external potential has 
a phase that helps to gain energy,
but they bid their time between jumps
staying close to the unperturbed periodic orbits
till the phase becomes favorable again.
As
we will see later, this can also be interpreted in the usual language of
whiskered tori and their heteroclinic intersections.
By choosing the time when to perform the jumps, it will be possible for
the orbits to keep on 
gaining energy.

Therefore, the main technical goal will be to compute perturbatively,
for high energy, the inner and the outer maps  $F$ and  $S$, show that
applying them alternatively we can construct sequences  $p_{k}$ as
in~(\ref{transitionchain}) and then, show that these orbits can be
shadowed by real orbits.

The existence of  $p_{k}$  will require some non-degeneracy assumptions
on the external potential
(namely, that there are times at which jumping produces a gain in
energy). It turns out that the gain in energy is expressed by an
integral---commonly termed the Poincar\'e function---which
depends on
the phase at which the jump takes place (relative to the phase of the
potential). If this function, as a function of the jumping time, is not
constant, it is indeed possible to make jumps that gain energy.
Rather remarkably, the same integral and the same condition appears in
J.N.~Mather's approach even if with a very different motivation.



\begin{rem}
It is interesting to note that the variational construction in 
\cite{Mather96} also involves jumps
separated by orbits that stay close to  $\Lambda $. Nevertheless,
his construction emphasizes a great deal that the
orbits satisfy minimizing properties. In the
construction here, it seems that
minimality does not play any role. On the other hand, we need regularity
conditions for the invariant sets (they are smooth tori) and we could
not consider jumps to Cantor sets, as the variational method can.
\end{rem}

\begin{rem}
We recall attention  to the fact that the problem has two
different
smallness parameters. One is how close is the metric to the
integrable one. Another one is the inverse of the energy.
For large values of the energy, the potential can be considered
as a perturbation of the geodesic flow.
Since the last perturbation is done on an unstable system, the problem
can be considered as {\em a priori} unstable 
in the notation of \cite{ChierchiaG94}.

We also note that there are two different time
scales involved. One is the time scale of the period of the perturbation
($\Or(1)$) and the second one is that of the period of the geodesic
($1/\sqrt{\! E}$), which is also a characteristic time of the homoclinic
trajectory.
\end{rem}




\subsection{The scaled problem}\label{subs:scaled}


In order to make the perturbative structure of the problem more apparent
we will scale the variables and the time.
Thus, we pick a (large) number  $E_{*}$ and introduce  $\varepsilon
=1/\sqrt{\! E_{*}}$.

Recall that the original Hamiltonian is
$H(p,q,t)=\frac{1}{2}g_{q}(p,p)+U(q,t)$, hence
$\varepsilon ^{2}H(p,q,t)=\frac{1}{2}g_{q}(\varepsilon
p,\varepsilon p)+\varepsilon ^{2}U(q,t)$.
If we denote
$\varepsilon p=\bar p$
 and consider the symplectic form
$\bar \Omega =d\bar p\wedge dq = \varepsilon \Omega $,
we note that  $q,\bar p$ are
conjugate variables in  $\bar \Omega $. We also introduce a new time
$\bar t =t/\varepsilon $.
We see that the equations
\begin{eqnarray*}
\frac{dp}{dt} &=& -\frac{\partial H}{\partial q} =
-\frac{1}{2} \frac{\partial g_{q}}{\partial
q}(p,p)-\frac{\partial U}{\partial q}(q,t), \\
\frac{dq}{dt} &=& \frac{\partial H}{\partial p} =
g_{q}(p,\cdot),
\end{eqnarray*}
are equivalent to
\begin{eqnarray*}
\frac{d\bar p}{d\bar t} &=& -\frac{1}{2} \frac{\partial g_{q}}{\partial
q}(\bar p,\bar p)-\varepsilon ^{2}\frac{\partial U}{\partial
q}(q,\bar t \varepsilon ), \\
\frac{dq}{d\bar t} &=& g_{q}(\bar p,\cdot),
\end{eqnarray*}
which are Hamiltonian equations in  $\bar \Omega $, for the time  $\bar
t $, with respect to the Hamiltonian
\begin{equation}
\label{rescaled}
\bar H_{\varepsilon }(\bar p,q,\bar t \varepsilon )
=\frac{1}{2}g_{q}(\bar
p,\bar p)+\varepsilon ^{2}U(q,\bar t\varepsilon ).
\end{equation}

We also introduce  $\bar E = E/E_{*}$.
For our purposes, it suffices to 
analyze a fixed range in scaled energies 
(which we will fix  arbitrarily to be $[1/2,2]$) 
and establish that for large enough $E_{*}$, 
we can find pseudo-orbits which are often
close to $\Lambda$  and whose energy increases from 
$\approx 1/2$ to $\approx 2$.  Then, using that the result is 
valid for all the large enough energies, 
we can construct a pseudo-orbit whose 
energy grows unboundedly.

 From now on and until further notice, we will drop the  bar from
the problem. We will refer to the  bar variables as the rescaled
variables and the original ones as the physical variables. Then the
Hamiltonian  $H_{\varepsilon }$ and all the function derived from it
will be
$1/\varepsilon $ periodic in time. In order to do this more apparent we
will use the notation given in~(\ref{rescaled}).
Now, we want to compute rescaled versions of the maps $F_{0}$ and
$S_{0}$.

In order to compute  $F_{0}$, we note that  $H_{0}$ is the
Hamiltonian for the geodesic
flow and that what we are doing is just considering the time
$1/\varepsilon $ in the region  $ \cup _{1/2\le E \le 2}  \Sigma _{ E}$.

We note that  in the calculation of the inner map,
$\varepsilon$ enters in two different ways. One is
as a parameter in the Hamiltonian (\ref{rescaled})
and another one is that  $1/\varepsilon$ is the
time that we have to follow the flow.
We introduce  the notation
$f^\delta_\varepsilon$  as  the time  $1/\delta$  map
of the Hamiltonian (\ref{rescaled}).
We will find it convenient to compute
$f^\varepsilon_\varepsilon$
as a perturbation of $f^\varepsilon_0$.
We will show that $f^\varepsilon_0$ can be understood
in terms of the geodesic flow and, then, the
perturbation theory required to pass from $f_{0}^{\varepsilon }$ to
$f_{\varepsilon }^{\varepsilon }$ is only regular perturbation theory
for~(\ref{rescaled}).

Note also that when we consider extended phase spaces as introduced
in section~\ref{subs:extended}, the coordinate $s$, being an angle of period
$1/\varepsilon$ moving at unit speed, is invariant under the time
$1/\varepsilon$ map. Hence, when we consider $1/\varepsilon$ time
maps, it suffices to consider only the coordinates in $T^{*}\torus ^{2}$.
Of course, when the flow is non-autonomous, the map could depend on $s$,
the phase at which the forcing happens.
 

We note that $f^\varepsilon_0$ leaves invariant the manifold
$\lao$. In the coordinates  $(E,\varphi )$,
this map restricted
to  $\Lambda $ is (compare to~(\ref{F0})):
\begin{eqnarray*}
f_{0}^{\varepsilon }: \Lambda  & \to & \Lambda \\
(E,\varphi ) & \mapsto & (E, \varphi +\sqrt{\! E}/\varepsilon )
\end{eqnarray*}
Moreover, $\Lambda $ is normally hyperbolic, and the
hyperbolicity properties of the manifold  $\lao$ under this map
can be easily ascertained:

\begin{enumerate}
\item
The manifold  $\lao$  is invariant under $f_{0}^{\varepsilon }$.
\item
The stable and unstable bundles do not depend on  $\varepsilon $.
(As stated in Remark~\ref{rem:scaling},
 they do not depend on $E$
when we identify the  energy surfaces using the rescaling of the flow.)
\item
The contraction and expansion factors can   be bounded
\begin{eqnarray*}
\norm{Df_{0}^{\varepsilon }|_{T_{\lao}W^{\st}_{\lao}}} &
\le & \alpha ^{\sqrt{\! E}/\varepsilon }  \\
\norm{D(f_{0}^{\varepsilon })^{-1}|_{T_{\lao}W^{\un}_{\lao}}} & \le &
\alpha ^{\sqrt{\! E}/\varepsilon } \end{eqnarray*}
where  $\alpha $ does not depend on  $\varepsilon $.
\end{enumerate}
In the next pages, we will study the perturbation theory for the
manifold  $\Lambda $ and the time  $1/\varepsilon $ map restricted
to it.




\subsection{The perturbed invariant manifold}\label{subs:perturbed}


Using the hyperbolicity properties of the manifold  $\Lambda $ for the
geodesic flow (see section~\ref{subs:hyperbolic}), we will apply  the
results of hyperbolic perturbation theory summarized in the
appendix~\ref{ap:hyperbolic}.

In order to do
perturbation theory for the manifold $\Lambda
$, it will be more convenient to use the flow rather than the time
$1/\varepsilon $ map. Notice that the Lyapunov exponents of the
unperturbed map are  $\pm \infty $. Even if this does not interfere with
stability (roughly, the larger the Lyapunov exponents are, the more
stable the system is), it is cumbersome to  write the arguments. Later
we will see that the computation of the inner  map  $F_{\varepsilon }$   can
also be done more easily in this framework.

As indicated in section~\ref{subs:extended}, we  find it
more convenient to  introduce the extra variable  $s$   and consider
the autonomous flow
\begin{eqnarray}
\dot p &=& -\frac{\partial H_{0}}{\partial q}(p,q)
-\delta \frac{\partial H_{1}}{\partial q}(p,q,s/{\cal T}) \nonumber \\
\dot q &=& \frac{\partial H_{0}}{\partial p}(p,q)
 + \delta \frac{\partial H_{1}}{\partial p}(p,q,s/{\cal T})
\label{flow} \\
\dot s &=& 1 \nonumber
\end{eqnarray}
defined on   the extended
phase space ${\bf T}^{*}\torus ^{2} \times {\cal T}\torus^{1}  $. This
problem is equivalent to our
original one if we set  $\delta =\varepsilon ^{2}$,  ${\cal T}
=1/\varepsilon$, and  $H_{1}(p,q,s/{\cal T})= U(q,\varepsilon s)$.

We will denote the flow of~(\ref{flow}) by 
$\tilde \Phi_{t, {\cal T},\delta}(p,q,s)
= (\Gamma ^{s,s+t}_{{\cal T}, \delta }(p,q),s+t)$,
where  $\Gamma ^{t,t'}_{{\cal T}, \delta }(p,q)$ is the non-autonomous flow.
Note that as usual
$
\Gamma^{t',t''}_{{\cal T}, \delta} \circ \Gamma^{t,t'}_{{\cal T}, \delta}
= \Gamma^{t,t''}_{{\cal T}, \delta}
$
in the domains where these compositions make sense.



We note that setting  $\delta =0$  in~(\ref{flow}) we have that
\[
\tilde \Lambda := \Lambda \times {\cal T}\torus^{1}\simeq
[E_{0}, \infty ) \times \torus^{1}\times {\cal T}\torus^{1}
\]
is a  manifold locally invariant for the flow. This manifold is also
normally hyperbolic in the sense of Definition~\ref{def:hyperbolic}.

Using Theorem~\ref{thm:persistence1} and
observation~\ref{obs:parameters}
after it, we have:

\begin{thm}
\label{thm:Fenichel}
Assume that we have a system  of equations as in~{\rm (\ref{flow})},
where the Hamiltonian  $H=H_{0}+\delta H_{1}$ is  ${\cal C}^{r}$,
$2 \le r<\infty $. Then, there exists a  $\delta ^{*} > 0$ such that for
$|\delta| <\delta ^{*}$, there is a ${ \cal C}^{r-1} $ function
$$
{\cal F}: [E_{0}+K\delta ,\infty )\times \torus^{1}\times
{\cal T} \torus^{1}\times (-\delta^{*},\delta ^{*})
\longrightarrow {\bf
T}^{*}\torus ^{2} \times {\cal T} \torus^{1}
$$
such that
\begin{equation}
\tilde \Lambda _{{\cal T}, \delta }=
{\cal F}\left([E_{0}+K\delta  ,\infty ) \times \torus^{1} \times
{\cal T}\torus^{1}\times \{ \delta \} \right)
\label{parametrization}
\end{equation}
is locally invariant for the flow of~{\rm(\ref{flow})}.

Therefore, $ \tilde \Lambda _{{\cal T}, \delta } $ 
is $\delta$ ${\cal C}^{r-2}$-close to 
$ \tilde \Lambda _{{\cal T}, 0 } =  \tilde \Lambda$.
\end{thm}


 From now on, we are going to fix  our attention to the case $\delta
=\varepsilon ^{2}$ and  ${\cal T}=1/\varepsilon $,  and we will call
$\tilde \Lambda _{\varepsilon } 
= \tilde \Lambda _{1/\varepsilon , \varepsilon ^{2}} $,
$\tilde \Phi_{t, \varepsilon}= \tilde \Phi_{t, 1/\varepsilon,\varepsilon^2}$
and $\Gamma ^{t,t'}_{\varepsilon}=
\Gamma ^{t,t'}_{1/\varepsilon , \varepsilon ^{2}} $.


\begin{rem}
Even if the Theorem~\ref{thm:Fenichel} only guarantees local invariance
for  $\tilde \Lambda _{\varepsilon }$, we will show later that KAM theory
will provide with invariant boundaries consisting of KAM tori.
Therefore, it is possible to take  $\tilde \Lambda _{\varepsilon }$
invariant. Since the results in hyperbolic theory for locally invariant 
manifolds are somewhat sharper for invariant 
manifolds (they include uniqueness statements and 
a geometric definition of stable and unstable manifolds),
this will allow us later to state slightly sharper
results. The main results in this paper can be obtained 
without this improvement, hence we will just 
develop it in remarks.
\end{rem}

\subsection{The perturbed inner map}

\subsubsection{Averaging theory}\label{subs:averaging}

First of all, let us note that in  $\tilde \Lambda $ the unperturbed
inner map is given by:
\begin{eqnarray*}
f_{0}^{\varepsilon }: \tilde \Lambda  & \to & \tilde \Lambda  \\
 (E,\varphi,s ) & \mapsto &  \left( E,\varphi +
\frac{\sqrt{\!  E}}{\varepsilon },s+\frac{1}{\varepsilon } \right)
  = \left( E,\varphi + \frac{\sqrt{\!  E}}{\varepsilon },s \right).
\end{eqnarray*}

In order to compute the inner map
$F:= f_{\varepsilon }^{\varepsilon } : \tilde \Lambda _{\varepsilon }\to
\tilde \Lambda _{\varepsilon }$, we note that  the external perturbation
not only is small but is also slow. In this situation,
we can apply the methods of the adiabatic perturbation theory (see, for
example \cite{ArnoldKN88,LochakM88}).

For a fixed  $s$, we denote
$\Lambda^s_\varepsilon \subset {\bf T}^{*}\torus^{2}$ the manifold
obtained by  fixing $s$ in  $\tilde \Lambda _{\varepsilon }$ given by
(\ref{parametrization}):
$$
(\Lambda ^{s}_{\varepsilon },s) =
{\cal F}\left( [E_0+K\varepsilon ^{2},\infty )
\times \torus ^{1}\times\{ s \}\times \{ \varepsilon ^{2} \} \right)
$$
By Theorem~\ref{thm:Fenichel},  $\Lambda ^{s}_{\varepsilon }$    is
$\varepsilon ^{2}$-close to the unperturbed
manifold $\Lambda $ in the  ${\cal C}^{r-2}$ sense. In particular,
if we denote by $\Omega_\varepsilon ^{s}$ the restriction of
the symplectic form $ \Omega$ to these manifolds, it
is a symplectic form.  We also have
$\Omega^s_\varepsilon = d \theta^s_\varepsilon$,
where $\theta^s_\varepsilon$ is the restriction of
the  symplectic potential form
to $\Lambda^s_\varepsilon$.

The classical results of adiabatic perturbation theory we want to quote
refer to time dependent Hamiltonian flows on a fixed manifold with a
fixed symplectic structure, whereas we have a time dependent manifold.
Thus, we introduce changes of variables that keep the manifold fixed and
study the flow induced in the fixed manifold. Since the Hamiltonian
character is important in adiabatic perturbation theory, we pay
attention to the Hamiltonian structure of the changes of variables.


Since  $\tilde \Lambda _{\varepsilon }$ is invariant by the
flow
$\tilde \Phi_{t, \varepsilon}(p,q,s)
= (\Gamma ^{s,s+t}_{\varepsilon}(p,q),s+t)$ of~(\ref{flow}), we have
that $\Gamma^{t,t'}_{\varepsilon } : \Upsilon
_{\varepsilon }^{t}\subset \Lambda^t_\varepsilon \to
\Lambda^{t'}_\varepsilon$ (where  $\Upsilon _{\varepsilon }^{t}$
excludes a neighborhood of order  $\varepsilon ^{2}$ outside the
boundary of  $\Lambda _{\varepsilon }^{t}$).
Moreover, this
flow transforms the symplectic structure in
one manifold to the one of the image
${\Gamma^{t,t'}_{\varepsilon }}^{*}  \Omega^t_\varepsilon =
\Omega^{t'}_\varepsilon$.
Furthermore,
it is an exact transformation, that is,
${\Gamma^{t,t'}_\varepsilon}^{*} \theta^t_\varepsilon =
\theta^{t'}_\varepsilon + d S^{t,t'}_\varepsilon$
where $S^{t,t'}_\varepsilon$ is a real valued function
in $\Lambda^{t'}_\varepsilon $ and the $d$ refers to
the exterior differential in that manifold.

Now, since the manifolds $\Lambda_\varepsilon^s$
are close to the standard one $\Lambda $  we can  find coordinate maps
$C^s_\varepsilon: \Lambda_\varepsilon^s \rightarrow \Lambda$.
We claim that it is possible to choose these
$C^s_\varepsilon $ in such a way that they
transform the symplectic form  into the standard one.
In effect,
if we push forward the symplectic forms $\Omega^s_\varepsilon$,
we obtain a family of symplectic forms in $\Lambda$ which are close to
$\Omega $. These symplectic forms are also exact.
Applying Moser's method \cite{Weinstein77}, we can  find maps
from $\Lambda$ to $\Lambda$ that
transform these symplectic forms into the
standard one. We will just redefine
the $C^s_\varepsilon$ to include the composition
with these mappings in $\Lambda$.
A proof that these maps can be chosen to be
${\cal C}^{r -2} $ jointly with the parameters can be found in
complete detail in
\cite{BanyagaLW96}.

If we now consider
$ C_{\varepsilon }^{t'} \circ \Gamma^{t,t'}_\varepsilon \circ
(C_{\varepsilon }^{t})^{-1}$ we see that it is a flow of exact
symplectic mappings
in $\Lambda$.  The Hamiltonian  $k_{\varepsilon }(E,\varphi ,\varepsilon
s)$ generating this flow is the  push-forward  by $C^s_\varepsilon$
of the Hamiltonian  $H_{\varepsilon }(p,q,s/{\cal T})=
H_{\varepsilon }(p,q,\varepsilon s) $  generating
the flow of~(\ref{flow}) ($ {\cal T} = 1/\varepsilon$).
In particular, it is a  ${\cal C}^{r-2}$ flow,   $1/\varepsilon $
periodic
and  it is an small perturbation of
the constant flow $ \dot E = 0,  \dot \varphi  = \sqrt{\! E}$ of
Hamiltonian $2/3 E^{3/2}$.

In this situation, we can apply the Adiabatic Invariance
Theorem.


\begin{thm} \label{thm:adiabatic}
Let  $k_{\varepsilon }(E,\varphi ,\varepsilon s)$ be a 
${\cal C}^{ n }$  Hamiltonian,  $1$-periodic in $\varphi $ and  $\varepsilon
s$,  such that $k_{\varepsilon }(E,\varphi ,\varepsilon s) =
\frac{2}{3}E^{3/2}+\varepsilon ^{2}k_{1}(E,\varphi
,\varepsilon s;\varepsilon )$.

Then, for any  $0< m < n $, and  $E>0$,  there exists a canonical
change of variables $(E,\varphi ,s) \mapsto (I,\psi ,s)$,
$1$-periodic in $\varphi $ and  $\varepsilon s$, which is
$\varepsilon ^{2}$-close to the identity in the  ${\cal C}^{ n - m }$
topology,
such that transforms the Hamiltonian system of Hamiltonian
$k_{\varepsilon
}(E,\varphi ,\varepsilon s)$ into a Hamiltonian system of
Hamiltonian $K_{\varepsilon }(I,\psi ,\varepsilon s)$. This new
Hamiltonian is a ${\cal C}^{ n - m }$ function of the form:
$$
K_{\varepsilon }(I,\psi ,\varepsilon s) = K^{0}_{\varepsilon
}(I,\varepsilon s) + \varepsilon ^{ m +1} K^{1}_{\varepsilon }(I,\psi
,\varepsilon s)
$$
where  $K^{0}_{\varepsilon }(I,\varepsilon s)=
\frac{2}{3}I^{3/2}+\Ord {1}(\varepsilon ^{2})$,
and the notation
$\Ord {1} (\varepsilon )$
means a function whose 
${\cal C}^1$ norm is $\Or(\varepsilon)$.
\end{thm}


\proof

The proof of this Theorem is standard. For more details and
applications of the analytic
case, one can see \cite{ArnoldKN88}.
We will just go over the proof to show that it works for finite
differentiable Hamilttonians.

Calling  $a$ the  action
conjugate of time  $s$, we have the  $2$-degrees of freedom
Hamiltonian
$a+k_{\varepsilon }(E,\varphi ,\varepsilon s)$, which has a fast angle
$\varphi $ and a slow one  $\varepsilon s$.

We look for a canonical change of variables which eliminates the fast
angle  $\varphi $.
The change will be obtained through a composition of
changes of variables.
Each of these changes will be generated through a
generating function of the form:
\begin{equation}\label{generating}
Ps+I\varphi +\varepsilon ^{q+2}S_{q}(I,\varphi ,\varepsilon s;\varepsilon)
\end{equation}
where  $S_{q}$ is   $1$-periodic on  $\varphi $  and
$\varepsilon s$.

In this way, through the implicit equations
\begin{eqnarray*}
E &=& I +\varepsilon ^{q+2}\frac{\partial S_{q}}{\partial \varphi }
(I,\varphi ,\varepsilon s;\varepsilon)\\
a &=& P +\varepsilon ^{q+3}\frac{\partial S_{q}}{\partial \varepsilon
s }(I,\varphi ,\varepsilon s;\varepsilon) \\
\psi &=& \varphi  +\varepsilon ^{q+2}\frac{\partial S_{q}}{\partial I}
(I,\varphi ,\varepsilon s;\varepsilon)
\end{eqnarray*}
 we obtain a canonical change of variables
 $(E,\varphi ,a,\varepsilon s)\to (I,\psi ,P,\varepsilon s)$, where

\begin{equation}\label{change}
(E,\varphi ,a)= (I,\psi ,P) +\varepsilon ^{q+2} \psi _{q} (I,\psi
,\varepsilon s;\varepsilon)
\end{equation}
which, by the implicit function theorem, has one degree less of
differentiability than its generating function~(\ref{generating}).

We will apply the following inductive Lemma:

\begin{lem}\label{lem:inductive}

Let us consider a Hamiltonian of the form

$$
a+K_{q}(E,\varphi ,\varepsilon s ;\varepsilon)
= a+K^{0}_{q}(E,\varepsilon s;\varepsilon ) + \varepsilon ^{q+2}
K^{1}_{q}(E,\varphi ,\varepsilon s;\varepsilon )
$$
where  $K^{0}_{q}= \frac{2}{3} E^{3/2}+ \Ord {1} (\varepsilon ^{2})$ is
${\cal C}^{n-q+1}$  and
$K^{1}_{q}$ is   ${\cal C}^{n-q}$.

Let us consider  $S_{q}(I,\varphi ,\varepsilon s;\varepsilon)$ the
function verifying

\begin{equation}\label{gene}
\frac{\partial }{\partial I}
K^{0}_{q}(I,\varepsilon s;\varepsilon)
\frac{\partial S_{q}}{\partial \varphi }(I,\varphi,\varepsilon s;\varepsilon)
+ K^{1}_{q}(I,\varphi,\varepsilon s;\varepsilon) 
= \overline{K}^{1}_{q}(I,\varepsilon s;\varepsilon)
\end{equation}
where
$$
\overline{ K}^{1}_{q}(I,\varepsilon s;\varepsilon) =
\int_{0}^{1}K^{1}_{q}(I,\varphi,\varepsilon s;\varepsilon)d\varphi .
$$

Then, the change~(\ref{change}) generated by~(\ref{generating})
transforms the Hamiltonian
$a+K_{q}(E,\varphi ,\varepsilon s ;\varepsilon ) $, into a
 Hamiltonian
$$
a+K_{q+1}(I,\psi  ,\varepsilon s ;\varepsilon)
= a+K^{0}_{q+1}(I,\varepsilon s;\varepsilon ) + \varepsilon ^{q+3}
K^{1}_{q+1}(I,\psi  ,\varepsilon s;\varepsilon )
$$
where
$$
K^{0}_{q+1} (I,\varepsilon s;\varepsilon)
= K^{0}_{q}(I,\varepsilon s;\varepsilon)
+\varepsilon ^{q+2}\overline{ K}^{1}_{q} (I,\varepsilon s;\varepsilon) 
= \frac{2}{3} I^{3/2}+ \Ord{ n - q }(\varepsilon ^{2})
$$
is  ${\cal C}^{n-q}$ and $K^{1}_{q}$ is   ${\cal
C}^{n-q-1}$.
\end{lem}

\proof

Note that
$S_q = \int \d \varphi \left(K^{1}_{q} -\overline{K}^{1}_{q}\right)/
\partial _{I} K^{0}_{q}$, as well as 
$\partial S_{q}/\partial q$ are ${\cal C}^{ n - q }$.
The new Hamiltonian is given by
\begin{eqnarray*}
&&
P+
\varepsilon ^{q+3}\frac{\partial S_{q}}{\partial \varepsilon s }
(I,\varphi ,\varepsilon s;\varepsilon )\\
&&
+K^{0}_{q}\left( I+\varepsilon ^{q+2}\frac{\partial S_{q}}{\partial
\varphi }(I,\varphi ,\varepsilon s;\varepsilon ),\varepsilon
s;\varepsilon \right) \\
&&
+\varepsilon ^{q+2}K^{1}_{q}\left( I+\varepsilon ^{q+2}\frac{\partial
S_{q}}{\partial \varphi }(I,\varphi ,\varepsilon s;\varepsilon )
,\varphi ,\varepsilon s;\varepsilon \right) \\
&&=
P+K^{0}_{q+1}(I,\varepsilon s;\varepsilon )
+\varepsilon ^{q+3}K^{1}_{q+1}(I,\psi  ,\varepsilon s;\varepsilon )
\end{eqnarray*}
where, Taylor expanding  $K_{q}^{0}$ and  $K_{q}^{1}$ and   using the
Definition~(\ref{gene}) of the generating function, we get:
$$
K^{0}_{q+1}(I,\varepsilon s;\varepsilon ) =
K^{0}_{q}(I,\varepsilon s;\varepsilon )
+ \varepsilon ^{q+2}
\overline {K}^{1}_{q}(I,\varepsilon s;\varepsilon )
$$
and
\begin{eqnarray*}
&&
\varepsilon ^{q+3}K^{1}_{q+1}(I,\psi  ,\varepsilon s;\varepsilon )
=
K^{0}_{q}\left(I+\varepsilon ^{q+2}\frac{\partial S_{q}}{\partial
\varphi }(I,\varphi ,\varepsilon s;\varepsilon )
,\varepsilon s;\varepsilon \right)\\
&&-K^{0}_{q}(I,\varepsilon s;\varepsilon )
-\frac{\partial }{\partial I}K^{0}_{q}(I,\varepsilon s;\varepsilon )
\varepsilon ^{q+2}\frac{\partial S_{q}}{\partial
\varphi } (I,\varphi ,\varepsilon s;\varepsilon )\\
&&
+\varepsilon ^{q+2}\left(
K^{1}_{q}\left(I+\varepsilon
^{q+2}\frac{\partial S_{q}}{\partial \varphi }
(I,\varphi ,\varepsilon s;\varepsilon ),\varphi ,\varepsilon
s;\varepsilon  \right)-
K^{1}_{q}(I,\varphi ,\varepsilon s;\varepsilon  )\right)\\
&&
+\varepsilon ^{q+3}\frac{\partial S_{q}}{\partial \varepsilon s }
(I,\varphi ,\varepsilon s;\varepsilon )
\end{eqnarray*}
where, in these formulas,  $\varphi $   has to be expressed in terms of
the variables  $(I,\psi ,\varepsilon s;\varepsilon)$   using the change of
variables~(\ref{change}).

Since $K^{0}_{q}$ is  ${\cal C}^{n-q+1}$ and $K^{1}_{q}$ is ${\cal
C}^{n-q}$, it is clear that
$S_{q}$, $\partial _{\varphi} S _{q}$ are ${\cal C}^{n-q}$, and the change of
variables~(\ref{change}) is ${\cal C}^{n-q-1}$. 
(Note that in the equation above, only the term 
$\varepsilon ^{q+3} \partial _{\varepsilon s } S_{q}$
is ${\cal C}^{n-q-1}$.) 
Then one has that  $K^{0}_{q+1}$ is ${\cal C}^{n-q}$
and   $K^{1}_{q+1}$ is  ${\cal C}^{n-q-1}$.\qed

To finish the proof of Theorem~\ref{thm:adiabatic}, we
only need to apply the inductive Lemma~\ref{lem:inductive}  for
$q=0,1,\ldots m -1$, and we obtain the desired result.
For  $q=0$, it is important to note that
$K_{0}^{0}(E,\varepsilon s;\varepsilon) =
\frac{2}{3}E^{3/2}$ is  ${\cal C}^{\infty }$ if  $E>0$, and
$K_{0}^{1}(E,\varphi ,\varepsilon s;\varepsilon ) = k^{1}(E,\varphi
,\varepsilon s;\varepsilon )$ is  ${\cal C}^{n}$. Then the last
Hamiltonian will be of class  ${\cal C}^{ n - m }$.\qed



\begin{lem}\label{lem:preKAM}
In the conditions of Theorem~\ref{thm:adiabatic} with  $ n =r-2$, the map
$f_{\varepsilon }^{\varepsilon }$
restricted to $\Lambda_{\varepsilon}^{0}$,
which is exact symplectic, can be
written in the coordinates  $(I,\psi )$ introduced in
Theorem~\ref{thm:adiabatic} as
\begin{equation}
f_{\varepsilon }^{\varepsilon }(I,\psi ) = \left( I,\psi
+\frac{1}{\varepsilon }A(I,\varepsilon )\right) + \varepsilon
^{ m }R(I,\psi ;\varepsilon ),
\label{eqn:preKAM}
\end{equation}
where  $A(I,\varepsilon )= \varepsilon \int_{0}^{1/\varepsilon }
D_{1}K_{\varepsilon }^{0}(I,\varepsilon s)ds = \sqrt{I}+\Ord{0}
(\varepsilon ^{2})$, and  $R$ is a  ${\cal C}^{r- m -4}$ function.
\end{lem}

\proof

Recall that  $f_{\varepsilon }^{\varepsilon }$  in the  $(I,\psi )$
coordinates is the
time $1/\varepsilon $ map of the  ${\cal C}^{r-2- m }$ Hamiltonian
$K_{\varepsilon }$ whose flow is  ${\cal C}^{r-3- m }$.
The flow, in these coordinates, is the flow of an integrable Hamiltonian
$K_{\varepsilon }^{0}$ plus some Hamiltonian of order  $\Or
(\varepsilon ^{ m +1})$. Hence, using variational equations, we obtain
that the time  $1/\varepsilon $ map differs from that of the
integrable part by an amount not larger than  $\varepsilon ^{ m }$  in the
${\cal C}^{ n -4- m }$ topology.
\qed

\subsubsection{K.A.M. theory} \label{subs:KAM}

We now recall  a quantitative version of KAM Theorem. The version below
is somewhat weaker than that of \cite{Herman85} (we do not use
fractional
regularities so we loose whole integer number of derivatives in the
conclusion while an arbitrary real number would suffice), but is enough
for our purposes.
We recall that a real number $\omega $ is called a Diophantine number
of exponent $\theta $ if there exists a constant $C>0$ such that
$\abs{\omega -p/q}\geq C/ q^{\theta +1}$ for all $p\in\integer$,
$q\in\natural$.

\begin{thm} \label{thm:KAM}
Let
 $f:[0,1]\times \torus^{1}\mapsto [0,1]\times \torus^{1}$ be an exact
symplectic  ${\cal C}^{l}$   map, with $l\ge 4$.

Assume that  $f= f_{0}+ \delta f_{1}$, where  $f_{0}(I,\psi ) = (I,
\psi +A(I))$,  $A$ is $ {\cal C}^{l}$,
$\displaystyle \abs{\frac{dA}{dI}}\ge M$, and
 $\norm{f_{1}}_{{\cal C}^{l}}\le 1$.

Then, if  $\delta ^{1/2}M^{-1} = \rho $ is sufficiently small,
for a set of  $\sigma $ of Diophantine numbers of exponent
$\theta = 5/4$, we can
find invariant tori which are the graph of  ${\cal C}^{l-3}$ functions
$u_{\sigma }$, the motion on them is  ${\cal C}^{l-3}$ conjugate to the
rotation by  $\sigma $,
and $\norm{u_{\sigma }}_{{\cal C}^{l-3}}\le\cte  \, \delta
^{1/2}$, and the tori cover the whole annulus except a set of measure
smaller than  $\cte \, M^{-1}\delta ^{1/2}$.

Moreover, if  $l\ge 6$ we can find expansions
 $u_{\sigma }= u^{0}_{\sigma }+\delta u^{1}_{\sigma }+ r_{\sigma }$,
with $\norm
{r}_{{\cal C}^{l-4}}\le \cte \, \delta ^{2} $, and
$\norm{u^{1}}_{{\cal C}^{l-4}}\le \cte $.

\end{thm}

Applying Theorem~\ref{thm:KAM} to the map
$f_{\varepsilon }^{\varepsilon }$ restricted to $\Lambda_{\varepsilon}^{0}$,
in the coordinates given in
Lemma~\ref{lem:preKAM}, we obtain KAM invariant tori of 
system~(\ref{flow}), as long as  this map is 
${\cal C}^l $ with $l:=r - m -4\ge 6$.
Note that according to (\ref{eqn:preKAM}), 
the frequencies of $f^\varepsilon_0$ 
are roughly $(1/\varepsilon) A(I,\varepsilon)$
with $A(I,0)$ the frequencies of the unperturbed
Hamiltonian flow. Hence,
the ${\cal C}^{l -4}$ distance
between invariant tori is not bigger than $\varepsilon ^{m/2+1}$.
We note that these invariant circles for $f_{\varepsilon }^{\varepsilon }$
correspond to  invariant two dimensional tori for 
the extended flow. 
An invariant circle  for $f^\varepsilon_\varepsilon$ with 
frequency $\omega$ corresponds to a two dimensional invariant torus 
for $\tilde \Phi_{t,\varepsilon}$ with frequency $(\omega, \varepsilon)$.

\begin{rem}
Note that these KAM tori that we have produced for the map
$f_{\varepsilon }^{\varepsilon }$ are
really whiskered tori for the extended flow  $\tilde \Phi
_{t,\varepsilon }$. They
could have been produced also by appealing to the Graff-Zehnder Theorem.

In particular, proceeding as in Zehnder~\cite{Zehnder75a,Zehnder75b}
we can obtain a normal form for the Hamiltonian  $H_{\varepsilon
}(p,q,\varepsilon s)$ in a neighborhood of these KAM tori:

\begin{eqnarray}
G(I,a,\varphi ,s,z^{s},z^{u}) &=& \omega I+a+\frac{\Gamma
}{2}I^{2}+<z^{s},\Omega (\varphi ,s)z^{u}>\nonumber \\
&+& g(I,\varphi ,s,z^{s},z^{u})   \label{normalform}
\end{eqnarray}

Such normal forms are commonly used
in the study of  inclination lemmas for whiskered tori.
However, we will perform our study of inclination lemmas
in the normal form  for whiskered tori
with  one dimensional whiskers
introduced in 
\cite{FontichM98} (Section~(4.1)).
This normal form does not require that the
motion on the tori satisfies Diophantine
conditions---only that it is an irrational rotation---,
and requires much less regularity.
\end{rem}

\begin{rem}
When the metric and the potential
are ${\cal C}^\infty$
or  ${\cal C}^\omega$, 
even if the argument using the 
hyperbolic invariant manifold only
allows to construct finitely differentiable 
tori, appealing to the results in 
\cite{Zehnder75a,Zehnder75b},
we can conclude that these tori we constructed 
are indeed ${\cal C}^\infty$
or  ${\cal C}^\omega$.
\end{rem}

\begin{rem}
Note that KAM tori produced by Theorem~\ref{thm:KAM} are of codimension  $1$
inside  $\tilde \Lambda _{\varepsilon }$. If we choose a submanifold
whose boundary consists of two KAM tori, this submanifold will be an
invariant manifold for the extended flow. The results of hyperbolic
perturbation theory of appendix~\ref{ap:hyperbolic} can be extended to
include uniqueness as is explained in observation~\ref{obs:uniqueness}
after Theorem~\ref{thm:persistence1}.
\end{rem}




Once we have the existence of the invariant tori of system~(\ref{flow}),
it is worthwhile to obtain some explicit approximations 
for them in the coordinate system given by the phases
 $(\varphi ,s)$ and the value of the Hamiltonian  $H_{\varepsilon }$.
(Note that since the hamiltonian $H_\varepsilon$ is 
close to the coordinate $E$, $(H_\varepsilon, \varphi,s)$
constitute a good system of coordinates.)

We will find it convenient to write
$$
U(q,\tau ) = \bar U(\tau ) + \tilde U(q,\tau )
$$
where the functions  $\bar U(\tau )$,
and  $\tilde U(q,\tau )$ are given
by

\begin{equation} \label{ubar}
\bar U(\tau ) = \int_{0}^{1}U(\Lambda _{1}(\varphi ),\tau )d\varphi ,
\qquad \tilde U(q ,\tau ) = U(q,\tau ) - \bar U(\tau ).
\end{equation}

This decomposition is natural  because of  the different scales
involving the problem. We are separating
explicitly the average  on the fast variables.
We call attention to the fact that  $\bar U(\tau )$, being independent
of  $q$, does not affect the dynamics.

\begin{lem}
\label{lem:torus}
Let $\omega$ be one of the frequencies $\sigma$ 
allowed in Theorem~\ref{thm:KAM}.
Then, in the coordinate system 
$(H_\varepsilon, \varphi,s)$, 
we can write the torus of frequency 
$(\omega, \varepsilon)$ as the 
graph of a function 
$G(\varphi, s; \varepsilon)$.
Moreover, we 
can write 
\begin{equation}
G(\varphi,s;\varepsilon)
= E + \varepsilon^2 {\bar U}(\varepsilon s)  
+ \varepsilon^3  \tilde g(\varphi,s;\varepsilon)  
+ \Ord{l-4}(\varepsilon^4)
\label{eq:torus}
\end{equation}
where  $E = \omega^2$,
$\tilde g(\varphi ,\tau;\varepsilon )$ is a
$1$-periodic in  $(\varphi ,\tau )$  function which verifies
\begin{equation}
\omega D_{1}\tilde g(\varphi ,\tau;\varepsilon)+\varepsilon
D_{2}\tilde g(\varphi ,\tau;\varepsilon ) =
D_{2}\tilde U(\Lambda_{1} ^{q}(\varphi ),\tau ) + 
\Ord{l-4}(\varepsilon^3)
\label{tildeg}
\end{equation}
and $||\tilde g( \cdot, \cdot; \varepsilon) ||_{ {\cal C}^{l -4} }$
is bounded uniformly in $\varepsilon$.

Furthermore, we can choose $ \tilde g$ in such a
way that $\tilde g = D_2  \tilde h$.
This  $\tilde h$ satisfies (obviously)
\begin{equation}
\omega D_{1}\tilde h(\varphi ,\tau;\varepsilon)+\varepsilon
D_{2}\tilde h(\varphi ,\tau ) =
\tilde U(\Lambda_{1} ^{q}(\varphi ),\tau ) + 
\Ord{l-4}(\varepsilon^3)
\label{tildeh}
\end{equation}
and $||\tilde h( \cdot, \cdot; \varepsilon) ||_{ {\cal C}^{l -4} }$
is bounded uniformly in $\varepsilon$.
\end{lem}

We call attention to the fact that the functions as
we have defined them are not unique. On the other
hand, as we will see later, the ambiguities 
only arise in subdominant terms.



\proof
We will first present a formal proof and 
then we will work out the relation 
with perturbative methods such
as Lindstedt-Poincar\'e, which are
somewhat subtle since the problem involves
singular perturbations.
(One frequency is much larger than the other.)

The KAM theorem \ref{thm:KAM} provides us 
with parameterizations
$$
(p(\psi,\varepsilon s),
q(\psi,\varepsilon s), s)
$$
of the 
torus  in terms of 
variables  $\psi,s$
which satisfy $\dot \psi = \omega$,
$\dot s = 1$.

These parameterizations are $\Ord{l-4}(\varepsilon^3)$
close
to constant when expressed in terms 
of the averaged variables.

We denote by
\begin{equation}
G(\psi ,\varepsilon s;\varepsilon )=
H_{\varepsilon }(p(\psi ,\varepsilon s),q(\psi ,\varepsilon s),
\varepsilon s)
\label{Gdefined}
\end{equation}
and note that the derivative with respect to the 
flow of this equation is
\begin{eqnarray}
{d \over dt} G\circ \Phi_{t,\varepsilon} |_{t = 0}  &=&
\omega D_{1}G
(\psi,\varepsilon s;\varepsilon )
+\varepsilon D_{2}G(\psi ,\varepsilon s;\varepsilon ) \nonumber\\
&=& \varepsilon ^{3}D_{2}U(
q(\psi,\varepsilon s ;\varepsilon ),\varepsilon s).
\label{Gequation}
\end{eqnarray}

We note that the first two terms of the 
averaging transformations are 
$E + \varepsilon^2 {\bar U } (\varepsilon s)$
and that, as a consequence of
the hyperbolic perturbation theory, 
the averaging method and the KAM theory,
the KAM tori are close to
an orbit $\Lambda$ of the  unperturbed system. 
If we perform this substitution in
(\ref{Gequation}), we obtain the desired result.
\qed


\begin{rem}
The previous calculation can be also 
understood as a modification  of
Lindstedt-Poincar\'e method. 
Since the Lindstedt-Poincar\'e method
is a commonly used tool 
in singularly perturbed systems, 
we thought it could be interesting to
some readers to develop a 
comparison. We refer to 
\cite{Gallavotti94} for a survey 
of Lindstedt methods for analytic systems 
that includes a  treatment of 
singularly perturbed systems through the use 
of tree-like diagrams.

Since we are considering a 
system with two time scales, the 
most standard method, which fixes the
frequency and, then, seeks 
parameterizations of tori 
with the prescribed frequency as 
expansions in powers of $\varepsilon$,
cannot work since the frequency 
dependence in $\varepsilon$ will cause
the composed frequency to  go 
through resonances on which we 
do not expect tori to exist.

Nevertheless, we will see that it 
is possible to compute  systematically
parameterizations $p(\psi, \varepsilon s; \varepsilon)$,
$q(\psi, \varepsilon s; \varepsilon)$ that satisfy the 
equations of motion to a very high accuracy and 
whose coefficients  are, furthermore, of moderate size.
Once we have that, the Newton method 
started on them will lead to a true solution 
which is close to these approximate solutions.
(See \cite{Zehnder75a,Zehnder75b})

If we seek a parameterization
of the torus with frequency
vector $(\omega,\varepsilon)$,
as above,
we obtain a system of equations 
\begin{equation}
\begin{array}{rcl}
\left[\omega D_1+\varepsilon D_2\right]\, p(\psi, \varepsilon s; \varepsilon) 
&=&
- D_q H_\varepsilon ( p(\psi, \varepsilon s;\varepsilon), 
q(\psi, \varepsilon s;\varepsilon), s )  \\
\left[\omega D_1+\varepsilon D_2\right]\, q(\psi, \varepsilon s; \varepsilon) 
&=&
\hphantom{-} D_p H_\varepsilon ( p(\psi, \varepsilon s; \varepsilon), 
q(\psi, \varepsilon s;\varepsilon), s )  
\end{array}
\label{Lindstedt}
\end{equation}

Even if, as we will soon see, 
it is a bad idea to try to obtain solutions 
that are just powers of $\varepsilon$ 
with coefficients that are functions only  of
the other variables,
it is quite feasible to obtain 
expansions  in powers of 
$\varepsilon$ 
with coefficients
that are functions of all
the variables---including $\varepsilon$---which solve 
(\ref{Lindstedt}) up to a high order 
power in $\varepsilon$ and such that
all the coefficients are of order $1$.
These coefficients are not unique since 
the term of a certain order is only defined 
up to terms of higher order.

The main observation is that,
given $\Gamma$ with 
$\int_\torus 
\Gamma(\psi, \varepsilon s;\varepsilon)  \, d\psi = 0
$ and smooth, the equation 
for $\eta$
\begin{equation}
[ \omega D_1 + \varepsilon D_2] \eta(\psi, \varepsilon s;\varepsilon) 
=  \Gamma(\psi, \varepsilon s;\varepsilon) 
\label{recursiveLindstedt}
\end{equation}
can be satisfied up to high order error in 
$\varepsilon$ by  functions whose 
size is comparable to $\Gamma$.
As it is well known, this is the homology equation and the 
Lindstedt series can be computed by recursively solving this
equation on expressions  that involve only previously 
computer quantities.

If we try to solve
(\ref{recursiveLindstedt}) 
using Fourier analysis,
we find that it is 
equivalent to
\begin{equation}
\hat \eta_{k_1,k_2} = 
\left( 2 \pi \i (\omega  k_1 + \varepsilon k_2 )\right)^{-1}
\hat \Gamma_{k_1,k_2}
\label{FourierLindstedt}
\end{equation}

If we choose $\eta$ in such a way that 
its Fourier coefficients with 
$|k|  \le \varepsilon^{-1/2} $
are obtained according to 
(\ref{FourierLindstedt}) and the 
other ones are zero,
we note that:
\begin{itemize}
\item[a)] If $\Gamma$ is  ${\cal C}^m$
then 
$$
| \hat \Gamma_{k_1,k_2}| \le  C  |k|^{-m} || \Gamma||_{ {\cal C}^m }.$$
Hence,  $\eta$ solves the equation 
(\ref{recursiveLindstedt}) up
to an error whose
${\cal C}^{l} $ norm 
can be bounded 
by 
$ C  || \Gamma||_{ {\cal C}^m} \sum_{|k| \ge \varepsilon^{1/2}} |k|^{l-m}$,
which 
can, in turn, be bounded 
by  $C || \Gamma||_{ {\cal C}^m} \varepsilon^{ (-l+m-2)/2 } $
when  $ l - m + 1  < -1$. 

\item[b)] Since $\Gamma$ has no 
Fourier coefficients  with $k_1 = 0$, 
then the denominators of 
(\ref{FourierLindstedt}) are 
uniformly bounded from below
and we have, using the same 
estimates as above, 
$|| \eta||_{{\cal C}^l} \le C  || \Gamma||_{ {\cal C}^m}$
when $ l -m + 1 < -1$.

\end{itemize}

By repeating this construction in all the steps 
that we have to solve 
(\ref{recursiveLindstedt}) in the 
calculation of the Lindstedt series, we obtain
functions of size bounded uniformly in 
$\varepsilon$ which satisfy  (\ref{Lindstedt}) up to
an error which can be bounded by a power of $\varepsilon$.
This power can be made arbitrarily high if 
we are considering systems that are 
differentiable enough.


Note that these approximate solutions---in contrast to those of the 
standard Lindstedt method---are not unique
since they include choices such as the level 
of truncation
(we took  $|k| \le \varepsilon^{-1/2} $
but could have made other choices).


The above procedure makes it 
clear that it is a bad idea solving 
the equations 
(\ref{recursiveLindstedt})
exactly. If we 
considered in 
(\ref{FourierLindstedt})
the coefficients  with $|k| \approx \varepsilon^{-1}$ 
or bigger we would indeed have to consider small 
divisors.  This is a reflection of 
the fact that there is no number 
$\omega$ such that 
$(\omega, \varepsilon)$ is a nonresonant 
vector for an interval of 
$\varepsilon$ around zero.

Once we have parameterizations 
that solve (\ref{Lindstedt}) 
with a very small error, we 
can apply  an appropriate version 
of KAM theorem to produce
an exact solution.
Indeed, this 
Lindstedt method is an alternative to
the averaging method that we
used in the main text.

We emphasize that for the applications that we have in mind here,
there suffices to compute only a finite number of terms to obtain
approximations to $\Or(\varepsilon^n)$. Hence, there is no need to
discuss convergence and we only need that the functions involved are
finitely differentiable. 
\end{rem}

\subsection{The perturbed outer map. Theoretical results}
\label{sec:hyperbolic-theoretical}

Now, we turn to the problem of  computing the map  $S$.
We will be mainly interested in computing the image under  $S$ of one of
the KAM circles invariant under  $F$. Moreover it will suffice to study
the change in energy, and, more importantly, characterize when this
change of energy produces a jump from a torus to another.

The theory developed in appendix \ref{ap:hyperbolic}
will show that the construction of the map $S$ 
can be carried out for small perturbations, 
giving rise to 
a differentiable 
map that also depends differentiably on 
parameters. Note that once we now that the map 
is differentiable with respect to 
parameters, we will be able to compute the derivatives
just matching powers.


In effect,  if we fix a  transversal section ${\cal S}$  to the flow
in a neighborhood of $\gamma$, 
we can consider 
${\cal V}_\varepsilon = W^{u}_{ \tilde \Lambda, \varepsilon} \cap
W^{u}_{ \tilde \Lambda, \varepsilon} \cap  {\cal S} $,
where 
$W^s_{\tilde \Lambda, \varepsilon}$ is
the stable manifold  under the perturbed flow
$\tilde \Phi_{t,\varepsilon}$ of $\tilde \Lambda_\epsilon$---the
manifold invariant under $\tilde \Phi_{t,\varepsilon}$ close
to $\Lambda$. (See section \ref{subs:hyperbolic} and
appendix  \ref{ap:hyperbolic}.)

Because of the smooth dependence on  parameters of 
the stable and unstable manifolds, 
for small enough  $\varepsilon$,
this ${\cal V}_\varepsilon$
is  a  smooth  manifold that depends smoothly on 
the parameter $\varepsilon$.
(See Theorem \ref{thm:persistence1} for details on
how smooth the flow may be chosen to be.)

On ${\cal V}_\varepsilon$ we can define  functions
$x_{\pm, \varepsilon}: {\cal V}_\varepsilon \to
 \tilde \Lambda_\varepsilon$. These 
functions are jointly smooth in the variables and in the 
parameters. 
For some precisions on the rather standard
method  to
define smooth functions on smooth  families of 
manifolds and for the degree of differentiability that one 
can expect in this situation, we refer
to Theorem~\ref{thm:persistence1}.
We note that, for small $\varepsilon$, we
can assume that, when 
$r$, the differentiability of 
the flow  satisfies $r < \infty$, 
all  the differentiability properties of 
the manifolds and of the dependence with respect to
parameter  are just $r$ since
the other limiting  regularities in the
general theory of hyperbolic systems are 
higher than $r$.

We argued already that 
$ x_{+,0}( {\cal V}_0) \times
 x_{-,0}( {\cal V}_0)  \subset \Lambda_0 \times \Lambda_0 $
is the graph of a function. 
Hence, for small $\varepsilon$, 
$ x_{+,\varepsilon}( {\cal V}_\varepsilon) \times
 x_{-,\varepsilon}( {\cal V}_\varepsilon)  \subset
\tilde \Lambda_\varepsilon \times \tilde \Lambda_\varepsilon $
will also be the graph of a smooth function,
and it will depend smoothly on parameters.

\medskip

We call attention to the fact that, as discussed in 
appendix~\ref{ap:hyperbolic}, the stable
and unstable manifolds for 
$ \tilde \Lambda_\varepsilon $ are foliated by the stable
and unstable manifolds of  the
points in $ \tilde \Lambda_\varepsilon$. For the tori produced in 
Theorem~\ref{thm:KAM}, these (un)stable manifolds
are called the whiskers.


As we know already from 
Lemma~\ref{lem:torus}  an approximate 
characterization of the KAM tori in
$\tilde
\Lambda_\varepsilon$ in terms of $H_\varepsilon$, we will be interested
in computing the change of $H_\varepsilon$ in the
outer map.

Due to the fact that  $\tilde \Lambda _{\varepsilon }$ is a hyperbolic
manifold and that its stable and unstable manifolds intersect, we know
that any point   on $\tilde \Lambda _{\varepsilon }$ is connected to
another point through some ``homoclinic" (to
$\tilde \Lambda _{\varepsilon }$) orbit.

The fact that $x_+ = S(x_-) $
for two points $x_{\pm} \in \tilde \Lambda_\varepsilon$ is,
by definition,
equivalent to the existence of a  point
$z \in {\bf T}^{*}\torus^{2}\times 
{\cal T}\torus ^{1}$,
for  ${\cal T}=1/\varepsilon $,
such that the trajectory
$\tilde \gamma (t)$ starting on it converges for $t \to \pm \infty$
exponentially fast to the trajectories
$\tilde \Lambda^\pm $ started in  $x_{\pm}$ respectively.
i.e.
$\tilde \Lambda^{\pm}(0) = x_\pm$
and
\begin{equation}\label{eq:asymptotics}
\dist ( \tilde \Lambda^\pm(t), \tilde \gamma(t) ) \le 
C e^{-\beta |t|}
\quad \mbox{as}  \quad t \to \pm \infty.
\end{equation}

We want to give criteria to decide whether   $x_{+}$  and
$x_{-}$ belong
to the same torus (and  $\tilde \gamma  $ would be  a homoclinic orbit
to
this torus), or if they are in different tori (and  $\tilde \gamma $ is
heteroclinic).






\subsection {The perturbed outer map. The Poincar\'e function.}
\label{subs:poincare}



The characterization of  orbits homoclinic to a torus will be done in two
steps.
First we will compute  $x_{+}=S(x_{-})$. From~(\ref{unperturbedouter}),
and using the fact that the perturbation is  $\Or(\varepsilon ^{2})$,
see~(\ref{rescaled}),  we already know
that $\varphi _{+}= \varphi _{-}+\Delta +O(\varepsilon ^{2})$, where
$\Delta =a_{+}-a_{-}$ is the phase shift. Moreover, the
asymptotic condition~(\ref{eq:asymptotics})
and the fact that $\dot s = 1$ even for the perturbation,
implies  $s_{+}=s_{-}=s$,
where   $s$ is the time coordinate of the homoclinic point  $z$.

The first goal of this section will be
to compute the leading term of
the difference 
$H_{\varepsilon }(x_{+})-H_{\varepsilon }(x_{-}) $ 
(which is zero in the unperturbed case). This is done in
Lemma~{\ref{lem:melnikov1}} using the classical Poincar\'e-Melnikov
theory.
In a second step, we need to decide whether  $S(x_{-})=x_{+}$ and
 $x_{-}$ lie in the same torus of those produced by Theorem~\ref{thm:KAM}.
Since the energy $H_{\varepsilon
}$ is not constant on the tori, it is not enough to compare
$H_{\varepsilon }(x_{+})$ with $H_{\varepsilon }(x_{-})$. Instead,
we will consider the point  $y$ belonging to the torus of
$x_{-}$
but with the same phases as  $x_{+}$ and  compute  $
H_{\varepsilon }(x_{+})-H_{\varepsilon }(y)$, which will be our
Melnikov function. See figure~\ref{pointy}.

Therefore, the main  goal of this section will be to give an explicit
expression of the difference  $H_{\varepsilon
}(x_{+})-H_{\varepsilon }(y)$.
To compute $H_{\varepsilon }(y)$
we will follow a general method,
which consists in using the
equations~(\ref{eq:torus}) 
that express  the torus
containing
$x_{-}$ as
a graph  of the energy $H_\varepsilon$ over the phases.
If we evaluate this function at the 
phases of  $x_{+}$, we obtain a Poincar\'e
function~(\ref{pmelnikov}) which gives the leading term of
$H_{\varepsilon }(x_{+})-H_{\varepsilon }(y)$~\cite{Treschev94a}.
This will be done in Lemma~\ref{lem:melnikov}.

For the system at hand, we can take advantage of the slow dynamics and
we can use the fact that the point  
$\tilde \Phi_{\Delta /\omega ,\varepsilon }(x_{-}) \equiv u $
 has the same phases  $(\varphi ,\varepsilon s)$
as  $y$ up to order  $\varepsilon $.
Using this fact, in   Lemma~\ref{lem:potmather} we
will  give an explicit formula for the leading term of the Poincar\'e
function
in terms of the potential  $U$ and the
unperturbed
geodesics, with no need to solve any small divisors equation to
obtain  $H_{\varepsilon }(y)$.
This explicit expression will be quite important to establish that, for
high enough energies---in the scaled variables for small enough
$\varepsilon $---the KAM tori have homoclinic and heteroclinic
intersections.



\begin{figure}
\centerline{\psfig{file=tori.eps,height=75mm,angle=-90}}
\caption{Illustration of the perturbed tori and the outer map.}
\label{pointy}
\end{figure}


\begin{lem}\label{lem:melnikov1}

Let $x_{-}$ and  $x_{+}$ be two points on  $\tilde
\Lambda_{\varepsilon } $ such that $x_{+}=S(x_{-})$, then
\begin{eqnarray}
H_\varepsilon(x_+)  - H_\varepsilon(x_-) &=&
\varepsilon^3 \lim_{(T_{1},T_{2})\to \infty}\left[
\int_{-T_{1}}^{T_{2}}\, dt \,
D_2 \tilde U( \gamma_{E}^{q}(t+\frac{\varphi_{0}}{\sqrt{\!E}}),
\varepsilon s+\varepsilon t)   \right. \nonumber \\
&&-\int_{-T_{1}}^{ 0 } \, dt \,
 D_2 \tilde U( \Lambda_{E}^{q}(t+\frac{\varphi_{0}+a_{-}} {\sqrt{\!E}}),
\varepsilon s+\varepsilon t)  \label{incre} \\
&&- \left. \int_{0}^{ T_{2} } \, dt \,
 D_2 \tilde U( \Lambda _{E}^{q}(t+\frac{\varphi_{0}+a_{+}}{\sqrt{\!E}}),
\varepsilon s+\varepsilon t)
\right]  + \Or(\varepsilon^{5}) \nonumber
\end{eqnarray}
where
$$
x_{\pm} = {\cal F}(E,\varphi _{\pm},s,\varepsilon ^{2}) =
\left(\las \left(\frac{\varphi _{0}+a_{\pm}} { \sqrt{\! E}} \right)
+ \Or (\varepsilon ^{2}),s\right)
$$
for some $\varphi _{0}\in \real$, where  ${\cal F}$   is introduced in
Theorem~{\rm \ref{thm:Fenichel}}, as a parameterization of the
invariant manifold  $\tilde \Lambda _{\varepsilon }$
and $\tilde U$, introduced in~{\rm(\ref{ubar})}, is the forcing potential minus 
its average on the periodic orbit $\Lambda$.
\end{lem}

\proof

Recall that if a trajectory $\tilde \lambda (t) = (\lambda
^{p}(t),\lambda ^{q}(t),s+t)$ satisfies (\ref{flow}) then:
$$
{d \over dt } H_\varepsilon \circ \tilde \lambda(t)
= \varepsilon^3 D_2 U( \lambda^q(t), \varepsilon s+\varepsilon t).
$$

Therefore, for any two trajectories
$\tilde \lambda= (\lambda ^{p},\lambda ^{q},s+t))$, $\tilde \mu=(\mu
^{p},\mu ^{q},r+t)$ of (\ref{flow}), we have, by the fundamental
theorem of Calculus,

\begin{eqnarray}
\lefteqn{H_\varepsilon(\tilde \lambda(T)) - H_\varepsilon(\tilde  \mu(T))
=  H_\varepsilon(\tilde \lambda(0)) - H_\varepsilon( \tilde \mu(0))}
   \label{calculus}  \\
&+&   \varepsilon^3 \int_0^T  dt \, D_2 U( \lambda^q(t), \varepsilon
s+\varepsilon t) -  \varepsilon^3  \int_0^T  dt \, D_2 U( \mu^q(t),
\varepsilon r+\varepsilon t) . \nonumber
\end{eqnarray}
As  $x_{+}=S(x_{-})$, we know that there exists
 $z \in {\bf T}^{*}\torus ^{2}\times {\cal T}\torus ^{1}$,  ${\cal
T}=1/\varepsilon $, such that the trajectory
$\tilde \gamma _{(\varepsilon )}$ starting
on
it and  $\tilde \Lambda _{+,(\varepsilon )}
$,  $\tilde \Lambda _{-,(\varepsilon )}$, the trajectories
starting on  $x_{+}$,  $x_{-}$,   verify~(\ref{eq:asymptotics}).
Therefore the three points have the same time coordinate  $s$.


Now we can use~(\ref{calculus}) and, by~(\ref{eq:asymptotics}),
 taking limits at $\pm \infty$ as appropriate,
\begin{eqnarray*}
0 &=&
H_\varepsilon(x_+)  - H_\varepsilon (z)  \\
&&\mbox{} + \lim_{T_{2}\to \infty}
\varepsilon^3
\int_{0}^{T_{2}}  \, dt \,
\left( D_2 U( \Lambda^{q}_{+,(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  -
 D_2 U( \gamma^{q}_{(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  \right)\\
 0 &=& H_\varepsilon(x_-)  - H_\varepsilon(z)  \\
&&\mbox{} + \lim_{T_{1}\to \infty}
 \varepsilon^3
\int_0^{-T_{1}}  \, dt \,
\left( D_2 U( \Lambda^{q}_{-,(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  -
 D_2 U( \gamma^{q}_{(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  \right).
\end{eqnarray*}

Subtracting these two equations
we obtain
\begin{eqnarray}
\lefteqn{H_{\varepsilon} (x_{+})  - H_{\varepsilon} (x_{-})=} \nonumber \\
&-& \varepsilon^{3} \lim_{(T_{1},T_{2})\to \infty}
\left[ \int_{0}^{T_{2}}  \, dt \,
\left( D_2 U( \Lambda^{q}_{+,(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  -
 D_2 U( \gamma^{q}_{(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  \right) \right.
\nonumber \\
&-& \left.
\int_{-T_{1}}^{0}  \, dt \,
\left( D_2 U( \Lambda^{q}_{-,(\varepsilon) }
(t), \varepsilon s+\varepsilon t)  -
 D_2 U( \gamma^{q}_{(\varepsilon )}
(t), \varepsilon s+\varepsilon t)  \right) \right].
\label{increment}
\end{eqnarray}
By~(\ref{eq:asymptotics}), these limits are
reached uniformly in $\varepsilon$. We also note
that the dependence of the trajectories on
$\varepsilon$  is uniform
on compact intervals of time.
Hence, at the expense  only of 
introducing an error of higher order in $\varepsilon$,
 we can  substitute in~(\ref{increment}) for
$\Lambda_{\pm,(\varepsilon )}$ and $\gamma_{(\varepsilon )}$ the
unperturbed orbits:
\begin{eqnarray*}
\Lambda_{\pm,(\varepsilon )}(t) &=& \Lambda_{E}\left(t +\frac{\varphi _{0}
+a_{\pm}}{\sqrt{\!E}}\right) +\Or (\varepsilon ^{2})\\
\gamma _{(\varepsilon )}(t) &=& \gamma _{E}\left(t+\frac{\varphi
_{0}}{\sqrt{\!E}}\right)+\Or (\varepsilon ^{2})
\end{eqnarray*}
with $\Lambda_{E}$
and  $\gamma _{E}$  the orbits constructed in  Theorem~\ref{thm:morseflow}.

We note that the right hand side of~(\ref{increment}) is linear in  $U$.
Hence if we use the decomposition  $U(q,\tau )=\bar U(\tau )+\tilde
U(q,\tau )$ given in~(\ref{ubar}), and observe that computing the right
hand side of~(\ref{increment}) in  $\bar U$ gives zero,
we obtain~(\ref{incre}). \qed



\begin{lem}\label{lem:melnikov}
Let  $y$ be a point 
with the phases of $x_{+}$
and which lies on the invariant torus for the perturbed flow which
contains $x_{-}$,
where
$$
x_{+} = {\cal F}(E,\varphi _{+},\varepsilon s,\varepsilon ^{2})
=\left(\las\left(\frac{\varphi_{0} +a_{+}}{ \sqrt{\! E} } \right)+
\Or (\varepsilon^{2}),\varepsilon s\right).
$$
Then:
\begin{eqnarray}
H_{\varepsilon }(x_{+})-H_{\varepsilon }(y) 
&=&\varepsilon^{3} \lim _{(T_{1},T_{2}) \to \infty } \Bigg[
\int_{-T_{1}}^{T_{2}}\ dt D_{2}
\tilde U(\gamma _{E}^{q}\left(t+\frac{\varphi _{0}
}{\sqrt{\!E}} \right) ,\varepsilon s+\varepsilon t) \nonumber \\
&&\hphantom{\varepsilon ^{3} \lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
- \tilde g (\varphi _{0}+a_{+}+\sqrt{\!E}\,T_{2},\varepsilon
s+\varepsilon T_{2};\varepsilon) \nonumber  \\
&&\hphantom{\varepsilon ^{3} \lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
- \tilde g (\varphi _{0}+a_{-}-\sqrt{\!E}\, T_{1},
\varepsilon s-\varepsilon T_{1};\varepsilon) \Bigg]
\nonumber \\
&&+ \Or (\varepsilon ^{5})\, . \label{melnikov}
\end{eqnarray}
where  $\tilde g$ is the function given in Lemma~\ref{lem:torus}
verifying~(\ref{tildeg}), associated to the invariant torus of the
perturbed flow which contains $x_{-}$.
\end{lem}
\proof

We  use Lemma~\ref{lem:melnikov1} for
$H_{\varepsilon }(x_{+})-H_{\varepsilon }(x_{-})$
and Lemma~\ref{lem:torus} for
$H_{\varepsilon }(x_{-})-H_{\varepsilon }(y)$:
\begin{eqnarray}
\lefteqn{H_\varepsilon(x_+)  - H_\varepsilon(y)
= H_\varepsilon(x_+)  - H_\varepsilon(x_{-})+
H_\varepsilon(x_-)  - H_\varepsilon(y)}\qquad \nonumber \\
&=&\varepsilon^3 \lim_{(T_{1},T_{2})\to \infty} \Bigg[
 \int_{-T_{1}}^{ T_{2} }
\, dt \,
D_2 \tilde U\left( \gamma_{E}^{q}\left(t+\frac{\varphi_{0}}{\sqrt{\!E}}\right),
\varepsilon s+\varepsilon t \right) \nonumber \\
&& \hphantom{\varepsilon^3 \lim_{(T_{1},T_{2})\to \infty} \Bigg[}
- \int_{-T_{1}}^{ 0 } \, dt \,
 D_2 \tilde U\left( \Lambda_{E}^{q}\left(t+\frac{\varphi_{0}+a_{-}}
{\sqrt{\!E}}\right), \varepsilon s+\varepsilon t \right)
\nonumber \\
&& \hphantom{\varepsilon^3 \lim_{(T_{1},T_{2})\to \infty} \Bigg[}
-  \int_{0}^{ T_{2} } \, dt \,
 D_2 \tilde U\left( \Lambda
_{E}^{q}\left(t+\frac{\varphi_{0}+a_{+}}{\sqrt{\!E}}\right),
\varepsilon s+\varepsilon t \right)
\nonumber \\
&& \hphantom{\varepsilon^3 \lim_{(T_{1},T_{2})\to \infty} \Bigg[}
+ \tilde g(\varphi _{0}+a_{-},\varepsilon s;\varepsilon)
-\tilde g(\varphi _{0}+a_{+},\varepsilon s;\varepsilon) \Bigg]  \nonumber \\
&&+ \Or(\varepsilon^{5}). 
\label{melnikov2}
\end{eqnarray}
Now, calling
$A_{-}(t)= \tilde g(\varphi _{0}+a_{-}+\sqrt{\!E}t,
\varepsilon s+\varepsilon t;\varepsilon)$,
we have,
using the functional equation~(\ref{tildeg}) verified by  $\tilde g$:
\begin{eqnarray*}
\dot A_{-}(t)
&=&\sqrt{\!E}D_{1}\tilde g(
\varphi _{0}+a_{-} +\sqrt{\! E}t,
\varepsilon s+\varepsilon t;\varepsilon) \\
& &\mbox{} +\varepsilon D_{2}\tilde g(\varphi _{0}+ a_{-}+\sqrt{\! E}t,
\varepsilon s+\varepsilon t;\varepsilon)\\
&=&
D_{2}\tilde U(\Lambda _{1}(\sqrt{\!E}t+\varphi _{0}+a_{-}),
\varepsilon s+\varepsilon t) 
+ \Or(\varepsilon^{3})\\ 
&=&
D_{2}\tilde U\left(\Lambda _{E}\left(t+\frac{\varphi _{0}
+a_{-}}{\sqrt{\!E}}\right),\varepsilon s+\varepsilon t\right)
+ \Or(\varepsilon^{3}) 
\end{eqnarray*}
and a similar identity holds for  $ A_{+}(t)=
\tilde g(\varphi _{0}+a_{+}+\sqrt{\!E}t, 
\varepsilon s+\varepsilon t;\varepsilon) $, which verifies:
\begin{eqnarray*}
\dot A_{+}(t)
& =&
D_{2}\tilde U\left(\Lambda _{E}\left(t+\frac{\varphi _{0}
+a_{+}}{\sqrt{\!E}}\right),\varepsilon s+\varepsilon t\right)
+ \Or(\varepsilon^{3}) 
\end{eqnarray*}


Then, using the fundamental theorem of Calculus, we have for any $T$:
$$
A_{\pm}(T)-A_{\pm}(0) = \int_{0}^{T}
\, dt \, D_{2}\tilde U\left(\Lambda _{E}\left(t+\frac{\varphi _{0}
+a_{\pm}}{\sqrt{\!E}}\right),\varepsilon s+\varepsilon t\right)
+ \Or(\varepsilon^{3}) 
$$
and using these identities to express the second and third
integrals in~(\ref{melnikov2})
with $T_{1}$ and  $T_{2}$  we obtain formula~(\ref{melnikov}).\qed

\begin{rem}
The function provided by Lemma~\ref{lem:melnikov}:
\begin{eqnarray}\label{melnikovintegral}
M(\varphi _{0},\varepsilon s,E;\varepsilon ) 
&=&  \lim _{(T_{1},T_{2})  \to \infty }\Bigg[
\int_{-T_{1}}^{T_{2}}\ dt
D_{2}\tilde U\left(\gamma _{E}^{q}\left(t+\frac{\varphi _{0}
}{\sqrt{\!E}} \right) ,\varepsilon s+\varepsilon t\right) \nonumber \\
&&\hphantom{\lim _{(T_{1},T_{2})  \to \infty }\Bigg[}
-  \tilde g(\varphi _{0}+a_{+}+\sqrt{\!E}T_{2},
\varepsilon s+\varepsilon T_{2};\varepsilon) \label{fmelnikov} \\
&&\hphantom{\lim _{(T_{1},T_{2})  \to \infty }\Bigg[}
+ \tilde g(\varphi _{0}+a_{-}-\sqrt{\!E}T_{1},
\varepsilon s-\varepsilon T_{1};\varepsilon) \Bigg] \nonumber
\end{eqnarray}
is usually called the Melnikov function associated to the perturbed
torus.
As
\begin{equation}
H_{\varepsilon }(x_{+}) -H_{\varepsilon }(y) = \varepsilon
^{3}M(\varphi _{0},\varepsilon s,E;\varepsilon ) + \Or (\varepsilon
^{5})
\end{equation}
the non-degenerate zeros of this function correspond to points of the
torus which have a homoclinic orbit.
\end{rem}

\begin{rem}
Note that in  (\ref{melnikovintegral}) in general,  neither the integral nor
the other terms reach a limit as  $T_1,T_2$, but rather
oscillate quasiperiodically. Only their combination converges. 

The meaning of this phenomenon can be clearly ascertained when we 
realize that $\tilde g$ measures the fact that the invariant torus is 
displaced. If we are interested in the 
intersections of the  manifolds of perturbed torus, we
 need to consider the changes induced in the 
stable manifolds of the perturbed torus, not on 
the unperturbed one.

We warn the reader that in many places in the literature, this 
term is omitted. This omission is incorrect, unless
special circumstances (e.g. symmetries, that the perturbation 
vanishes on the torus, etc.) justify it.
\end{rem}


As a matter of fact, the zeros of the Melnikov function are critical
points of the Melnikov potential, see~\cite{DelshamsR97a}, also called
Poincar\'e function, defined by:
\begin{eqnarray}
\pot (\varphi _{0},\varepsilon s,E;\varepsilon )
&=& \lim _{(T_{1},T_{2}) \to \infty }\Bigg[
\int_{-T_{1}}^{T_{2}}\ dt \, \tilde U\left(\gamma
_{E}^{q}\left(t+\frac{\varphi _{0}
}{\sqrt{\!E}} \right) ,\varepsilon s+\varepsilon t\right) \nonumber \\
&& \hphantom{\lim _{(T_{1},T_{2}) \to \infty }\Bigg[}
-  \tilde h(\varphi _{0}+a_{+}+\sqrt{\!E}T_{2},\varepsilon
s+\varepsilon T_{2};\varepsilon) \label{pmelnikov}  \\
&& \hphantom{\lim _{(T_{1},T_{2}) \to \infty }\Bigg[}
+ \tilde h (\varphi _{0}+a_{-}-\sqrt{\!E}T_{1},
\varepsilon s-\varepsilon T_{1}; \varepsilon) \Bigg] \nonumber
\end{eqnarray}
where  $D_{2}\tilde h =\tilde g$  and $\tilde h$
verifies~(\ref{tildeh}).

The Poincar\'e function satisfies the following properties:

\begin{enumerate}

\item
$M(\varphi _{0},\varepsilon s,E;\varepsilon )= D_{2}
\pot (\varphi _{0},\varepsilon s,E;\varepsilon )$

Note that the uniform convergence of the difference of 
two integrals by (\ref{eq:asymptotics})
readily justifies the taking of derivatives term 
by term.

\item
$\pot (\varphi _{0},\varepsilon s,E;\varepsilon )$  is
$1/\varepsilon $-periodic in  $s$.

\item
For any  $u\in \real$ one has:
$$
\pot (\varphi _{0}+\sqrt{\!E}u,\varepsilon s+\varepsilon u,E;\varepsilon ) =
\pot (\varphi _{0},\varepsilon s,E;\varepsilon )
$$
and, taking  $u=-\varphi _{0}/\sqrt{\!E}$:
$$
\pot(\varphi _{0},\varepsilon s,E;\varepsilon ) = \pot (0,\varepsilon
(s-\varphi _{0}/\sqrt{\!E}),E;\varepsilon ),
$$
that is,  $\pot$ is a   $\sqrt{\!E}/\varepsilon $-periodic function of
$\varphi _{0}$.
\end{enumerate}

In the following Lemma we are going to give an approximation of the
Poincar\'e function
$\pot (\varphi _{0},\varepsilon s,E;\varepsilon )$.

\begin{lem}\label{lem:potmather}
\begin{equation}\label{potmather}
\pot (\varphi _{0},\varepsilon s ,E;\varepsilon ) 
= \frac{1}{\sqrt{\!E}} {\cal L}
\left( \varepsilon \left(s-\frac{\varphi _{0}}{\sqrt{\!E}}\right) \right)+
\Ord {2} (\varepsilon )
\end{equation}
where
\begin{equation}
{\cal L}(\tau ) = \lim_{(T_{1},T_{2})\to
\infty }\left[ \int_{-T_{1}}^{+T_{2}}\ dt \ \tilde U (\gamma
_{1}(t),\tau )
- \int_{-T_{1}+a_{-}}^{+T_{2}+a_{+}}\ dt \ \tilde U (\Lambda
_{1}(t),\tau )\right]
\end{equation}
\end{lem}

\proof
In order to obtain the first order terms in the Poincar\'e function
$\pot$ we write~(\ref{pmelnikov}) as
\begin{eqnarray*}
\pot (0 ,\tau,E;\varepsilon ) 
&=&  \lim _{(T_{1},T_{2}) \to \infty } \Bigg[
\int_{-T_{1}}^{T_{2}}\ dt \, \tilde U(\gamma
_{E}^{q}(t),\tau+\varepsilon t) \\
&&  \hphantom{\lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
- \tilde h(a_{+}+\sqrt{\!E}\,T_{2},\tau+\varepsilon T_{2};\varepsilon)
 + \tilde h(a_{+},\tau;\varepsilon)\\
&&  \hphantom{\lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
+ \tilde h (a_{-}-\sqrt{\!E}\,T_{1},\tau-\varepsilon T_{1};\varepsilon)
- \tilde h (a_{-},\tau;\varepsilon)\\
&&  \hphantom{\lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
- \tilde h(a_{+},\tau;\varepsilon)
 + \tilde h (a_{-}+\Delta ,\tau+\varepsilon \Delta /\sqrt{\!E};\varepsilon)\\
&&  \hphantom{\lim _{(T_{1},T_{2}) \to \infty } \Bigg[}
-  \tilde h (a_{-}+\Delta ,\tau+\varepsilon \Delta /\sqrt{\!E};\varepsilon)
 +\tilde h(a_{-},\tau;\varepsilon) \Bigg] .
\end{eqnarray*}

The fourth line in this expression is of order  $\varepsilon$ in the
${\cal C}^{1}$ norm   due to the
fact that  $\tilde h(\cdot, \cdot;\varepsilon)$
is a bounded function with bounded derivatives,
(see Lemma \ref{lem:torus})
and  $a_{-}+\Delta =a_{+}$.
In order to obtain integral expressions for the other three, we only
need to use the fundamental Theorem of Calculus and the functional
equation~(\ref{tildeh}) verified by  $\tilde h$.
Then we obtain:
\begin{eqnarray*}
\lefteqn { \pot (0,\tau,E;\varepsilon )} \\
&=& \lim_{(T_{1},T_{2})\to \infty} \Bigg[ 
\int_{-T_{1}}^{ 0 } \, dt \,
\tilde U( \gamma _{E}^{q}(t),\tau+\varepsilon t)-
\tilde U\left( \Lambda _{E}^{q}\left(t+\frac{a_{-}}{\sqrt{\!E}}\right), \tau
+\varepsilon t\right) \\
&& \hphantom{\lim_{(T_{1},T_{2})\to \infty} \Bigg[}
+  \int_{0}^{ T_{2} } \, dt \,
\tilde U( \gamma _{E}^{q}(t), \tau+\varepsilon t)
-\tilde U\left( \Lambda_{E}^{q}\left(t+\frac{a_{+}}{\sqrt{\!E}}\right),\tau
+\varepsilon t\right) \\
&& \hphantom{\lim_{(T_{1},T_{2})\to \infty} \Bigg[}
- \int_{0}^{\frac{\Delta }{\sqrt{\! E}}}
dt \, \tilde U\left(\Lambda^{q}_{E}\left(t+\frac{a_{-}}
{\sqrt{\!E}}\right),\tau +\varepsilon t\right)\Bigg] +\Or (\varepsilon ),
\end{eqnarray*}
now, using the rescaling properties~(\ref{rescaling}), and changing the
variable  $u=\sqrt{\!E}t$:
\begin{eqnarray*}
\lefteqn {\pot (0,\tau,E;\varepsilon )= } \\
&&\frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})\to \infty}\Bigg[
\int_{-T_{1}}^{ 0 } \, du \,
\tilde U\left( \gamma _{1}^{q}(u),\tau+
\frac{\varepsilon u}{\sqrt{\!E}}\right)
-\tilde U\left( \Lambda _{1}^{q}(u+a_{-}), \tau+
\frac{\varepsilon u}{\sqrt{\!E}}\right) \\
&&\hphantom{\frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})}\Bigg[}\;
+\int_{0}^{ T_{2} } \, du \,
\tilde U\left( \gamma _{1}^{q}(u), \tau+
\frac{\varepsilon u}{\sqrt{\!E}}\right)
-\tilde U\left( \Lambda _{1}^{q}(u+a_{+}), \tau+
\frac{\varepsilon u}{\sqrt{\!E}}\right)\nonumber \\
&&\hphantom{\frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})}\Bigg[}\;
- \int_{0}^{\Delta }
du \, \tilde U\left(\Lambda^{q}_{1}(u+a_{-}),\tau
+\frac{\varepsilon u}{\sqrt{\!E}}\right)\Bigg]
+\Or (\varepsilon ),\\
\end{eqnarray*}
and taking the dominant terms in  $\varepsilon $:
\begin{eqnarray*}
\lefteqn{\pot (0,\tau,E;\varepsilon )}\\ 
&=& \frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})\to \infty}\Bigg[
\int_{-T_{1} \sqrt{\!E}}^{ 0 } \, du \,
\tilde U( \gamma _{1}^{q}(u),\tau)
-\tilde U( \Lambda _{1}^{q}(u+a_{-}), \tau) \nonumber \\
&& \hphantom{\frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})\to \infty}\Bigg[}
+\int_{0}^{ T_{2}\sqrt{\!E} } \, du \,
\tilde U( \gamma _{1}^{q}(u), \tau )
-\tilde U( \Lambda _{1}^{q}(u+a_{+}), \tau)\nonumber \\
&& \hphantom{\frac{1}{\sqrt{\!E}} \lim_{(T_{1},T_{2})\to \infty}\Bigg[}
- \int_{0}^{\Delta }
du \, \tilde U(\Lambda^{q}_{1}(u+a_{-}),\tau)\Bigg] +
\frac{1}{\sqrt{\!E}} R (\tau
,\varepsilon ) +\Or (\varepsilon )\\
&=&\frac{1}{\sqrt{\!E}} {\cal L}(\tau )
+\frac{1}{\sqrt{\!E}}R(\tau ,\varepsilon )+ \Or (\varepsilon ).
\end{eqnarray*}

Where $R(\tau,\varepsilon)$ is defined so that the above is 
an identity. Note that it only involves  
the difference of integrals whose integrands
have second arguments that are slightly different.

One can bound  $R(\tau ,\varepsilon )$, using
the properties~(\ref{eq:asymptotics}) and the fact
that $\tilde U(q,r)$ is a periodic function with respect to its second
variable, as
$$
\abs{R(\tau ,\varepsilon )} \le K\varepsilon  \left( \int_{-\infty
}^{+\infty
}\ du \ e^{-\beta _{0}\abs{u } } + \int_{0}^{\Delta }\ du \right)  \le C
\varepsilon.
$$
Similarly, one can bound the first and second derivatives because one can take
derivatives under the integral sign (the convergence of the integrand
is exponentially fast) and then, similar cancelations than those used
above, establish the result.

Then taking  $\tau = \varepsilon (s-\varphi _{0}/\sqrt{\!E})$, we
have the Lemma.\qed

\begin{prop}
Given a metric that satisfies the genericity conditions of
Theorem~{\rm\ref{thm:morseflow}}, the set of periodic potentials for which
the function  ${\cal L}(\tau )$ in Lemma~{\ref{lem:potmather}} is
identically constant is a  ${\cal C}^{l}$ closed subspace of infinite
codimension for  $l\ge 0$.
\end{prop}

\proof

We note that, for every  $\tau $  $\tau'$, the mapping  
$U \mapsto {\cal L}(\tau) - {\cal L}(\tau')$
is a continuous linear functional  map if we give  $U$ the  ${\cal
C}^{l}$ topology,  $l\ge 0$. This functional is non trivial as can be
observed by noting that, since  $\Lambda $ and  $\gamma $ do not
coincide, it is possible to choose  potentials $U$ with support near
$\gamma $ so that the functional does not vanish. \qed


\subsection{Transition chains and transition lemmas} \label{subs:transition}

We recall that  according to 
\cite{Arnold64, ArnoldA68},
a transition chain
for a Hamiltonian flow is
a sequence of transition tori
such that the unstable manifold of
one intersects transversally the
stable manifold of the next.

The  definition of transition tori 
in \cite{Arnold64} is topological,
but for our purposes we note that
it has been shown in 
\cite{FontichM98}
(we will discuss it in
Lemma~\ref{lambdalemma})
that all whiskered tori with 
one dimensional  whiskers and 
with irrational motion are 
transition tori. This includes
the tori produced applying  
Theorem~\ref{thm:KAM}
to $f_\varepsilon^\varepsilon$ in our problem.

The importance of transition chains is that
there are orbits that follow them closely.




So,  we will verify that we 
can find a
sequence of tori  obtained by
applying 
Theorem~\ref{thm:KAM}
to $f_\varepsilon^\varepsilon$ 
and such that 
the stable manifold of one crosses
transversally the  unstable manifold of the previous one.
Then, we will discuss some small modifications 
needed to the standard arguments (they 
only apply to finite sequences) that show that 
indeed there are orbits that follow them.



We note that, 
in the notation that we have introduced in this paper,
the assertion that the unstable  manifold of 
a torus contained in $\Lambda$
intersects the unstable manifold of
another one is equivalent to the assertion 
that the image of the first torus 
under the outer map $S$ intersects the second.

We will refer to the tori obtained
applying  Theorem~\ref{thm:KAM} to 
$f_\varepsilon^\varepsilon$ simply as
KAM tori.


\begin{lem}\label{lem:nontrivialpot}
Assume that $r$ is sufficiently big. 
If the function  ${\cal L}(\tau )$
is not constant,  we can find
$K > 0$
such that for $\varepsilon$ sufficiently 
small, 
given a KAM torus $\Tau$, 
we can find  other KAM tori 
$\Taup$, $\Taum$  such that 
\begin{eqnarray*}
W^u_\Tau \itr W^s_\Taup, && 
W^u_\Tau \itr W^s_\Taum\\
\min H_\varepsilon( \Tau) &\ge& \max H_\varepsilon (\Taum) + K \varepsilon^3 \\
\max H_\varepsilon( \Tau) &\le& \min H_\varepsilon (\Taup) - K \varepsilon^3
\end{eqnarray*}
\end{lem}


\proof

Observe that, since ${\cal L}$ is periodic and $C^2$, 
if it is not constant, we can find 
two numbers $\tau_\pm$ such that 
${\cal L}'(\tau_+) > 0 $,
${\cal L}'(\tau_-) < 0 $,
${\cal L}''(\tau_\pm) \ne 0 $.
Since ${\cal L}$ is ${\cal C}^2$ the same inequalities 
are true for small intervals around
$\tau_\pm$.

We study the dynamics of the manifold $\Lambda$
using the coordinates $H_\varepsilon$, $\varphi$.


Since ${\cal L}$ approximates in the ${\cal C}^2$ sense
the Poincar\'e function, and
the derivative of this function 
measures the increase in $H_\varepsilon$ under the map $S$,
it follows that 
for small enough $\varepsilon$, given any KAM torus
$\Tau$, its image under $S$ has to include segments $\rho_\pm$
(corresponding to the intervals around
$\tau_\pm$  above) 
such that
$\max H_\varepsilon (\rho_\pm) -  \min H_\varepsilon (\rho_\pm) 
\ge K_1 \varepsilon^3$.
On the other hand, the projection of these intervals 
over the $\varphi$  variable  has a length not more 
that $K_2 \varepsilon$.

We note that in the averaged coordinates, the KAM tori 
are not more that $\varepsilon^{m/2 +1} $ apart and that they 
correspond very approximately to surfaces of constant action.
Hence,  in the original coordinates, they will be 
graphs of  functions in the $\varphi, H_\varepsilon$ 
coordinates
which are not more than $\varepsilon^{m/2 +1} $ apart
in the ${\cal C}^{l -4}$ sense.

Since the interpretation of the  function $M$ 
(See figure~\ref{pointy}) was the increment
in energy over a torus of the map $S$, 
we see that the image of one torus has 
to cross two KAM tori, one of  higher energy 
and another one of lower energy.

Moreover, this intersection has to be 
transversal. The fact that 
${\cal L}''( \tau_\pm) \ne 0$ implies 
that the derivative of the gain in energy
with respect to the angle is 
bounded from below by a constant times
$\varepsilon^3$. 
That is, if we express the torus $\Tau$,
$S(\Tau)$ and $\Taup,\Taum$ as 
graphs of functions $\Psi, \Psi_S, 
\Psi_\pm$ respectively, 
we have
$|\Psi_S' - \Psi_\pm'| \ge K \varepsilon^3 $
in a neighborhood of the intersection.
This implies that the intersection of the 
tori is transversal.



\begin{figure}
\centerline{\psfig{file=cross.eps,bbllx=60pt,bblly=400pt,bburx=320pt,bbury=180pt}}
%BoundingBox: 60 450 320 130
\vskip220pt
\caption{Illustration of the action of the map $S$ on a torus $\tau$.}
\label{crossing}
\end{figure}

As an immediate consequence, we have:


\begin{lem}\label{lem:symbolicdynamics}
Assume that the metric $g$ satisfies the assumptions of 
Theorem~\ref{thm:morseflow} and that $V$ is such that 
${\cal L} $ is not constant. Assume moreover that 
both $g$ and $V$ are ${\cal C}^{15}$.
Then, there exist
$M>0$,  $\alpha >0$, such that if
$$
I_{i} = [E_{-}^{i} , E_{+}^{i}]\quad i=1,\ldots
$$
is any sequence of intervals such that
\begin{eqnarray*}
E^{i}_{-} &\ge & M \\
(E_{+}^{i}-E_{-}^{i}) & \ge & M (E_{+}^{i})^{-\alpha }
\end{eqnarray*}
Then, we can find  a
sequence $\{\Tau_i\}$ of KAM tori such that 
\begin{eqnarray*}
&&H_\varepsilon(\Tau_{i}) \cap I_i \ne \emptyset \\
&& W^{s}_{{\cal T} _{i+1}} \itr W^{u}_{{\cal T} _{i}}.
\end{eqnarray*}
\end{lem}

Our next goal is to show that the pseudo orbits
obtained by interspeding the KAM homoclinic jumps 
with the motion along the torus can be 
shadowed by true orbits of 
the system. As it is usual in the literature 
for Arnol'd diffusion, the key step is to 
find an appropriate inclination 
lemma (also called sometimes $\lambda$-lemma).

In the literature, one can find very sharp 
inclination lemmas---including even some estimates 
of the times needed to do the shadowing---for 
analytic maps, when the rotation is Diophantine, 
in \cite{Marco96,Cresson97,Valdinoci98}.
(Related results appear in 
\cite{ChierchiaG94}).
The result that  we have found best adapted to our 
purposes is that of 
\cite{FontichM98} for whiskered tori 
with one dimensional strong (un)stable 
directions---as is the case 
in the problem we are considering---,  which works for ${\cal  C}^1$
maps and only requires that the torus 
has an irrational rotation.

A particular case of the results of 
\cite{FontichM98} is:


\begin{lem} \label{lambdalemma}
Let $f$ be a ${\cal C}^1$ symplectic  mapping 
in a $2(d+1)$ symplectic manifold. 
Assume that the map leaves invariant 
a ${\cal C}^1$ $d$-dimensional torus $\Tau$
and that the motion on the torus is 
an irrational rotation.
Let $\Gamma$ be a $d+1$ manifold 
intersecting $W^s_\Tau$ transversally.

Then
$$
W^{u}_{\Tau} \subset \overline { \bigcup_{i > 0} f^i (\Gamma) }
$$
\end{lem}

An immediate consequence of this is that any finite 
transition chain 
can be shadowed by a true  orbit. 
The argument for infinite chains requires some elementary 
point set topology.




\begin{lem}\label{lem:infinitechain}
Let  $\{ {\cal T} _{i} \}_{i=1}^{\infty }$ be a sequence of
transition tori.
Given  $\{ \varepsilon _{i} \}_{i=1}^{\infty }$ a sequence   of
strictly  positive numbers, we can find a point  $P$  and a
increasing sequence of numbers  $T_{i}$ such that
$$
\Phi _{T_{i}}(P) \in N_{\varepsilon _{i}} ({\cal T} _{i})
$$
where  $N_{\varepsilon _{i}}({\cal T} _{i})$ is a  neighborhood of size
$\varepsilon _{i}$ of the torus ${\cal T} _{i}$.
\end{lem}

\proof

Let  $x\in W^{s}_{{\cal T} _{1}}$.
We can find a closed ball  $B_{1}$, centered on  $x$, and such that
\begin{equation}\label{intersection}
\Phi _{T_{1}}(B_{1})\subset N_{\varepsilon _{1}}({\cal T} _{1}).
\end{equation}
By the  Inclination Lemma~\ref{lambdalemma}
$$
W^{s}_{{\cal T} _{2}}\cap B_{1}\ne \emptyset.
$$
Hence, we can find a closed ball  $B_{2}\subset B_{1}$, centered in a
point
in $W^{s}_{{\cal T} _{2}}$ such that, besides
satisfying~(\ref{intersection}):
$$
\Phi _{T_{2}}(B_{2})\subset N_{\varepsilon _{2}}({\cal T} _{2}).
$$

Proceeding by induction, we can find a sequence of closed balls
\begin{eqnarray*}
&& B_{i} \subset  B_{i-1} \subset \cdots \subset B_{1} \\
&& \Phi _{T_{j}}(B_{i}) \subset N_{\varepsilon _{j}}({\cal T} _{j}), \quad
i \le j .
\end{eqnarray*}

Since the balls are compact,  $\cap B_{i}\ne \emptyset$. A point  $P$ in
the intersection satisfies the required property.\qed

Putting together Lemma~\ref{lem:symbolicdynamics} 
and Lemma~\ref{lem:infinitechain}, 
we obtain the following result, 
which clearly implies 
Theorem~\ref{unboundedorbit}.


\begin{thm}
\label{thm:main}
Assume that the metric $g$ satisfies the assumptions of 
Theorem~\ref{thm:morseflow} and that $V$ is such that 
${\cal L} $ is not constant. Assume moreover that 
both $g$ and $V$ are ${\cal C}^{15}$.
Then, there exist
$M>0$,  $\alpha >0$, such that if
$$
I_{i} = [E_{-}^{i} , E_{+}^{i}]\quad i=1,\ldots
$$
is any sequence of intervals such that
\begin{eqnarray*}
E^{i}_{-} &\ge & M \\
(E_{+}^{i}-E_{-}^{i}) & \ge & M (E_{+}^{i})^{-\alpha }
\end{eqnarray*}

Then, we can find  an
orbit $p(t),q(t)$ of
the Hamiltonian flow and  an increasing sequence of times  $t_{1}<t_{2}<
\cdots <t_{n}<\cdots$, such that
$$
H_{\varepsilon }(p(t_{i}),q(t_{i}),t_{i}) \in I_{i} .
$$
and  $(p(t_i), q(t_i)) $ is in a 
neighborhood of size $M (E_{-}^{i})^{-2}$ of the 
periodic orbit $\Lambda$.
\end{thm}

\noindent{\bf Note:} By assuming more differentiability in the
hypothesis of the theorem, we can get  $\alpha $ to be arbitrarily large.

\begin{rem}
A question that has often been asked us and, which is indeed
quite relevant for physical applications, is what is the measure of
the diffusing orbits 
and what is the speed of diffusion predicted by this mechanism.


We do not know at the moment of this writing how to  produce 
a set of positive measure of diffusing orbits. Of 
course, the mechanism described here is presumably not the 
only mechanism that contributes to diffusion.

A heuristic argument---which at the moment we cannot even raise
as a conjecture---suggests
that the orbit following 
the mechanism studied in this paper can perform 
$ \approx  E^{1/2} $ homoclinic excursions in 
a unit of time and in each of them it can gain 
$ \approx E^{-3/2}$ energy. Hence, the energy can 
increment in a unit of time a quantity proportional to
the inverse if the energy.  Hence, we expect that our
orbits satisfy $E(t) \approx t^{1/2} $.

Note that this argument implicitly assumes that the 
proportion of times that are favorable for the 
jump  and indeed the average gain in energy per jump 
reach a limit as the energy grows and that the 
time that one needs to bid preparing for the next 
jump is a fixed proportion of the total time.


The  rigorous justification (and indeed a non-rigorous 
but reliable assessment)  of this assumptions  seems like 
a daunting task, but we hope some reader may be motivated 
to investigate this question.
\end{rem}

\subsection*{Acknowledgements}

We thank J.N.~Mather for communicating his results and for
encouragement.
This work has been partially supported by the NATO grant CRG950273.
Research by A.D. and T.M.S. is also supported by
EC grant ERBCHRXCT-940460, the Spanish grant
DGICYT PB94-0215 and the Catalan grant CIRIT 1996SGR--00105.
Research by R.L. is also supported by NSF grants.
We also thank TICAM, UPC  and IMA for invitations that made possible
these collaborations.

\appendix
\section{Appendix: Brief summary of  hyperbolicity theory}
\label{ap:hyperbolic}

In this section we collect some of the results from
the rich theory of  hyperbolic (or normally hyperbolic or
further qualifications)
invariant manifolds.

The results we present are quite standard and can be
found in many places, (indeed the theory seems 
to have been developed several times)  so we will
just highlight some of the
more subtle points of the conclusions
which affect some of the statements of the theorems
we will prove. 
We just recommend
\cite{Fenichel74,HirschP68,Wiggins94}
as readable and complete references.
Another version---somewhat more demanding in notation
and style---is in \cite{HirschPS77}. Yet another 
point of view can be found in \cite{SackerS74} and 
following papers.
We refer to
\cite{Wiggins94}
for a discussion of original references.

We discuss only three aspects: the regularity
properties of invariant manifolds and foliations,  their
persistence properties and the smooth dependence on parameters.

We will present here the theory for flows. Using the standard suspension
trick, all the general results for flows imply corresponding results for
invertible maps. There are aspects of the theory of hyperbolic non-invertible 
maps without flow
counterparts, but the aspects of the theory we are discussing are
identical for flows and for maps. The  theory of non-invertible maps is
still 
somewhat incomplete even in the aspects we discuss here.


\begin{defi}
\label{def:hyperbolic}
Let   $M$ be a manifold and  $\Phi _{t}$ a  ${\cal C}^{r}$,  $r\ge 1$,
flow on it.
We say that a manifold $\Lambda \subset  M$---possibly with
boundary---invariant under $\Phi _{t}$ is hyperbolic
when there is a  bundle decomposition
\[
TM = T\Lambda \oplus E^s \oplus E^u
\]
invariant under the flow,
and numbers $C > 0 $, $0 < \beta < \alpha $,   such that for  $x \in
\Lambda $
\begin{eqnarray}
\label{eqn:bounds}
\nonumber
v \in  E^s_x &\iff& |D\Phi_t(x) v |  \le   C e^{-\alpha t }|v|\quad
\forall \ t > 0 \\
v \in  E^u_x &\iff& |D\Phi_t(x) v |  \le   C e^{ \alpha t }|v| \ \quad
\forall \ t < 0 \\ \nonumber
v \in  T_{x}\Lambda &\iff& |D\Phi_t(x) v |  \le   C e^{ \beta |t| }|v| \quad
\forall \ t
\end{eqnarray}
\end{defi}

\begin{rem}
In this paper, we will refer to (\ref{eqn:bounds}) as saying that the
manifold is ``hyperbolic". In some references where more precision is
needed, names as  $\alpha -\beta $ hyperbolic or normally hyperbolic are
used.

The  hypothesis (\ref{eqn:bounds}) are often referred
as saying that the bundle decomposition satisfies
exponential dichotomies.
\end{rem}

\begin{rem}
There are two different
ways of developing hyperbolicity theory. One is,
as we stated, to
assume  that the constants in
(\ref{eqn:bounds}) are uniform in the bundle.
Another one is to assume bounds such as those in
(\ref{eqn:bounds}) along an orbit and that the
ratios along several constants along the orbit
are bounded.
The first method is the basis of
\cite{HirschP68} and \cite{HirschPS77}.
The second one was used in \cite{Fenichel74,Fenichel77}.

Clearly, the hypothesis of the bundle approach imply those
of the orbit method. The difference in the bounds can be particularly
significative in systems in which a geometric structure implies
relations between expansion and contraction rates along an orbit but not
on a bundle.
One example of this situation is the study of the horospheric
foliation in geodesic flows in manifolds of negative curvature
(\cite{HurderK90}).
Moreover, the study of individual orbits leads naturally to the
non-uniform hyperbolic theory  \cite{Pesin76,Pesin77}.

For the applications we have in mind, we do
not need the sharper results, so that we
will state the results in the somewhat simpler
language of bundles.
\end{rem}

\begin{rem}
Note that if the  inequalities
(\ref{eqn:bounds}) are established for
$|t | \le T$ with $T$ sufficiently large to overcome the constant $C$
(i.e. $C e ^{\beta-\alpha T }< 1 $),
then we can recover the definition we have given
because we can bound 
$|| D \Phi_{n T + s } || \le || D \Phi_T||^n  \, \cdot \, || D \Phi_s||$.

This  observation is useful when we want to study the persistence 
of these structures for sufficiently small perturbations.
\end{rem}

\begin{rem}
Similarly, we note that, by redefining the metric in  $M$, one can get
rid of the constant $C$ in the definition.  A metric
satisfying this is called the adapted metric or
sometimes, specially in the East, Lyapunov metric. We refer to 
the general references above.
\end{rem}

\begin{rem}
\label{rem:perturbedbundle}
If we construct an splitting between bundles in such
a way that the bundles are not assumed to be invariant
but that they satisfy the inequalities in
(\ref{eqn:bounds}) for $|t| \le T$,
with $T$ large enough with respect to  $C$,  $\alpha $ and  $\beta $,
then one can construct invariant bundles
that satisfy similar inequalities with slightly worse constants.
\end{rem}

Intuitively, Definition~\ref{def:hyperbolic}
means that the normal infinitesimal perturbations
grow faster (either in the future or in the past) than the infinitesimal
perturbations along the manifold.

The first result is about the existence of invariant stable and unstable
manifolds for  hyperbolic manifolds.

\begin{thm}
\label{thm:regularity}
Let $\Lambda$ be a compact hyperbolic manifold (possibly with boundary)
for the ${\cal C}^{r}$ flow $\Phi_t$,
satisfying Definition~\ref{def:hyperbolic}.
Then, there exists a sufficiently small
neighborhood $U$, and a sufficiently small  $\delta >0$, such that:
\begin{enumerate}
\item
The manifold $\Lambda$ is ${\cal C}^{\min (r,r_{1}-\delta )}$,
where  $r_{1}= \alpha /\beta $.
\item
For any $x$ in $\Lambda$, the set
\begin{eqnarray*}
W^s_x &=&
\{ y  \in U : \dist( \Phi_t(y), \Phi_t(x) )
\le C e^{(-\alpha + \delta) t} \ \ {\rm for } \ \ t  > 0  \}\\
&=&
\{ y  \in U : \dist( \Phi_t(y), \Phi_t(x) )
\le C e^{(-\beta - \delta) t} \ \ {\rm for } \ \  t  > 0  \}
\end{eqnarray*}
is a ${\cal C}^{r} $ manifold and $T_x W^s_x  = E^s_x$.
\item
The bundles  $E_{x}^{s}$ are  ${\cal C}^{\min (r,r_{0}-\delta )}$ in
$x$, where $ r_0 = (\alpha  - \beta)/\beta $, and
\begin{eqnarray*}
W^s_\Lambda &=&
\{ y \in U  : \dist( \Phi_t(y), \Lambda )
\le C e^{(-\alpha + \delta) t} \ \ {\rm for } \ \ t  > 0  \}\\
&=&
\{ y  \in U : \dist( \Phi_t(y), \Lambda ) )
\le C e^{(-\beta - \delta) t} \ \ {\rm for } \ \ t  > 0  \}
\end{eqnarray*}
is a  ${\cal C}^{\min (r,r_{0}-\delta )}$ manifold. Moreover
$T_x W^s_\Lambda  = E^s_x$. Finally
$$
W_{\Lambda}^{s} = \cup_{x \in \Lambda } W^s_x .
$$
Moreover, we can find a $\rho > 0 $ sufficiently small
and a ${\cal C}^{\min (r,r_{0}-\delta )}$   diffeomorphism from  the
bundle of balls of radius $\rho$ in $E_{\Lambda}^{s}$  to
$W^s_\Lambda \cap U$.
\end{enumerate}
\end{thm}
An analogous theorem can be stated for  $W^{u}_{\Lambda }$
considering the flow generated by $-X$.

The last part of the conclusions state, roughly, that all
the orbits that approach fast enough the manifold $\Lambda$,
approach an orbit in $\Lambda$. Moreover,  for
the points approaching $\Lambda$ fast
enough and  in a sufficiently small
neighborhood of $\Lambda$, the point whose orbit is approached
is a well defined function in $W^s_\Lambda$ and is
${\cal C}^{\min (r,r_{0}-\delta )}$.

We point that compactness enters only mildly in the assumptions.
We only need that the  flow is uniformly ${\cal C}^r$ in a
neighborhood of $\Lambda$.
\begin{rem}
When  $\beta =0$,  $r_{0}$ and  $r_{1}$ have zero denominator. This
cannot be interpreted as  $\infty $ without care. Even if  $r=\infty
,\omega $, we cannot conclude that  $\min(r,r_{0})=\infty $ and that the
manifolds are  ${\cal C}^{\infty }$ or  ${\cal C}^{\omega }$. The best
that can be said is that there are  ${\cal C}^{k}$ manifolds for every
$k$. There are examples where the  ${\cal C}^{\infty }$ conclusions are
false even for polynomial perturbations.
\end{rem}

\begin{rem}
\label{rem:notmore}
We emphasize that, even if the manifolds $W ^s_x$ are as
smooth as the flow, the dependence on $x$  is not
claimed to be smoother than $r_0$,
which depends on
the contraction factors in the tangent and
(un)stable  bundles. Indeed, it is sometimes
the case that these bounds are sharp in
${\cal C}^{r_0}$ open sets. Similarly, the regularity of
the manifold $\Lambda$ and that of $W^s_\Lambda$
can be sharp even if the flow is assumed to be
analytic. An example for maps can be obtained setting
$$
f : \torus ^{2}\times \real  \mapsto  \torus ^{2}\times \real
$$
given by
$$
f(x,y)=\left( \left(\begin{array}{cc}2 & 1 \\ 1 & 1\end{array}\right)x,
h(x)+\frac{1}{50}y \right).
$$
with $h: \torus^2 \mapsto \real$ a conveniently chosen 
trigonometric polynomial.
Using the standard suspension trick, similar 
examples can be obtained for flows. More examples in 
this line and a more detailed analysis can be found in 
\cite{Llave92}.
\end{rem}

\begin{rem}
Even if the above examples show that the 
regularity numbers $r_0, r_1$ cannot be improved in general,
it is possible to obtain sharper results if 
we introduce more parameters  to characterize 
the exponential rates of the different 
bundles.  Here we have used only $\alpha $ and 
$\beta$, but one obtains sharper results if 
one introduces different parameters for 
the contraction rate of the unstable bundle in the 
past and the stable bundle in the future.
\end{rem}



The above theorem has as a corollary the smooth
dependence on parameters of the (un)stable manifolds.
The trick is completely elementary and  will be used
later several times.

\begin{cor}
\label{cor:dependencemanifold}
Assume that $\Phi_{t,\varepsilon}$ is a family of flows which is
jointly ${\cal C}^r$ in all its variables (the base point $x$, the time
$t$ and the parameter $\varepsilon$) and that for all the values of
the parameter in a ball
$\Phi_{t,\varepsilon}$ leaves invariant the manifold $\Lambda$.
Then, for sufficiently small $|\varepsilon|$, it is  possible
to apply the Theorem~\ref{thm:regularity}.
Moreover, the manifolds $W_{\Lambda,\varepsilon}^s$,
$W_{x,\varepsilon}^s$  are ${\cal C}^{\min(r,r_0 - \delta)}$ jointly in
$x$ and  $\varepsilon $.
\end{cor}

The idea of the proof is very simple. We just consider
the extended flow
$\tilde \Phi_t (x,\varepsilon ) = ( \Phi_{t,\varepsilon}(x), \varepsilon)$,
on $M \times B$
with $B$ a sufficiently small ball in $\varepsilon$.
It is easy to check that the manifold $\Lambda \times B$ is
invariant  for the flow and that, for  a finite time
the flow satisfies the exponential dichotomy bounds in the
stable and unstable subspaces.
Using  Remark~\ref{rem:perturbedbundle}
we conclude that there are invariant bundles with very close
constants.
A moment's reflection shows that the dependence of  manifolds of
the extended system on the base point gives the dependence on
parameters and the base point in the original system.\qed

\begin{rem}
We note that the dependence on parameters cannot be more
differentiable than the dependence on the base point and,
indeed, the examples alluded to in
Remark~\ref{rem:notmore} can be easily made into examples in
which the dependence  with respect to parameters
is optimal. Take, for example
$\Phi_{t,\varepsilon}(x )  = \Phi_t( x - \varepsilon v ) + \varepsilon v $
so that the invariant objects of the
$\Phi_{t,\varepsilon}(x ) $ are
just translates  by $\varepsilon v $ of the invariant objects
for $\Phi_t$ and, therefore, the dependence on
parameters is the same as the  space dependence in
the original problem.

This is in sharp contrast with the results of the usual implicit
function theorem so that the formulations of these problems in
terms of implicit function theorems need to involve
specialized implicit function theorems that  do not have
the same properties as the usual one.
\end{rem}

Now, we continue to discuss persistence.
Roughly, we state that any perturbation of a system
admitting a hyperbolic manifold has 
to carry another hyperbolic  invariant manifold
which is a perturbation of that of the original 
system.

\begin{thm}
\label{thm:persistence1}
Let $\Lambda \subset M $---not necessarily compact---be hyperbolic
for  the flow $\Phi_t$ generated
by the vector field $X$, which is uniformly ${\cal C}^r$ in
a neighborhood  $U$ of $\Lambda$ such that
$\dist(M \setminus U ,\Lambda) > 0$. 
Let $\Psi_t$ be   the flow generated by another
vector field $Y$ which is ${\cal C}^r$ and sufficiently ${\cal C}^1$ close to
$X$. Then, we can find a manifold $\Gamma$  which is
hyperbolic for $Y$ and ${\cal C}^{\min(r,r_1 -\delta )}$ close to $\Lambda$.

The constants in 
Definition~\ref{def:hyperbolic}
for $\Gamma$ are arbitrarily close
to those of $\Lambda$ if $Y$  is sufficiently
${\cal C}^1$ close to $X$.

The manifold $\Gamma$ is the only ${\cal C}^1$ manifold,
${\cal C}^0 $ close to $\Lambda$ and invariant under the flow of
$Y$.
\end{thm}

There are several extensions of this result that can be
readily obtained.  We will just  sketch the method
of proof and refer to the sources mentioned above.

\begin{enumerate}
\item \label{obs:parameters}
Similarly to Corollary~\ref{cor:dependencemanifold}, one can obtain
smooth dependence on parameters in
Theorem~\ref{thm:persistence1} by   extending the system by another one
with trivial dynamics. Again, we obtain only $\min(r,r_{1}-\delta )$
regularity and this is optimal in examples.


A convenient way of formulating this smooth dependence on
parameters is using the implicit function theorem
and  finding a ${\cal C}^{\min(r,r_{1} - \delta})$ mapping
${\cal F}: \Lambda \times B\to M$ in such a way
that ${\cal F} (\Lambda, \varepsilon)  = \Lambda_\varepsilon$, and
${\cal F}( \cdot,0)$ is the identity.

\item \label{obs:onemanifold}
Using the remark  above,
given a family of flows, we can  use the map
${\cal F}$  to identify all the invariant manifolds
of all the flows.
Extending the mapping ${\cal F}$ to a neighborhood
and changing coordinates to it, we obtain
that we can reduce the study of a family of
flows to the problem of a family of flows which
 preserve a common manifold.
This is precisely the case considered in
Corollary~\ref{cor:dependencemanifold}.

Hence, we can obtain that there is a ${\cal C}^{\min(r,r_{1}-\delta )}$ mapping
$R: W^s_\Lambda \times B \to M$
in such a way that
$R( W^s_\Lambda, \varepsilon) =
W^s_{\Lambda_{\varepsilon }, \varepsilon}$,
$R(\cdot, \varepsilon )|_{\Lambda} = {\cal F}(.,\varepsilon )$,
$R(W^s_x,\varepsilon) = W^s_{{\cal F}(x,\varepsilon ),\varepsilon}$.

\item \label{obs:boundary}
It is also possible to discuss persistence of manifolds with boundary
and  locally invariant manifolds.

The idea is that we can extend the flow  to a globally defined
one, with ${\cal C}^r$ bounds which are close to the ones of
the original problem and with bounds on the bundles which are
also close to the ones we assumed and which agrees with our
original flow in the points of the original manifold
which are sufficiently far  from the boundary.

Then, we can apply Theorem~\ref{thm:persistence1} to the extended
system. The invariant manifold for the extended system
will be locally invariant for the original one.

\item \label{obs:uniqueness}
Even if
Theorem~\ref{thm:persistence1}  includes uniqueness in its
conclusions and, therefore the manifold produced is unique
(under appropriate conditions)
for the extended system, the  extension process is not
unique and  the manifold produced does depend on the
extension used. Hence, one cannot claim uniqueness
for the locally invariant manifold produced for the original
system.

On the other hand, it follows from the uniqueness conclusions of
Theorem~\ref{thm:regularity}, that
all the orbits that remain in a sufficiently small   neighborhood of
$\Lambda$ and away from the boundary should be present in
all the extensions that do not modify the  vector field
away from this neighborhood of the boundary.

Similarly, note that the  definition of
stable manifold   of a point $x$  or
of  a manifold $\Lambda$ involves  discussing what happens
for arbitrarily large times of the time in the evolution.
Such long time orbits depend on the extension used if the orbit
of $x$ is not contained in the manifold $\Lambda$ away from
the boundary.

On the other hand, for the orbits that indeed remain inside
of $\Lambda$, the definition identifies  the points of
the stable manifold. Hence, the germs of these
stable manifolds have to agree in all the extensions.

\item
The above extension process can be combined with the dependence on
parameters. We just remark  that  given a family of
perturbations, one can perform the extension in such a
way that it depends smoothly on parameters.
(The extension only involves elements such as cut-off functions,
mappings to identify spaces etc., that can be used  for all
the values of the parameter.)

Again, the extension process is not unique
and the smooth dependence on parameters should be interpreted as
the possibility of finding a map ${\cal F}$ or $R$ so
that its range produces the invariant manifolds.

As before, we note  that the orbits that are contained in  a small
neighborhood  of $\Lambda$ away from the boundaries and
the germs of  their stable and unstable manifolds should be
present in all the extended systems.
\end{enumerate}

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