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inegrable billiard tables, inverse spectral problem
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\title{ Liouville billiard tables and  an inverse spectral result}  
\author{G.Popov and P.Topalov\footnote{partially supported by  
MESC Grants no. MM-810/98 and MM-1003/00}}  
  
\date{}  
\maketitle  
\thispagestyle{empty}  
  
\begin{abstract}   
\noindent   
We consider a class of billiard tables $(X,g)$, where $X$ is a smooth  
compact manifold of dimension 2 with smooth boundary $\partial X$   
and $g$ is a  
smooth  Riemannian metric on $X$,  the billiard  
flow of which is completely integrable.   
The billiard table  $(X,g)$ is  defined by means of a special double   
cover with two branched points and it admits a group of isometries  
$G \cong {\bf Z}_2 \times{\bf Z}_2$. Its boundary can be  
characterized by the string property, namely, the sum of distances  
from any point of $\partial X$ to the branched points is constant.   
 We provide  examples of such  
billiard tables in the plane (elliptical regions), on the sphere ${\bf  
S}^2$, on the hyperbolic space ${\bf H}^2$, and on quadrics.   
The main result is that the spectrum of the corresponding   
Laplace-Beltrami operator with Robin boundary conditions involving   
a smooth function $K$ on $\partial X$  determines uniquely the function  
$K$ provided that $K$  is   
invariant under the action of $G$ .    
  
\vspace{0.3cm}  
\noindent  
Subject classification: 58J50, 70H06   
\end{abstract}   
  
\section{Introduction}  
\setcounter{equation}{0}  
  This paper is concerned with  a class of smooth compact Riemannian  
manifolds of dimension two  with smooth  boundaries,   
 the billiard flows of which are  
completely integrable. We call them {\it Liouville billiard  
tables  of classical type}. For such  billiard  
tables  we prove that the   
following  inverse spectral result is true:  
  
Let $(X,g)$ be a closed Riemannian manifolds,   
${\rm dim}\, X = 2$,  with a $C^\infty$ boundary   
$\Gamma := \partial X\ne\emptyset$.    
Let $\Delta$ be the ``positive'' Laplace-Beltrami operator on $(X,g)$.   
Given  a real-valued function $K\in C^{\infty}(\Gamma)$,   
we  consider the  operator $\Delta$  with a domain of  
definition    
\[  
\{u\in H^2(X):\, \frac{\partial u}{\partial n}|_\Gamma = K  
u|_\Gamma\}\, ,  
\]   
where $n(x)$, $x\in \Gamma$, is the inward unit normal to  
$\Gamma$ with respect to the metric $g$. We denote this operator by  
$\Delta_K$.   
This is a selfadjoint operator in $L^2(X)$ with  
discrete spectrum   
\[  
{\rm Spec}\,  \Delta_K:=\{ \lambda_1 \le \lambda_2 \le \cdots  
\}\, ,  
\]  
where each eigenvalue $\lambda=\lambda_j$  is repeated according to its  
multiplicity, and it solves the spectral problem  
\begin{equation}  
\left\{  
\begin{array}{rcll}  
\Delta u\ &=& \ \la u \, \quad \mbox{in}\ $X$\, , \\  
\frac{\partial u}{\partial n}|_\Gamma \  &=& \ Ku|_\Gamma \, .  
\end{array}  
\right.  
                                         \label{thespectrum}  
\end{equation}  
The manifolds we define admit a group of isometries $G$ isomorphic to  
${\bf Z}_2\times{\bf Z}_2$.  We denote by $\mbox{Symm}_G\, (\Gamma)$  
the set of all real-valued functions $K\in C^{\infty}(\Gamma)$ which  
are invariant  
under the induced action of $G$ on $\Gamma$. Consider the map   
\begin{equation}  
\mbox{Symm}_G\, (\Gamma)\ \longrightarrow \ {\bf R}^{\bf N}\, ,  
                                        \label{themap}  
\end{equation}  
assigning to each $K\in\mbox{Symm}_G\, (\Gamma)$ the spectrum    
${\rm Spec}\,  (\Delta_K)$ of the boundary value problem (\ref{thespectrum}).   
Guillemin and Melrose \cite{GM} have proved that for elliptical  
regions in ${\bf R}^2$ ($\Gamma$ is ellipse)   
the map (\ref{themap}) is one-to-one  
(injective). We generalize their result for   
Liouville billiard tables  of classical type (see Definition  
\ref{lb-class}).   
  
Our main result is:   
\begin{Th}\hspace{-2.mm}.  
Let $(X,g)$ be a Liouville billiard table of classical type. Then the  
map (\ref{themap}) is one-to-one (injective).  
                                                 \label{MainTh}   
\end{Th}  
By billiard table  we mean a compact connected Riemannian  
manifold $(X,g)$ of dimension 2 with smooth boundary. The  
corresponding dynamical system is the billiard flow.   
Denote by $H:T^\ast X  \rightarrow {\bf R}$   
the Hamiltonian  corresponding to the metric $g$ via the  
Legendre transformation.   
It will be  shown that   Hamiltonian  systems  corresponding to   
Liouville billiard tables are integrable. Hereafter   
integrable means that there exists a real-valued function $I\in  
C^\infty(T^\ast  X \setminus 0)$     
which is constant on each (broken)  
bicharacteristic of $H$, and such that the differentials    
$dH$ and $dI$ are linearly  
independent at almost any  $\varrho \in T^\ast X\setminus 0$.  A (broken)  
bicharacteristic  $\gamma$ of $H$ is a map   
\[  
\gamma:\ [0,T)\setminus P \ \longrightarrow \ T^\ast \displaystyle  
\mathop X^\circ \setminus  
0  ,\   
\displaystyle \mathop X ^\circ = X\setminus \Gamma\, ,   
\]   
where $P$ is either a finite  
subset $0 < t_1 < t_2 <\cdots < t_N < T$   of $[0,T)$ or the empty  
set, such that on each connected component of $[0,T)\setminus P$ the  
curve   
$\gamma(t) = (x(t),\xi(t))$ is an integral curve of the Hamiltonian vector  
field $X_H$ of $H$ in  $T^\ast \displaystyle \mathop X ^\circ  \setminus 0$  
and for $0<t \in P$ we have $x(t) = x(t-0)=x(t+0)\in \Gamma$ and  
\[  
\xi(t -0)_{|T_{x(t)}\Gamma} =    
\xi(t + 0)_{|T_{x(t)}\Gamma}\quad , \quad   
\langle \xi(t + 0), n(x(t))\rangle > 0\, .  
\]  
Recall from \cite{Hor}, Sect. 24.3 , that for each   
$\varrho \in \displaystyle T^\ast\mathop X ^\circ  \setminus 0$  
there exists at least one generalized bicharacteristic $\gamma$ starting from   
$\varrho$ such that any compact arc of $\gamma$   
can be approximated uniformly by (broken) bicharacteristics.   
Hence, $I$ is constant on any such generalized bicharacteristic.    
In other words, $I$ is invariant under the ``generalized billiard  
flow''. Note that in the analytic case for each   
$\displaystyle \varrho \in T^\ast \mathop X ^\circ  
\setminus 0$ there is only one generalized bicharacteristic issuing from  
$\varrho$.   
  
  
Elliptical regions in ${\bf R}^2$ give a particular case  
of integrable billiard tables. We shall  give several other examples  
on   the sphere $ {\bf S}^2$, the hyperbolic space ${\bf H}^2$,    
and on quadrics.   
There is a  common property for  all of them,   
namely, the existence of a special double cover with   
two branched points (see Proposition  \ref{FactorizationLemma}).  
It is more or less known that the corresponding    
billiard flows are integrable.   The novelty here is that we propose  a  
general construction  for such billiard tables and    
we give explicit  formulae for the  covering maps.   
We show  also that   
the  Liouville billiard tables in the examples are of classical type   
(Definition \ref{lb-class}). In particular, we obtain that    
Theorem 1 holds for all of them.   
  
We prove that the  boundary  
$\Gamma$ of any  Liouville billiard table  has the string   
property, namely, the sum of distances  
from any point of $\Gamma$ to the branched points is constant.   
In particular elliptical regions are the only Liouville billiard  
tables in ${\bf R}^2$. In addition, a class of the  billiard   
tables we define satisfies the so-called {\em strong evolution property}  
(see Proposition \ref{StrongEvolutionProperty}).   
  
To prove Theorem 1 we use a  result of Guillemin and Melrose in   
\cite{GM} and  \cite{GM1}  concerning   
the singularities of the distribution  
\begin{equation}  
Z_K(t)=\displaystyle \sum_{\lambda \in {\rm Spec}\,  (\Delta_K)}\, \cos  
(t\sqrt \lambda) = {\rm tr}\, (\cos(t\sqrt {\Delta_K})) \, ,   
\quad t>0 \, .  
                                           \label{trace}  
\end{equation}  
It is well known  that the singular support of $Z_K(t),\ t>0$, is contained  
in the length spectrum ${\cal L}(X,g)$ of the corresponding billiard  
table. Recall that  ${\cal L}(X,g)$ consists of the lengths of all  
closed generalized geodesics of $(X,g)$.   
Suppose now  that there is an  ``invariant circle'' $S$   
of the billiard ball map $B$ in the co-ball bundle $B^\ast \Gamma$   
 such that the rotation  
number of the map  $B:S \rightarrow  S$ is rational. Then the  
generalized geodesics issuing from $S$ are all closed and we denote by  
$\ell$ the corresponding common minimal length. Assume that $S$ is a  
``clean''  submanifold of $B^\ast \Gamma$ and that there are no other  
closed geodesics with the same length $\ell$. Then  the integral   
\begin{equation}  
M \ =\ \int\limits_{S}\frac{K}{\cos\phi}d\mu\, .   
                          \label{sp-invariant}  
\end{equation}  
can be recovered from the leading term of  
the asymptotic expansion of $\sigma(t) = Z_K(t) - Z_0(t)$   
at $t=\ell$, where $Z_0$ is the trace (\ref{trace}) corresponding to   
the Neumann problem   (see \cite{GM}, Theorem 4.2.).    
Here  $\phi$ is the angle between the initial vector of the  
corresponding  geodesic issuing form $S$   
and the inward normal to $\Gamma$   
at the initial point of the geodesic. The measure  
$\mu$ on $S$ coincides (up to multiplication with a constant) with  
the Leray form.   
  
To prove Theorem 1 we look for an infinite sequence of such   
``invariant circles'' $S_j$ approaching the glancing manifold as   
$j\to \infty$ (the   
boundary $\Gamma$ is strictly geodesically convex).  The main  
difficulty is to find $S_j$ so that the corresponding  lengths $\ell_j$ are  
all ``simple'' in the length spectrum of $(X,g)$,   
which means that if $\gamma$ is a  
closed generalized geodesic of length $\ell_j$ then $\gamma$ is a  
broken geodesic issuing from $S_j$.   
To do this  we use essentially the properties of  
the corresponding billiard ball map  (see  
Sect. 4). In this way, we recover the  integral  
invariants $M_j$ on $S_j$ in (\ref{sp-invariant})   
from the spectrum of $\Delta_K$.   
Moreover, we obtain a  simple formula for   
$M_j$  in terms of the functions $f$ and $q$   
defining the Liouville billiard table (see  (\ref{invariant})).   
This allows us to recover $K$ from the sequence $M_j$.   
  
An interesting and difficult question is if  similar   
results are valid for dimensions   
$\ge 3$. A construction for integrable billiard tables close to the  
one  in this paper could be done for dimensions $\ge 3$. The corresponding  
results will be published elsewhere.     
  
The paper is organized as follows: In Sect. 2 we give a general  
construction of Liouville billiard tables. In Sect. 3 we consider  
several examples on ${\bf S}^2$,  ${\bf H}^2$ and on quadrics. In Sect. 4  
we investigate the corresponding billiard ball map. The proof of the  
main theorem is given in Sect. 5.   
  
  
  
  
  
  
  
  
  
  
  
  
  
\section{Liouville billiard tables}\label{LB-section}  
\setcounter{equation}{0}  
In the present section we define a class of  
completely integrable billiard tables of dimension $2$.  
The construction we propose is influenced from the  
classification theorems of  {\it Liouville surfaces} given in  
\cite{Kiyo1,Kiyo2,Kolokol,B-F,M-T}  
and the classical examples of integrable billiards described  
in Section \ref{Examples} (see  
\cite{Birkhoff, Poritsky, Bolotin, Drizzi, Veselov, C-S, Tabach}).   
By definition, a Liouville surface is a  complete $2$-dimensional  
Riemannian manifold without boundary the geodesic flow of which  admits   
a quadratic in velocities integral functionally independent of the  
energy integral. The idea to use  special covers first appears in   
\cite{Whittaker,Kolokol} (see Section \ref{Examples} and \cite{B-F} for  
a complete list of references).  
Anyway, to our best knowledge,  
similar construction of integrable billiard tables has not been 
documented in the literature.   
  
We consider two functions $f\in C^{\infty}({\bf R})$, $f(x+1)=f(x)$,  
and $q\in C^{\infty}([-N,N])$, $N>0$, such that  
\begin{itemize}  
\item[(H$_1$)] $f$ is  even, $f>0$ if $x\notin  {1\over 2}{\bf Z}$, and        
$f(0)=f(1/2)=0$;  
\item[(H$_2$)] $q$ is even, $q<0$ if $y\ne 0$,  $q(0)=0$ and $q^{''}(0)<0$;  
\item[(H$_3$)] $f^{(2k)}(l/2)=(-1)^kq^{(2k)}(0)$,  $l=0,1$,    
for every natural $k\in{\bf N}$.  
\end{itemize}  
  
In particular,  
if  $f\sim \sum_{k=1}^{\infty}\ f_kx^{2k}$ is the Taylor expansion of $f$  
at $0$,  then,   by  (H$_3$),  the Taylor expansion of $q$ at $0$  
is   $q\sim \sum_{k=1}^{\infty}\ (-1)^k f_kx^{2k}$.  
  
Consider the quadratic forms  
\begin{eqnarray}   
dg^2&=&(f(x)-q(y))(dx^2+dy^2)\label{themetric}\\  
dI^2&=&(f(x)-q(y))(q(y)dx^2+f(x)dy^2)\label{theintegral}  
\end{eqnarray}  
defined on the cylinder $C= {\bf T}^1\times[-N,N]$,   
${\bf T}^1\eqdef{\bf R}/{\bf Z}$.   
  
The involution $\sigma : (x,y)\mapsto(-x,-y)$ induces an involution  
of the cylinder $C$, that will be denoted by   
$\sigma$ as well. We  identify the points $m$ and $\sigma(m)$  
on the cylinder and denote by $\C\eqdef C/\sigma$ the topological quotient  
space. Let $\pi : C\to\C$ be the corresponding projection.  
A point $x\in C$   
is called {\em singular} if $\pi^{-1}(\pi(x))=x$, otherwise it is   
a {\em regular} point of $\pi$. Obviously, the  singular points are   
$F_1=\pi(0,0)$ and $F_1=\pi(1/2,0)$.   
Denote by  ${\bf D}^2$  the unit disk $\{x_1^2+x_2^2\le 1\}$ in ${\bf R}^2$.  
\begin{Prop}\hspace{-2.mm}.\label{FactorizationLemma}  
Suppose that $f$ and $g$ satisfy {\em (H$_1$)$\div$(H$_3$)}.   
Then the quotient space $\C$ is homeomorphic to  
the unit disk ${\bf D}^2$.  
There exist an unique differential structure on $\C$ such that the  
projection $\pi: C\to\C$ is a smooth map, $\pi$ is a local diffeomorphism  
in the regular points,  and the push-forward $\pi_*g$ gives a smooth  
Riemannian metric.  
The push-forward of the form $I$ is also smooth.   
\end{Prop}  
\begin{Remark}\hspace{-2.mm}.\label{explanation}  
The uniqueness of the differential structure on $\C$ means that if  
$(\C,{\cal D}_1)$ and $(\C,{\cal D}_2)$ are two differential structures  
on the quotient space $\C$ satisfying the conditions of   
Proposition \ref{FactorizationLemma},  
then the identity map $\mathop{\rm id} : \C\to\C$ is a diffeomorphism of      
the manifolds $(\C,{\cal D}_1)$ and $(\C,{\cal D}_2)$.   
In this way we obtain  an unique Riemannian manifold,  that we denote   
 by $(X,\pi_\ast g)$.   
\end{Remark}  
We denote by $\Gamma$ the boundary $\partial X$  
of $X$. We  give the following:    
\begin{Def}\hspace{-2.mm}.  
The billiard table $(X,\pi_*g)$   
is said to be Liouville.   
\end{Def}  
To prove Proposition \ref{FactorizationLemma} we need the next simple lemma  
which  will be  useful also in the next section.  
Denote by $V\subset{\bf R}^2$  a neighborhood of the origin and by   
$U$  the square   
$(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)$,   
$\varepsilon>0$ in ${\bf R}^2$. We denote by $(x,y)$ and $(r,s)$ the  
coordinates in $U$ and $V$ respectively.   
\begin{Lemma}\hspace{-2.mm}.\label{smoothforward}  
Let $\Phi : U\to V$ be a smooth mapping such that  
$\sigma^\ast \Phi=\Phi$ and   
$\Phi^{-1}(0,0)=(0,0)$. Suppose that for each    
$(r,s)\ne (0,0)$, the set  $\Phi^{-1}(r,s)$ consists of  
exactly two  
points and the differential $d\Phi|_{(x,y)}$ is non-degenerate at each  
of them.  
Consider the quadratic form (\ref{themetric}),  where $f$ and $q$  
are smooth functions on the interval $(-\varepsilon,\varepsilon)$,   
$f\ge 0$ and $q\le 0$ and suppose that  
the push-forward $d{\tilde g}^2=\Phi_\ast(dg^2)$ is a well-defined  
smooth Riemannian metric on $\Phi(U)$. Then the next conditions are satisfied  
\begin{itemize}  
\item[(A$_1$)] $f$ is even, $f>0$ if $x\ne 0$, $f(0)=0$, $f''(0)>0$,  
\item[(A$_2$)] $q$ is even, $q<0$ if $y\ne 0$, $q(0)=0$,  
\item[(A$_3$)] $f^{(2k)}(0)=(-1)^k q^{(2k)}(0)$, $k\in{\bf N}$.  
\end{itemize}  
Moreover, there exist neighborhoods $V_0\subset\Phi(U)$ and  
$W_0\subset{\bf R}^2=\{(u,v)\}$ of the origin in ${\bf R}^2$ and   
a diffeomorphism $K : V_0\to W_0$,   
such that the map ${\tilde K}$ given by the  
diagram   
$$  
\begin{array}{ccccc}  
   &                           &\Phi^{-1}(V_0)         &\subset&U\\  
   &\stackrel{\tilde K}\swarrow              &\downarrow\lefteqn{\Phi}&  &  
\downarrow\lefteqn{\Phi} \\  
W_0&\stackrel{K}{\longleftarrow}&V_0                    &\subset&\Phi(U)  
\end{array}  
$$  
satisfies ${\tilde K}(x,y)\eqdef(K\circ\Phi)(x,y)=(x^2-y^2,2xy)$.  
In other words, setting   
$z=x+iy$ and $w=u+iv$ we obtain  ${\tilde K}(z)=z^2$.  
Conversely, if the mapping $\Phi$ is given by    
$\Phi : z\mapsto w=z^2$ and $(A_1)\div(A_3)$ are satisfied,   
then $\Phi_\ast g$ is well-defined and smooth,  
and the push-forward of the quadratic form (\ref{theintegral})   
is also well-defined and smooth.  
\end{Lemma}  
{\em Proof.}  
Suppose first that   
the push-forward $d{\tilde g}^2$ is well-defined and smooth. Then  
$\sigma^\ast(\Phi^\ast{\tilde g})=\Phi^\ast{\tilde g}$, and   
$\sigma^\ast g=g$. The last equality shows that $f(-x)-q(-y)=f(x)-q(y)$,  
hence,  $f$ and $q$ are even functions. Moreover, $f(0)=q(0)=0$ and   
$f>0$ if $x\ne 0$, and $q<0$ if $y\ne 0$, since  $d{\tilde g}^2$ is a  
Riemannian metric.   
On the other hand, it is clear that   
the push-forward $\tilde g$ is well-defined and smooth on   
$\Phi(U)\setminus(0,0)$, provided $f$ and $q$ are even.  
  
  
  
Suppose that $\tilde g$ is smooth in $\Phi(U)$. Consider a    
conformal coordinate system  
$\{(u',v')\}$ in a neighborhood $V_0$ of the point $(r,s)=(0,0)$. In  
other words,   
$d{\tilde g}^2=\mu(u',v')(du'^2+dv'^2)$ in these coordinates.    
Let $K_1 : V_0\to W_0'$ be  
the corresponding transition function, $K_1(0,0)=(0,0)$, and   
${\widetilde K}_1\eqdef K_1\circ\Phi$.   
Since both  $(x,y)$ and $(u',v')$ are conformal local coordinates of the  
metric $dg^2$ in  $U\setminus(0,0)$,  the mapping  
${\widetilde K}_1$ is conformal in $U\setminus(0,0)$.   
Taking the right orientation and introducing the complex variables  
$z= x+iy$ and $w = u'+iv'$,  we identify ${\widetilde K}_1$ with a   
holomorphic function $p(z)$.   
Then $p(-z)=p(z)$, and we have  $p(z)=p_1(z^2)$, where  
$p_1(w)$ is holomorphic and  $p_1(0)=0$.  
If  $\frac{dp_1}{dw}(0)=0$ then  for each  $w_0'\ne 0$  close to $0$ the  
preimage   
$p^{-1}(w_0')$ would consist  
of more than two points which is not  allowed by assumption.   
Hence, $\frac{dp_1}{dw}(0)\ne 0$ and shrinking $W_0'$ if necessary we  
obtain a   
biholomorphic map $p_1: W_0\to W_0'$, where   
$W_0$ is a  neighborhood of the origin  in $\bf C$.  
It is clear that in the coordinates $w=u+iv$ (with transition function      
$w=p_1^{-1}(w')$) the map $\Phi$ is given by  $z\mapsto w=z^2$.  
Let $d{\tilde g}^2=\la(u,v)(du^2+dv^2)=\la(w,{\bar w})dwd{\bar w}$ be  
the Riemannian metric ${\tilde g}$ in the new chart.  
We have $g=\Phi^\ast{\tilde g}=4\la(w,{\bar w})|z|^2dzd{\bar z}$.  
Hence,  
\begin{equation}\label{obstacle}  
f(x)-q(y)=4\la(u,v)(x^2+y^2),  
\end{equation}  
where $\la(u,v)$ is a smooth positive function,  
$u=x^2-y^2$, $v=2xy$.  
Let   
\[  
\la\sim\la^{(1)}+(\la^{(2)}_1u+\la^{(2)}_2v)+\cdots\ ,\   
f\sim f_0+f_1x^2+\cdots\ , \   
q\sim q_0+q_1y^2+\cdots  
\]   
be the Taylor  
expansion of the corresponding functions in the points $u=v=0$,  and   
$x=0$ and $y=0$ respectively. Substituting these series in  
(\ref{obstacle}) and comparing  the coefficients of the homogeneous  
terms of  $x$ and $y$ we obtain that $f_0=q_0$,  
$f_1=-q_1=4\la^{(1)}>0$ and $f_k=(-1)^kq_k$. Therefore, conditions  
$(A_1)\div(A_3)$ are satisfied.  
  
Conversely, suppose that $(A_1)\div(A_3)$ are satisfied   
and let $\Phi$ be given by  $z\mapsto w=z^2$. It is clear that  
the metric $d{\tilde g}^2=\la(u,v)(du^2+dv^2)$ is well-defined and   
smooth on $\Phi(U)\setminus 0$ and   
$4\la(u,v)=\frac{f(x)-q(y)}{x^2+y^2}$.  
We have  
\[  
4\la(u,v)=f_1+f_2(x^2-y^2)+o(|z|^2) =f_1+f_2u+o(|w|),  
\]  
where $\la(0,0)=f_1/4$.   
Hence, $\la$  is differentiable at $(u,v)=(0,0)$.  
The proof that $\la$ is smooth is straightforward.  
The same arguments show that the push-forward   
$\Phi_\ast I$ gives a  smooth form in a neighborhood of the zero, and  
we complete the proof of   
Lemma \ref{smoothforward}.  \finishproof  
  
A variant of the last lemma is proved in \cite{Kiyo3}.  
  
\noindent{\em Proof of Proposition \ref{FactorizationLemma}.}  
The first statement of the proposition is obvious.  
The points $A=(0,0)$ and $B=(1/2,0)$ on the cylinder $C$ are fixed points  
of the involution $\sigma$. Denote by $D^2_r$ the disk $\{x^2+y^2<r^2\}$,  
where $r$ is a fixed number, $0<r<\min\{1/2,N\}$, and  
consider the map $\Phi : z\mapsto w=z^2$,  
where $z=x+iy$ and $w=u+iv$.  
$\Phi$ maps $D^2_r$ onto a neighborhood of $w=0$.  
The coordinates $\{(u,v)\}$ give a differential structure 
in a neighborhood of  
the branched point $\pi(A)$. It follows from   
(H$_1$)$\div$(H$_3$) and Lemma \ref{smoothforward} that the push-forwards   
$\Phi_*g$ and $\Phi_*I$ are smooth and $\Phi_*g$ is positive definite.  
The same construction gives a differential structure in a neighborhood  
of the other branched point.  
The other points of the cylinder $C$ are regular  
and the differential structure is induced from the differential structure  
on the cylinder $C$.  
It follows from Lemma \ref{smoothforward} that the differential structure  
described above is unique in the sense of Remark \ref{explanation}.  
Proposition \ref{FactorizationLemma} is proved. \finishproof  
\begin{Prop}\hspace{-2.mm}.\label{Integrability}  
Liouville billiard tables are integrable.  
\end{Prop}  
{\em Proof.}  
Obviously the  
metric $\pi_*g$ has Liouville form (\ref{themetric})  
in a neighborhood of any regular point of the cover  
$\pi$. It is easy to see that the form $\pi_*I$ gives   
a first integral of the geodesic flow of the metric $\pi_\ast g$   
in a sufficiently small neighborhood of any regular point.  
Indeed, on the cover, $\pi_\ast g$  
and $\pi_\ast I$ are given respectively by   
(\ref{themetric}) and (\ref{theintegral})  
 in the coordinates $(x,y)$. As functions on the cotangent bundle   
$T^\ast C$ they have the form  
\begin{eqnarray}   
H(x,y,p_1,p_2)&=&\displaystyle\frac{p_1^2+p_2^2}{f(x)-q(y)} \label{hamilton}\\  
I(x,y,p_1,p_2)&=& \displaystyle\frac{q(y)p_1^2  
+ f(x)p_2^2}{f(x)-q(y)}\, .  \label{integral}  
\end{eqnarray}  
The functions  $H$ and $I$ are in involution and their differentials  
are  linearly independent almost everywhere.  
Hence, the geodesic flow of the Riemannian metric $\pi_*g$  
is completely integrable.  
Let $\xi\in T_x X$, $x\in\Gamma$, and let $n(x)$ be the inward   
unit normal to $\Gamma$ at the point $x$. Define an involution  
$\tau :TX|_{\Gamma}\to TX|_{\Gamma}$ by   
$(x,\xi)\mapsto(x,\xi-2g(\xi,n(x))n(x))$.  
It follows from (\ref{themetric}) and (\ref{theintegral}) that  
$\tau$ preserves the values of $g$ and $I$.  
This completes the proof of  Proposition \ref{Integrability}.  
\finishproof  
  
\begin{Remark}\hspace{-2.mm}.  
The corresponding analytic variants of Proposition 
\ref{FactorizationLemma},   
\ref{Integrability} and Lemma \ref{smoothforward} are also true.  
\end{Remark}  
From now on we consider Liouville billiard tables which ``resemble''   
that  of the ellipse (Sect. 3.1).  
We impose the following additional  conditions:  
\begin{itemize}  
\item[(H$_4$)] the boundary $\Gamma$ of $X$  is   
locally geodesically convex which amounts  to $\frac{dq}{dy}(N)<0$;   
\item[(H$_5$)] $f(x)=f(1/2-x)$ and $f$ is strictly monotone  
on the interval $[0,1/4]$;  
\item[(H$_6$)] $f$ and $q$ are analytic and their critical  
points  are non-degenerate.   
\end{itemize}  
\begin{Def}\hspace{-2mm}.   
Liouville billiard tables   
satisfying  conditions (H$_4$)$\div$(H$_6$)   
are said to be  {\it of classical type}.  
                                                   \label{lb-class}  
\end{Def}  
\begin{Remark}\hspace{-2mm}.  
One of the consequences of (H$_5$) is that there is a group  
$G\cong{\bf Z}_2\times{\bf Z}_2$ acting  on $(X,g)$ by isometries.  
\end{Remark}  
Indeed, the involutions $\si_1 : (x,y)\mapsto (x,-y)$ and  
$\si_2 : (x,y)\mapsto (1/2-x,y)$  induce isometries  
of the Liouville billiard table $(X,\pi_{*}g)$ which   
will be denoted  by the same letters. Consider  the group of isometries   
$G$  generated  
by $\si_1$ and $\si_2$.  
The set of fixed points of $\si_1$ ($\si_2$) can be parameterized by a  
geodesic arc $\ga_1$ ($\ga_2$) which links two different points   
of $\Gamma$ and it is orthogonal to $\Gamma$  at these points.   
In the case of the ellipse, $\gamma_1$ and   
$\gamma_2$ are the corresponding axes.   
  
Liouville billiard tables possess  the so called string property. Namely,  
we have:  
\begin{Prop}\hspace{-2.mm}.\label{StringProperty}  
Any broken geodesic in a Liouville billiard table starting from   
the singular point $F_1$  
($F_2$) passes through    
$F_2$ ($F_1$) after one reflection  at  
$\Gamma$. Moreover, the sum of distances from any point of $\Gamma$  
to  $F_1$ and  $F_2$ is constant.   
\end{Prop}  
The proposition will be proved at the end of  Sect. 4.2.   

  
We give examples of Liouville billiard tables of classical type in the  
next Section.  
  
  
  
  
  
  
  
\section{Integrable billiard tables on surfaces of constant curvature and  
quadrics}  
\label{Examples}  
The aim of this section is to provide several examples of Liouville  
billiard tables of classical type.   
It is more or less known that the corresponding   
billiard flows are integrable. The main point here is that we give  
explicit  formulae for the covering maps, the metric and the integral.   
\subsection{Elliptical billiard tables in the Euclidean space.}  
Consider the mapping $\Phi : {\bf T}^1\times{\bf R}\to{\bf R}^2$, given by:  
\[  
\left\{  
\begin{array}{ccc}  
u&=&\ep\cosh 2\pi y\cos 2\pi x,\\  
v&=&\ep\sinh 2\pi y\sin 2\pi x,  
\end{array}  
\right.  
\]  
where $(x,y)\in{\bf T}^1\times{\bf R}$ and $(u,v)\in{\bf R}^2$,  
$\ep>0$ (see \cite{Whittaker}).  
The images of the coordinate lines $y=y_0=\const\ne 0$ are  
confocal ellipses on the plane $\{(u,v)\}$ with foci $F_1=(-\ep,0)$ and  
$F_2=(\ep,0)$,   
while the images of the coordinate lines $x=x_0=\const\ne\frac{k}{2}$,  
$k\in{\bf Z}$, are confocal hyperbolas with the same foci $F_1$ and $F_2$.  
The mapping $\Phi$ is a double cover of the plane ${\bf R}^2$ with  
branched points $F_1$ and $F_2$.  
Let $ds^2=du^2+dv^2$ be the Euclidean metric on ${\bf R}^2$. The pull-back  
$g:= \Phi^{*}s$ of the metric $s$  has a   
Liouville form  
\begin{equation}\label{themetr-ellipse}  
dg^2=(f(x)-q(y))(dx^2+dy^2),  
\end{equation}   
where $f(x)=  4\ep^2\pi^2\sin^2 2\pi x$ and $q(y)= -4\ep^2\pi^2\sinh^2  
2\pi y$.  
The quadratic form  
\begin{equation}\label{theint-ellipse}  
dI^2\ :=\ (f(x)-q(y))(q(y)dx^2+f(x)dy^2)  
\end{equation}  
is a first integral of the geodesic flow of the ``metric'' $g$. Note that the  
quadratic forms $g$ and $I$ vanish at the points $(\frac{k}{2},0)$,  
$k\in{\bf Z}$. It follows from Proposition \ref{smoothforward},   
\ref{Integrability} and Lemma \ref{FactorizationLemma} that the  
push-forward $\widetilde I$   
of the form $I$ gives a smooth integral of the elliptical  
billiard tables in the interior of the ellipses $\{y=\const\ne 0\}$.  
Moreover, ${\widetilde I}$ is an integral of any billiard table,   
the boundary of which consists of curves from the confocal family   
described above.  
  
  
  
  
  
  
\subsection{Integrable billiard tables on the sphere.}\label{S^2}  
Integrable billiard tables on ${\bf S}^2$ and ${\bf H}^2$ have been    
considered earlier in \cite{C-S,Bolotin}. Here we give explicit formulae  
for  the corresponding branched covers.  
 
  
Denote by  ${\bf S}^2= \{x^2+y^2+z^2=1\}$ the unit sphere embedded  
in the Euclidean space ${\bf R}^3$.  
The metric $g_1$ is the restriction of the Euclidean one  
$dg_0^2=dx^2+dy^2+dz^2$ on ${\bf S}^2$. The coordinates $\{(x,y)\}$ of the  
unit disk ${\bf D}^2_1= \{x^2+y^2<1\}$ give a parameterization of the positive  
half ($z>0$) of the sphere ${\bf S}^2$ and we can rewrite the metric $g_1$  
in these coordinates. Let us consider the branched cover  
of the unit disk  
$\Phi_1 : {\bf R}/2\pi{\bf Z}\times (-\frac{1}{k},\frac{1}{k})\to{\bf D}^2_1$  
given by   
\[  
\left\{  
\begin{array}{ccc}  
x&=&\frac{k}{\sqrt{1+k^2}}\sqrt{1+v^2}\cos u,\\  
y&=&kv\sin u,  
\end{array}  
\right.  
\]  
where $k$ is a positive constant.  
The pull-back ${\widetilde g}_1\eqdef\Phi_1^{*}g_1$ has Liouville form  
\[  
d{\widetilde g}_1^2=k^2(f(u)-q(v))\left(\frac{du^2}{1+k^2\sin^2u}+  
\frac{dv^2}{(1+v^2)(1-k^2v^2)}\right),  
\]  
where $f(u)= \sin^2u$ and $q(v)= -v^2$. After an obvious change of   
the variables the metric ${\widetilde g}_1$ takes form (\ref{themetric}).  
Propositions \ref{FactorizationLemma}, \ref{Integrability} and  
Lemma \ref{smoothforward} show that the interior of any curve   
$\{v=\const>0\}$ is  a Liouville billiard table. It is easy to see  
that   all these billiard tables are of classical type.   
  
  
  
  
\subsection{Integrable billiard tables on the Hyperbolic space.}  
Consider the hyperboloid of two sheets ${\bf H}^2= \{-x^2-y^2+z^2=1\}$  
embedded in the Minkowski space ${\bf R}^{2,1}$.  
The metric $g_{-1}$ of the Hyperbolic space is the restriction of  
the Minkowski metric $dg_0^2=dx^2+dy^2-dz^2$ on ${\bf H}^2$.  
The coordinates $\{(x,y)\}$ give a parameterization of the positive  
sheet ($z>0$) of the hyperboloid ${\bf H}^2$. Consider the branched  
cover   
of the plane  
$\Phi_{-1} : {\bf R}/2\pi{\bf Z}\times{\bf R}\to{\bf R}^2$  
given by   
\[  
\left\{  
\begin{array}{ccc}  
x&=&\frac{k}{\sqrt{1-k^2}}\sqrt{1+v^2}\cos u,\\  
y&=&kv\sin u,  
\end{array}  
\right.  
\]  
where $k$ is a constant in the interval $0<k<1$.  
The pull-back ${\widetilde g}_{-1}= \Phi_{-1}^{*}g_{-1}$ has a Liouville form  
\[  
d{\widetilde g}_{-1}^2=k^2(f(u)-q(v))\left(\frac{du^2}{1-k^2\sin^2u}+  
\frac{dv^2}{(1+v^2)(1+k^2v^2)}\right),  
\]  
where $f(u)= \sin^2u$ and $q(v)= -v^2$. As in \ref{S^2} we obtain that   
the billiard tables inside the curves $\{v=\const\ne 0\}$ are   
Liouville billiard tables of classical type.  
  
  
  
  
\subsection{Integrable billiard tables  on quadratic surfaces.}  
Another class of integrable billiard tables appears on quadrics.   
Consider, for example, the ellipsoid $E= \{\frac{x^2}{a}+\frac{y^2}{b}+  
\frac{z^2}{c}=1\}$, where $0<c<b<a$. Denote by $E_\la$ the corresponding  
$1$-parameter family of confocal quadrics, i.e.,  
$E_\la= \{\frac{x^2}{a-\la}+\frac{y^2}{b-\la}+\frac{z^2}{c-\la}=1\}$.  
Fix a real number $\la_2^0\in (c,b)$ and consider the hyperboloid  
of one sheet $E_{\la_2^0}$. It can be easily seen that $E_{\la_2^0}$  
intersects the ellipsoid $E$ in two curves $S_1$ and $S_1^{'}$.  
These curves are boundaries of two isometric regions $D_1$ and  
$D_1^{'}$ on the ellipsoid, diffeomorphic to the unit disk.  
It turns out  that the billiard flows in $D_1$ and $D_1^{'}$   
(equipped with the induced  
Euclidean metric from ${\bf R}^3$) are integrable.  
Fix a real number $\la_3^0\in(b,a)$. The quadric $E_{\la_2^0}$ is  
a hyperboloid of two sheets that intersects the ellipsoid $E$ in two  
curves $S_2$ and $S_2^{'}$. As in the previous case these curves bound  
two isometric regions $D_2$ and $D_2^{'}$ on the ellipsoid. The corresponding  
billiard tables are integrable (see also \cite{C-S}).  
It is easy to see  that the same construction gives integrable billiard  
tables on every fixed hyperboloid of two sheets $E_{\la_2^0}$. The boundaries  
of the billiard tables are given by the intersection curves   
of the hyperboloid $E_{\la_2^0}$ with the ellipsoids from   
the confocal family $E_\la$.  
  
Denote by $(\la_1,\la_2,\la_3)$ the standard elliptical coordinates  
$\la_1<c<\la_2<b<\la_3<a$.  
The relation between the elliptical coordinates $\la_i$ and the  
Cartesian coordinates $x$, $y$ and $z$ is given by the formula  
\[  
\left\{  
\begin{array}{ccc}  
x^2&=&\frac{(a-\la_1)(a-\la_2)(a-\la_3)}{(a-b)(a-c)},\\  
y^2&=&\frac{(b-\la_1)(b-\la_2)(b-\la_3)}{(b-a)(b-c)},\\  
z^2&=&\frac{(c-\la_1)(c-\la_2)(c-\la_3)}{(c-a)(c-b)}.  
\end{array}  
\right.  
\]  
Taking $\la_1=  c-\te_1^2$, $\la_2=  c\cos^2\te_2+b\sin^2\te_2$  
and $\la_3=  b\cos^2\te_3+a\sin^2\te_3$ we obtain a  
single-valued mapping $(\te_1,\te_2,\te_3)\mapsto(x,y,z)$   
given by the formulae   
\begin{eqnarray*}  
x&=&(\sqrt{((a-c)+\theta_1^2)((a-c)\cos^2\theta_2+  
(a-b)\sin\theta_2)}\cos\theta_3)/\sqrt{a-c},\\  
y&=&\sqrt{(a-c)+\theta_1^2}\cos\theta_2\sin\theta_3,\\  
z&=&(\theta_1\sin\theta_2  
\sqrt{(b-c)\cos^2\theta_3+(a-c)\sin^2\theta_3})/\sqrt{a-c}.  
\end{eqnarray*}   
This mapping is a branched cover of the plane ${\bf R}^3$   
(see \cite{Kiyo3}).  
It is not hard to see that this cover gives a double branched  
cover of the billiard tables described above.  
We give the formulas for the corresponding metrics on the cover,  
while the integral is given by the push-down of (\ref{theintegral}).   
\begin{itemize}  
\item[a)]{\bf Integrable billiard tables on the ellipsoid.}  
We have two types of billiard tables on the ellipsoid:  
\begin{itemize}  
\item[1. ]   
Setting $\theta_1=\const$, $u=\theta_3$ and $v=\sqrt{b-c}\sin\theta_2$   
we obtain a branched cover of the first family of integrable  
billiard tables on the ellipsoid. In coordinates ${(u,v)}$ we obtain  
\begin{eqnarray*}  
dg^2&=&(f(u)-q(u))\left(\frac{(b\cos^2u+a\sin^2u)du^2}  
{(b\cos^2u+a\sin^2u)-c}+\right.\\  
&+&\left.\frac{(b-v^2)dv^2}{(a-(b-v^2))((b-v^2)-c)}\right),  
\end{eqnarray*}  
where $f(u)= (a-b)\sin^2u$, $q(v)= -v^2$ and $-\sqrt{b-c}<v<\sqrt{b-c}$;  
\item[2.]   
Taking $\theta_1=\const$, $u=\theta_2$ and $v=\sqrt{a-b}\sin\theta_3$  
we obtain a cover of the second family of billiard tables on the ellipsoid.  
On the cover the metric is  
\begin{eqnarray*}  
dg^2&=&(f(u)-q(u))\left(\frac{(c\sin^2u+b\cos^2u)du^2}  
{a-(c\sin^2u+b\cos^2u)}+\right.\\  
&+&\left.\frac{(b+v^2)dv^2}{(a-(b+v^2))((b+v^2)-c)}\right),  
\end{eqnarray*}  
where $f(u)= (b-c)\sin^2u$, $q(v)= -v^2$ and $-\sqrt{a-b}<v<\sqrt{a-b}$.  
\end{itemize}  
\item[b)]{\bf Integrable billiard tables on the hyperboloid of two sheets.}  
Fix $\al\in(b,a)$ and consider the hyperboloid of two sheets  
given by the equation $\la_3=\const$.  
Taking $u=\theta_2$ and $v=\theta_1$ we obtain a branched cover  
of the family of billiard tables on the hyperboloid of two sheets.   
On the cover we have  
\begin{eqnarray*}  
dg^2&=&(f(u)-q(u))\left(\frac{(\al-(c\sin^2u+b\cos^2u))du^2}  
{a-(c\sin^2u+b\cos^2u)}+\right.\\  
&+&\left.\frac{(\al-(c-v^2))dv^2}{(1+v^2)(b-(c-v^2))}\right),  
\end{eqnarray*}  
where $f(u)= (b-c)\sin^2u$, $q(v)= -v^2$ and $-1<v<1$.  
\end{itemize}  
It is easy to see that   
\begin{Prop}\hspace{-2mm}.  
The Liouville billiard tables  considered in Sect. \ref{Examples} are of  
classical type.  
\end{Prop}  
Applying Theorem \ref{MainTh} to the billiard tables considered in this  
section we obtain:  
\begin{Coro}\hspace{-2mm}.  
The map (\ref{themap}) is one-to-one for each of   
the billiard tables 3.1-3.4.    
\end{Coro}  
  
  
  
  
  
  
  
  
  
\section{Properties of the billiard ball map}  
  
Let $(X,{\tilde g})$ be a billiard table with boundary $\Gamma$.   
Consider the co-ball bundle   
$B^\ast \Gamma =\{(x,\xi)\in T^\ast \Gamma: \|\xi\|_x\le 1\}$, where  
the quadratic with respect to $\xi\in T_x^\ast \Gamma$ Hamiltonian   
$(x,\xi)\to  \|\xi\|_x^2$ is given by  the Legendre transformation of the   
induced Riemannian metric on $\Gamma$. Denote by    
$\displaystyle \mathop{B ^\ast }^\circ \Gamma$   
the interior of $B^\ast \Gamma$.  Let   
$S^\ast X   =\{(x,\xi)\in T^\ast X: {\tilde H}(x,\xi)= 1\} $ be the   
co-sphere bundle of $X$, where $\tilde H$ stands for    
the Hamiltonian function on $T^\ast X$ corresponding  
to the Riemannian metric $\tilde g$ via the Legendre transformation. Set   
$\Sigma^+ = \{(x,\xi)\in S^\ast X|_\Gamma: \langle \xi,  
n(x)\rangle >0\}$,  $n(x)$ being  the unit inward normal to $\Gamma$  
at $x$.  
   
The corresponding billiard ball map $B$ is defined  as     
follows (\cite{Birkhoff}):   
Take $(x,\xi)$ in the interior of $B^\ast \Gamma$ and denote  
by $\xi_+ \in \Sigma_x^+$ the  co-vector  uniquely  determined  by  
$\xi_+|_{T_x \Gamma} = \xi$. Consider the integral  
curve  $\exp(tX_{\tilde H})(x,\xi_+)$,  of the Hamiltonian vector field   
$X_{\tilde H}$ starting at $(x,\xi_+)$. If it intersects transversally    
$S^\ast X|_\Gamma$ at a time $t_1>0$ and lies entirely in the   
interior $\displaystyle {S^\ast }  \mathop X ^\circ$ of $S^\ast  X$   
for $t\in (0,t_1)$, we set   
$(y,\eta_-)=J(x,\xi_+) =\exp(t_1X_{\tilde H})(x,\xi_+)$,   
and define $B(x,\xi)=(y,\eta)$,  
where $\eta = \eta_-|_{T_y \Gamma}$.    
We  denote by $\widetilde B^\ast \Gamma$ the set  
of all such points $(x,\xi)$. As in \cite{kn:MM} we can write $B$ in an  
invariant form as follows. Consider  the pull-back   
$\omega_1$ in $T^{*}X|_\Gamma$ of   
the symplectic form $\omega$ in $T^{*}X$ via the inclusion map.   
The projection	along the characteristics of $\omega_1$ induces a  
smooth map  $\pi_1: S^{*}X|_\Gamma \to B^{*}\Gamma$ and  
we denote by   
$\displaystyle \pi_1^+: \mathop {B^{*} }^\circ \Gamma \to \Sigma^+$ an inverse  
map to $\pi_1$. Then we can write $B=\pi_1\circ J \circ \pi_1^+$.   
In this way we obtain a symplectic map   
$B:\widetilde B^\ast \Gamma \rightarrow B^\ast \Gamma $ which is  
analytic if the billiard table is analytic.   
We extend $B$ as the identity mapping on the  
boundary $S^\ast \Gamma$ of $B^\ast \Gamma$.   
  
  
  
\subsection{The phase portrait of the integral}\label{Portrait}  
Suppose now that $(X,{\tilde g})$ is a Liouville billiard table where  
${\tilde g}=\pi_*(g)$.   
We identify the  boundary $\Gamma$ of a Liouville billiard table   
with the circle $\{(x,N):\, x\in{\bf T}^1\}$   
on the cylinder $C$ with   
coordinates $\{(x,y)\}$.   
Consider the natural coordinates $\{(x,y,p_1,p_2)\}$ of the  
cotangent bundle $T^{*}C$ and set  $\Gamma_N = \{y=N\}\subset T^{*}C$.   
Then $\Gamma_N$ is diffeomorphic to $T^{*} X|_{\Gamma}$ and using    
(\ref{hamilton}) we identify    
\[  
S^\ast X|_\Gamma \cong   
\{(x,N,p_1,p_2)\in \Gamma_N:  p_1^2 + p_2^2 = f(x) -  
q(N)\}\, .   
\]   
Moreover, we   
identify $T^\ast \Gamma$ with $T^\ast {\bf T}^1$ via the inclusion map   
$(x,p_1) \longmapsto (x,N;p_1,0)$.   
The billiard ball map $B : {\widetilde B}^\ast\Gamma\to B^\ast\Gamma$  
preserves the  symplectic form $\om_0 = dp_1\wedge dx$ of $T^\ast \Gamma$.  
It is easy to see  that the characteristics of the pull-back  
$\omega|_{\Gamma_N}$  to $\Gamma_N$  of  
the symplectic form  $\om=dp_1\wedge dx+dp_2\wedge dy$  
are spanned by the vectors $\frac{\partial}{\partial p_2}$.   
Then $\pi_1(x,N,p_1,p_2) = (x,p_1)$ on $S^\ast X|_\Gamma$,   
\[  
B^\ast \Gamma \cong \{(x,p_1)\in  
T^\ast{\bf T}^1: p_1^2 - f(x) + q(N)\le 0\}\, ,  
\]   
and  the map   
$\pi_1^+:B^\ast \Gamma \to \Sigma^+$ is given by    
\[  
(x,p_1)\longmapsto \left(x,N;p_1,-\sqrt{f(x)-q(N)-p_1^2}\, \right)\, .  
\]  
Consider the function  ${\cal I} = I\circ \pi_1^+$ on $B^\ast \Gamma$,  
where $I$ is the integral (\ref{integral}). In the coordinates  
$\{(x,p_1)\}$  
it is given by    
\begin{equation}  
{\cal I}(x,p_1)=f(x)-p_1^2\, .   
                           \label{integral-bill}  
\end{equation}   
By construction, ${\cal I}$ is a smooth function in   
$ \displaystyle\mathop {B^\ast}^\circ \Gamma$   
and it is analytic if the Liouville billiard is of  
classical type.   
  
  
  
\begin{Lemma}\hspace{-2.mm}.  
The function ${\cal I}(x,p_1)=f(x)-p_1^2$ is constant on the trajectories  
of the billiard ball map $B$.  
\end{Lemma}  
The lemma follows immediately from Proposition \ref{Integrability}.   
  
  
It is easy to see  that  
$q(N)\le {\cal I}(x,p_1)\le f(x)$.   
Fix a value $h$ of the integral ${\cal I}(x,p_1)$,  
$q(N)\le h \le \max f$ and denote by $S_h=\{{\cal I}(x,p_1)=h\}$   
the corresponding  
level set in $B^\ast \Gamma$.  
The set of glancing points of the  
billiard ball map coincides with the set of constant level  
$\{{\cal I}(x,p_1)=q(N)\}$.  
  
Consider now the critical points of the integral ${\cal I}(x,p_1)$ in $B^\ast  
\Gamma$. Using (H$_1$)   
(and (H$_5$) if the Liouville billiard table is of classical type),  
we easily obtain:  
\begin{Prop}\hspace{-2.mm}. \label{Morse}  
The critical points of the integral ${\cal I}(x,p_1)$ are given by  
$P_i=(x_i,0)$,   
where $x_i$ are the critical points of $f$.  
The point $P_i$ is non-degenerate in Morse sense if and only if   
$f^{''}(x_i)\ne 0$.  
If $f^{''}(x_i)>0$, then $P_i$ is  non-degenerate  of index $1$   
(``hyperbolic'' point).  
If $f^{''}(x_i)<0$, then $P_i$ is  non-degenerate  of index $2$   
(``elliptic'' point).  
The points $A_1 = (0,0)$ and $A_2 = (1/2,0)$ are non-degenerate  of  
index $1$. Each  $h\in (q(N),0)$ is a  regular value of the   
integral ${\cal I}(x,p_1)$ and the corresponding  
level set $S_h$ consist of two circles. The level set $S_0$ consists of  
the critical points $A_1$ and $A_2$ and four different arcs   
connecting them and containing only    
regular points of the integral ${\cal I}(x,p_1)$.  
Moreover,  if  the Liouville billiard table   
is of classical type then   
each   
$h\in (0,f(1/4))$ is a  regular value of  ${\cal I}(x,p_1)$,   
the corresponding level set $S_h$ consists of two circles, and   
$S_{f(1/4)}$ consists of two non-degenerate critical points  
of index $2$.  
\end{Prop}  
  
If $f(x)-h>0$ for $x\in [a,b]$,   
then $p_1=\pm\sqrt{f(x)-h}$ is a parameterization  
of $S_h$, and we set  $S_h^{\pm}[a,b] = \{(x,\pm\sqrt{f(x)-h}): x\in [a,b]\}$.  
In a neighborhood of these curves we can introduce local  
coordinates $\{(x,{\cal I})\}$.   
The pull-back of $\omega_0$ to $S_h^{\pm}[a,b]$  
has the form   
\[  
\om_0\ =\ dp_1\wedge dx\ =\   
\pm\left(\frac{dx}{2\sqrt{f(x)-{\cal I}}}\right)\wedge d{\cal I}.  
\]  
We consider the following  1-form on $S_h$:  
\[  
\la_h\eqdef\left\{  
\begin{array}{l}  
\frac{dx}{\sqrt{f(x)-h}},\ p_1>0,\\  
-\frac{dx}{\sqrt{f(x)-h}},\ p_1<0.  
\end{array}  
\right.  
\]  
If  $h$ is a regular value of the integral ${\cal I}(x,p_1)$,  
the form $\la_h$ gives a  smooth $1$-form on $S_h$  
which  is invariant under the Hamiltonian flow of ${\cal I}$.  
$\la_h$ is called  {\it Leray's form}.   
It is easy to see  that the Leray's form $\la_h$ is invariant with  
respect to any symplectic transformation  defined in a tubular  
neighborhood of $S_h$ and preserving  the function  
${\cal I}(x, p_1)$.   
We supply $S_h$ with orientation by means of the Leray's form.  
  
  
  
  
  
  
\subsection{Rotation function for  Liouville billiard tables of  
classical type}\label{Sec:details}  
Fix a regular value $h$ of the integral ${\cal I}(x,p_1)$ and consider   
a connected component  $S_h^0 \cong {\bf T}^1$ of the level set $S_h$.   
Suppose that the billiard ball map $B$ is defined on $S_h^0$  
and preserves it inducing a diffeomorphism $B|_{S_h^0}:S_h^0 \to S_h^0$.  
The Leray's form $\la_h$ provides $S_h^0$ with a smooth positive   
measure which  is invariant with respect  to the map $B|_{S_h^0}$.   
By means of  $\la_h$ we  introduce  
a periodic coordinate $\{s \mod \mu_0\}$ ($\mu_0=\int_{S_h^0}\la_h$) on   
$S_h^0$ such that $ds=\la_h$. It is clear that $B|_{S_h^0}$ takes  
$s$ to $s+\mu$ where $\mu$ is a constant on $S_h^0$.  
The number $\rho|_{S_h^0}=\mu/\mu_0$ is the {\em rotation number} of the  
map $B|_{S_h^0}$.   
The rotation number $\rho|_{S_h^0}$ depends on the choice of   
the orientation on $S_h^0$ and it is defined modulo $1$.  
It is clear that $\rho|_{S_h^0}=\int_{\widehat{PQ}}\la_h/\int_{S_h^0}\la_h$,   
were $P$ is an arbitrary point on $S_h^0$, $Q=B(P)$, and $\widehat{PQ}$ is   
an arc in $S_h^0$ connecting $P$ and $Q$.  
  
  
  
  
  
We will prove (see Proposition \ref{RotFunc1})   
that there is at most a finite set $\Xi(q)$  
of values $h_1,...,h_l\in(q(N),0)$ such that the billiard ball map $B$  
is well defined on $S_h$ for each $h\in (q(N),0)\setminus \Xi(q)$ but it   
is not defined on the level sets $S_{h_i}$. Geometrically this  means  
that the geodesics  issuing from the sets $S_{h_i}$ do not  reach   
the boundary $\Gamma$ again and they stay forever in   
the interior of the billiard table.   
If $h\in(q(N),0)\setminus\Xi(q)$ then the billiard ball map $B$   
preserves the connected components of the level sets $S_h$.  
Moreover, the involution    
$(x,p_1) \stackrel{\imath}{\to} (x,-p_1)$ interchanges the connected   
components of $S_h$ and we have $\imath B\imath = B^{-1}$.   
In particular, the  
rotation number of the restriction of $B$ to each of the components of  
$S_h$ is the same and we   
denote it by  $\rho^-(h)$.  
If $0<h<f(1/4)$ then the billiard ball map interchanges the components of the  
level sets $S_h$. In this case we consider the square $B^2$ of the   
billiard ball map and denote the corresponding rotation function  
by $\rho^+(h)$, $h\in(0,f(1/4))$. In what follows we consider separately  
the cases $h\in(q(N),0)$, $h\in(0,f(1/4))$, and $h=0$.   
  
Given a value $h$ of the restriction of integral $I$ on   
$S^\ast C$, we consider   
the invariant set of bicharacteristic  flow $T_h=\{H=1,I = h\} \subset  
S^\ast C$.  Passing to $TC$ by the Legendre  
transformation, we determine  $T_h=\{g=1,I=h\} \subset T C$   
(in the coordinates $(x,y)$ on $C$)  
by the system   
$$  
\left\{  
\begin{array}{ccc}  
g(v,v)&=&(f(x)-q(y))({\dot x}^2+{\dot y}^2)=1,\\  
I(v,v)&=&(f(x)-q(y))(q(y){\dot x}^2+f(x){\dot y}^2)=h,  
\end{array}  
\right.  
$$  
where $v={\dot x}\partial_x+{\dot y}\partial_y$  
denotes the velocity of a geodesic curve  in  $T_h$.  
This yields  
$$  
\left\{  
\begin{array}{ccc}  
{\dot x}&=&\pm\frac{\sqrt{f(x)-h}}{f(x)-q(y)},\\  
{\dot y}&=&\pm\frac{\sqrt{h-q(y)}}{f(x)-q(y)}.  
\end{array}  
\right.  
$$  
It is easy to see that this dynamical system  corresponds to the projections  
on the base $C$ of the bicharacteristics   lying  in the invariant set $T_h$.  
Changing the parameterization we obtain  
\begin{equation}  
\left\{  
\begin{array}{rcll}  
\frac{dx}{d\tau}&=&\pm\sqrt{f(x)-h},\\  
\frac{dy}{d\tau}&=&\pm\sqrt{h-q(y)}.  
\end{array}  
\right.  
                                      \label{eq:caustic}  
\end{equation}  
If $h<q(N)$, then the corresponding geodesics never touch the boundary of  
the billiard table.  
  
\vspace{0.5cm}  
  
\noindent{\bf Case A).} Suppose that $q(N)<h<0$.  
Consider  the caustics of the bicharacteristics lying in $T_h$,  
i.e. the envelope of the geodesics which are projections of  
bicharacteristics in $T_h$. Note that the geodesics tangent to a  
caustic of $T_h$ may never reach the boundary $\Gamma$.   
It follows from (\ref{eq:caustic}) that   
 the caustics corresponding to  $T_h$, $q(N)<h<0$,  can be identified with  
the curves $\{y=y(h)\}$ on $C$,  
where $y(h)$ are the positive solutions of the equation $q(y)=h$.  
Denote by $y_m(h)$ the maximal of them and suppose that   
$y_m$ is not a critical point of the function $q(y)$.    
Then $\{y=y_m(h)\}$ is a caustic corresponding to the  broken geodesics  
issuing from the invariant set $S_h \subset   
\displaystyle\mathop {B^\ast}^\circ \Gamma$,  
of the billiard ball map. Indeed, for each $\nu\in S_h$,  the broken   
bicharacteristic $\gamma$ issuing from $\nu$ ($\gamma(0) =  
\pi^+(\nu))$ lies in $T_h$. Consider the corresponding broken geodesic  
$(x(\tau),y(\tau))$, $y(0)=N$,  parameterized by $\tau\ge 0$.   
In view of  
(\ref{eq:caustic}) it lies in the annulus $y_m \le y(\tau) \le N$.   
Moreover,   $dx/d\tau  
(\tau)>0$ for each $\tau \ge 0$,  and there exist $\tau_1> \tau(h)>0$   
such that $dy/d\tau (\tau(h))=0$,   
$dy/d\tau (\tau)<0$ for $\tau\in[0,\tau(h))$,  $dy/d\tau (\tau)>0$ in   
$\tau\in(\tau(h),\tau_1]$ and $y(\tau_1)=N$. By the same argument to any  
other solution $y_0$ of $q(y)=h$ correspond  caustics for $T_h$ such that  
the geodesics tangent to $\{y=y_0\}$ never reach the boundary  
($y(\tau)$   
oscillates between two different zeros of $q(y)=h)$.   
  
Consider  the rotation function $\rho^{-}(h)$, $q(N)<h<0$,   
of the billiard ball map corresponding to the geodesics issuing from  
$S_h$  the  caustic of which is given by $\{y=y_m\}$.  
\begin{Prop}\hspace{-2.mm}.\label{RotFunc1}  
The rotation function $\rho^{-}(h)$, $q(N)<h<0$,    
of the billiard ball map  is   
\[  
\rho^{-}(h)=\frac{2\int\limits_{y_m(h)}^N\frac{dy}{\sqrt{h-q(y)}}}  
{\int\limits_0^1\frac{dx}{\sqrt{f(x)-h}}}.  
\]  
\end{Prop}  
\noindent{\em Proof.}  
Denote by $P_1\in\Gamma$ the point with coordinates $(0,N)$ on  
the cylinder $C$. Consider the solution  $\ga(\tau)  
=(x(\tau),y(\tau))$   
of  
the system  
$$  
\left\{  
\begin{array}{ccc}  
\frac{dx}{d\tau}&=&\sqrt{f(x)-h},\\  
\frac{dy}{d\tau}&=&-\sqrt{h-q(y)}  
\end{array}  
\right.  
$$  
with initial data $\ga(0)=P_1$.   
The curve $\gamma$ touches the caustic $\{y=y_m(h)\}$ at   
\[  
\tau(h)=\int\limits_{y_m(h)}^N\frac{dy}{\sqrt{h-q(y)}}\, .  
\]  
Denote by $Q_1$ the point $\ga(\tau(h))$ and consider    
the solution  $\bar\gamma(\tau)$   
of the system  
$$  
\left\{  
\begin{array}{ccc}  
\frac{dx}{d\tau}&=&\sqrt{f(x)-h},\\  
\frac{dy}{d\tau}&=&\sqrt{h-q(y)},  
\end{array}  
\right.  
$$  
with initial data $\bar\gamma(0)=Q_1$.  
This curve intersects the boundary $\Gamma$ at a point $P_2=(x(h),N)$.  
In the coordinate chart $\{(x,I)\}$ (defined in a neighborhood of  
the connected component of $S_h$), the billiard  
ball map takes the point $(0,h)$ to $(x(h),h)$.  
Therefore, the rotation function is  
\begin{equation}\label{Rot1}  
\rho^{-}(h)=\frac{\int_0^{x(h)}\la_h}{\int_0^1\la_h},  
\end{equation}  
where $\la_h$ is the Leray form.  
It follows from $(\ref{eq:caustic})$ that  
$2\tau(h)=\int\limits_0^{x(h)}\frac{dx}{\sqrt{f(x)-h}}$ which   
completes the proof of Proposition \ref{RotFunc1}. \finishproof  
  
We consider a subset    
$\Xi(q)=\{h_1,\ldots,h_l\in(q(N),0)\}$  of the set of critical points of  
$q$,  defined recurrently  as follows:    
We take $h_1 =  q(N_1)$, where $N_1\in (0,N)$ is the closest to $N$  
critical point of $q$ (if such a point doesn't exist we  
set $\Xi(q)=\emptyset$). Then given $h_k$, $k\ge 1$, we  
define by recurrence  $h_{k+1} =  q(N_{k+1})$,  where  
$N_{k+1}<N_k$ is the closest to $N_k$ critical point of the function $q$  
on the interval $(0,N)$ such that $q(N_{k+1})> h_k$.  
If such a point $N_{k+1}$ doesn't exist we  set  
$\Xi(q)=\{h_1,\ldots,h_k\in(q(N),0)\}$.  
\begin{Prop}\hspace{-2.mm}.\label{RotBihav1}  
The rotation function $\rho^{-}(h)$ is strictly increasing  in a  
neighborhood of the point $q(N)$ and $\lim\limits_{h\to h(N)+0}\rho^{-}(h)=0$.  
The rotation function is analytic on the set    
$(q(N),0)\setminus\Xi(q)$, and  
$\lim\limits_{h\to h_i\pm 0}\rho^{-}(h)=+\infty$ for every $h_i\in\Xi(q)$.  
\end{Prop}  
\noindent{\em Proof.}  
It is clear that $y_m(h)\to N$ as $h\to q(N)+0$, hence,   
$\lim\limits_{h\to q(N)+0}\rho^-=0$.  
We have  
\[  
\tau(h) = \int\limits_{y_m(h)}^N\frac{dy}{\sqrt{h-q(y)}}  
=-2\int\limits_0^{\sqrt{h-q(N)}}\frac{dy}{dq}(h-w^2)dw,  
\]  
where $w^2=h-q(y)$.  
Hence,  
\[  
\tau^{'}(h)=-\frac{dy}{dq}(q(N))\frac{1}{\sqrt{h-q(N)}}  
-2\int\limits_0^{\sqrt{h-q(N)}}\frac{d^2y}{dq^2}(h-w^2)dw.  
\]  
This shows that $\tau^{'}(h)\to+\infty$ as $h\to q(N)+0$.  
We have  
\[  
\frac{d\rho^-}{dh}(h)=  
\frac{2\tau^{'}(h)\int\limits_0^1\frac{dx}{\sqrt{f(x)-h}}-  
\tau(h)\int\limits_0^1\frac{dx}{(f(x)-h)^{3/2}}}  
{\left(\int\limits_0^1\frac{dx}{\sqrt{f(x)-h}}\right)^2}.  
\]  
Therefore, $(\rho^{-})^\prime (h)>0$ if $h$ is sufficiently close to $q(N)$.  
  
Consider the closest to $N$ critical point $N_1$ of the function $q$  
and the  critical value $h_1=q(N_1)$.  
It is clear that $q(y)$ is strictly monotone on the interval $[N_1,N]$.  
Moreover, in view of (H$_6$), there exists $\delta >0$ such that   
\[  
\frac{dy}{dq}(h)=\frac{R(h)}{\sqrt{h_1-h}}\, , \quad h \in  
(h_1-\de,h_1]\, ,  
\]  
where the function $R(h)$ is continuous   
and  $R(h)<c_1<0$ on   
$(h_1-\de,h_1]$.  
We have  
\[  
\begin{array}{ccc}  
\tau(h)&=&\int\limits_{y_m(h)}^{N_1+\de}\frac{dy}{\sqrt{h-q(y)}}+  
\int\limits_{N_1+\de}^N\frac{dy}{\sqrt{h-q(y)}}\\  
&=&-2\int\limits_0^{\sqrt{h-q(N_1+\de)}}\frac{dy}{dq}(h-w^2)dw+r_1(h)\\  
&=&-2\int\limits_0^{\sqrt{h-q(N_1+\de)}}\frac{R(h-w^2)}{\sqrt{(h_1-h)+w^2}}dw+  
r_1(h),  
\end{array}  
\]  
where $r_1(h)$ is smooth in a neighborhood of $h_1$.  
Hence,  
\[  
\tau(h)>  
-2c_1\int\limits_0^{\sqrt{h-q(N_1+\de)}}\frac{1}{\sqrt{(h_1-h)+w^2}}dw=  
c_1\ln(h_1-h)+O(1).  
\]  
Therefore, $\lim\limits_{h\to h_1-0}\rho^{-}(h)=+\infty$.  
Analogously we prove $\lim\limits_{h\to h_1+0}\rho^{-}(h)=+\infty$.  
The same arguments can be applied to any of the points of $\Xi(q)$.  
This completes the proof of Proposition \ref{RotBihav1}. \finishproof  
  
\vspace{0.5cm}  
  
\noindent{\bf Case B).} Suppose that  $0<h<f(1/4)$.  
In this case the caustics of the set $T_h$ can be identified with  
the curves $\{x=x(h)\}$, where $x(h)$ are the solutions of the equation  
$f(x)=h$ in the interval $(0,1/2)$.  
Denote by $x_m(h)$ the minimal of them.  
The billiard ball map changes the connected components of the level set $S_h$.  
Denote by $\rho^{+}(h)$ the rotation function of the  
square of the billiard ball map.  
  
\begin{Prop}\hspace{-2.mm}.\label{RotFunc2} The rotation function   
$\rho^{+}(h)$ of the iterated billiard ball map  
on the interval $0<h<f(1/4)$ is  
\begin{equation}\label{Rot2}  
\rho^{+}(h)=\frac{\int\limits_{-N}^N\frac{dy}{\sqrt{h-q(y)}}}  
{\int\limits_{x_m(h)}^{1/2-x_m(h)}\frac{dx}{\sqrt{f(x)-h}}}.  
\end{equation}  
\end{Prop}  
\noindent{\em Proof.}  
Suppose that the iterated billiard ball map takes the  
point $(x_m(h),h)$ to $(x^{'}(h),h)$.  
As in the proof of Proposition \ref{RotFunc1} we show  
that  
\[  
2\int\limits_{-N}^N\frac{dy}{\sqrt{h-q(y)}}=  
2k\int\limits_{x_m(h)}^{1/2-x_m(h)}\frac{dx}{\sqrt{f(x)-h}}  
\pm\int\limits_{x_m(h)}^{x^{'}(h)}\frac{dx}{\sqrt{f(x)-h}},  
\]  
where $k$ is an integer.  
Therefore, the rotation function is given by  
formula (\ref{Rot2}). Proposition \ref{RotFunc2} is proved. \finishproof  
\begin{Coro}\hspace{-2.mm}.  
 The rotation function $\rho^+(h)$ is analytic on the interval $(0,f(1/4))$.  
\end{Coro}  
\begin{Remark}\hspace{-2.mm}.\label{RotBihav2}   
 Let $f\sim f(1/4)-\al_1(x-1/4)^2-\al_2(x-1/4)^4+...$ be  
the Taylor expansion of $f$ at $x=1/4$.  
As in the proof of Proposition \ref{RotBihav1} it can be seen that  
$\lim\limits_{h\to 0-0}\rho^{-}=1/2$,  
$\lim\limits_{h\to 0+0}\rho^{+}=1$ and  
$\lim\limits_{h\to f(1/4)-0}\rho^{+}=  
(\sqrt{\al_1}\int\limits_{-N}^N\frac{dy}{\sqrt{f(1/4)-q(y)}})/\pi$.  
\end{Remark}  
The existence and the exact values of these limits will not be used  
later  and we omit the proof.  
  
  
\vspace{0.5cm}  
  
\noindent{\bf Case C).} Let  $h=0$.   
Consider the coordinates $\{(x,p_1)\}$ on $T^*\Gamma$ described  
in Sect. \ref{Portrait}. The level set $S_0=\{{\cal I}(x,p_1)=0\}$   
consists of the points $A_1=(0,0)$ and $A_2=(0,1/2)$, and  
the arcs $S_0^\pm(0,1/2)=\{(x,\pm\sqrt{f(x)}) : x\in(0,1/2)\}$ and  
$S_0^\pm(1/2,1)=\{(x,\pm\sqrt{f(x)}) : x\in(1/2,1)\}$.   
The billiard ball map $B$ is defined and analytic in a regular neighborhood  
of the singular level $S_0$. It is clear that $B$ maps $S_0^+(0,1/2)$ to   
$S_0^+(1/2,1)$ and preserves the Leray's form $\la_0$ defined on them.   
Using $\la_0$ we introduce the variable $s=\int_{1/2}^x\la_0$ and  
identify $S_0^+(0,1/2)$ with $\bf R$. It is clear that $\la_0=ds$,   
and therefore, the square $B^2$ of the billiard ball map  
is simply a translation $s\mapsto s+\al$, where $\al$ is a constant.  
If $\al=0$ we obtain that all geodesics issuing from $S_0^+(0,1/2)$  
are periodic with the same primary length. If $\al\ne 0$   
the geodesics issuing from $S_0^+(0,1/2)$ are not periodic.   
We have proved  
\begin{Lemma}\hspace{-2.mm}.\label{singularlevel}  
For each $T>0$ there are at most finitely many $L\in(0,T)$ such  
that $L$ is the length of a closed broken geodesic issuing from   
$S_0$.  
\end{Lemma}  
  
\noindent  
{\em Proof of Proposition \ref{StringProperty}.}  
Consider a broken geodesic $\gamma$ in $X$ starting from  $F_1$ and denote by  
$P\in \Gamma$ its first point of contact with the boundary. Suppose  
that $\gamma$ is different from $\gamma_1$. Then the  
intersection points of $\gamma$ with   
$B^\ast\Gamma$ lie  in $S_0$ and they are different from  the critical  
points $A_1$ and $A_2$. In the coordinates $(x,y)$ in $C$ we have   
$F_1=(0,0)$, $F_2=(1/2,0)$, and we can suppose that    
$P=(x_1,N)$ with $0< x_1< 1/2$ (the case  $P=(x_1,-N)$, $0< x_1< 1/2$,  
is treated in the same way). Then $\gamma$ is given by  
the   solution  $\bar\gamma(\tau)= (x(\tau),y(\tau))$   
of the system  
$$  
\left\{  
\begin{array}{ccc}  
\frac{dx}{d\tau}&=&\sqrt{f(x)},\\  
\frac{dy}{d\tau}&=&\pm\sqrt{-q(y)},  
\end{array}  
\right.  
$$  
such that $\lim_{\tau\to -\infty}\bar\gamma(\tau )=(0,0)$, and there  
exists $\tau_0\in {\bf R}$ such that $y(\tau_0)=N$, $y'(\tau)>0$ for   
$\tau<\tau_0$ and   $y'(\tau)<0$ for $\tau>\tau_0$. We have to prove  
that  $\lim_{\tau\to +\infty}\bar\gamma(\tau )=(1/2,0)$. Using that   
$-q(y)= Cy^2(1 + O(y^2))$, $C>0$, near $y=0$, we obtain that   
$\displaystyle\lim_{\tau\to +\infty}y(\tau )=0$.   
In the same way we prove that   
$\displaystyle\lim_{\tau\to +\infty}x(\tau )=1/2$. Differentiating  
with respect to $P\in \Gamma$ we prove that the sum of distances from  
$P\in \Gamma$ to $F_1$ and $F_2$ is   
$|\displaystyle\widehat{F_1P}| +|\displaystyle\widehat{F_2 P}|  =  
C_0$, where $C_0>0$ is a positive constant. \finishproof  
 
Suppose that (H$_4$) holds. 
As we have seen in \ref{Sec:details} (Case A), the caustics of  
the billiard trajectories can be identified with the curves  
$\{y=y(h)\}$ on  $C$, where $y(h)$ is a positive solution of  
the equation $q(y)=h$ and $h$ is a fixed real number in  
$(q(N),0)$. 
If $h$ is close to $q(N)$,  the curve $\{y=y(h)\}$ (defined uniquely) 
is a boundary of another Liouville billiard table $(
X_h,g|_{X_h})$, 
$X_h \subset X$, defined by functions $f_h$ and 
$q_h$ such that  $f_h \equiv f$ and $q_h$  
is given by the restriction of  
the function $q$  on the interval $[-y(h),y(h)]$. 
It is clear that $X_h$ and $X$ share the same caustics 
$\Ga_{h'}=\{y=y(h')\} \subset X_h$, $h<h'$, near the  
boundary $\Ga_h$ of $X_h$. 
Hence, we have proved the following:   
\begin{Prop}\hspace{-2mm}{.}\label{StrongEvolutionProperty}  
Near the boundary, Liouville billiard tables 
satisfying (H$_4$)  
inherit from the ellipses  
the so-called {\em strong evolution} property.  
It means that {\em a caustic of a caustic is again a caustic}.   
\end{Prop}  

  
  
  
  
  
  
\subsection{Length spectrum}  
Let $P_1$ be a point in  $\Gamma$ with coordinates $(x_1,N)$ on the cylinder  
$C$. Fix a value $h \notin \Xi_(q)$ of the integral  
${\cal I}(x,p_1)$ of the billiard ball map   
and consider the geodesic $\ga_{(P_1,h)}(t)$ such that  
$\ga_{(P_1,h)}(0)=P_1$, $I(\gamma(0),\dot\ga(0))=h$ and  
$g(\dot\ga(0),\partial_x)\ge 0$. Denote by $P_2=(x_2,N)$  
the next  point of intersection of $\ga$ with $\Gamma$.  
Let $l(P_1,h)$ be the length  
of the geodesic segment of $\ga$ connecting $P_1$ with $P_2$.  
We are interested in the behavior of $l(P_1,h)$ as $h$ approaches the  
exceptional set $\Xi_(q)$.   
  
  
\begin{Lemma}\hspace{-2.mm}.\label{Lengths} We have:   
\begin{itemize}  
\item[a)] $l(P_1,h)\to\infty$ as $h$ tends to $h_i\pm 0$, $h_i\in \Xi(q)$;  
\item[b)] For each $0<\ep<h_1$ there is $C_{\ep}>0$ such that   
$l(P_1,h)>C_{\ep}$ for each  $h>q(N)+\ep$.  
\end{itemize}  
\end{Lemma}  
\noindent{\em Proof. }  
Suppose first that $h<0$.  
It follows from $\ref{eq:caustic}$ that  
\begin{equation}  
\begin{array}{rcll}   
l(P_1,h)&\eqdef&\int\limits_0^{\tau_0}  
\left\{(f(x)-q(y))\left(\left(\frac{dx}{d\tau}\right)^2+  
\left(\frac{dy}{d\tau}\right)^2\right)\right\}^{\frac{1}{2}}d\tau\\  
&=&\int\limits_0^{\tau_0}  
\{(f(x)-q(y))(|f(x)-h|+|h-q(y)|)\}^{\frac{1}{2}}d\tau\\  
&=&\int\limits_0^{\tau_0}(f(x)-q(y))d\tau=\int\limits_0^{\tau_0}f(x)d\tau-  
\int\limits_0^{\tau_0}q(y)d\tau\\  
&=&\int\limits_{x_1}^{x_2}\frac{f(x)}{\sqrt{f(x)-h}}dx-  
2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dx\\  
&\ge&-2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dy.  
\end{array}  
                                                   \label{LengthFormula}  
\end{equation}  
Similarly, if $h>0$, then  
$l(P_1,h)\ge-\int\limits_{-N}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$.  
The integral  
$l_0^{-}(h)\eqdef-2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$  
is a positive continuous function of $h$ on $[q(N)-\ep,0]\setminus\Xi(q)$.  
Moreover, it can be easily seen that  
$\lim\limits_{h\to h_i\pm 0}l_0^{-}=\infty$ and  
$\lim\limits_{h\to 0-0}l_0^{-}=l_0>0$.  
The integral  
$l_0^{+}(h)\eqdef-\int\limits_{-N}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$ is  
a positive continuous function of $h$ on $[0,f(1/4)]$.  
This completes the proof of  Lemma \ref{Lengths}. \finishproof  
  
Let $h\notin\Xi(q)$ be a regular value of the integral   
${\cal I}(x,p_1)$ and let  
the rotation number $\rho^{\pm}(h)\in {\bf Q}$ be rational.   
Then for each  $\nu \in S_h$ there is a  closed broken  
bicharacteristic issuing from $\nu$ and we denote by  $\gamma_\nu$ the  
corresponding primitive broken closed geodesic.   
It follows from the definition of the rotation number (see the  
beginning of Sect. 4) that the number  
of vertices of $\gamma_\nu$ is independent of the choice of $\nu$    
on $S_h$ and  the  length function $\nu \to L(\gamma_\nu)$ is  
continuous in $S_h$.     
Since  the length spectrum ${\cal L}(X,g)$ (the set of lengths of all closed  
generalized geodesics) of the   billiard table $(X,g)$ has measure $0$  
in ${\bf R}$, we obtain that  the continuous function    
$\nu \to L(\gamma_\nu)\in {\cal L}(X,g)$  
is constant on each connected component of $S_h$. If $h<0$ and the broken  
closed geodesic $[0,L(\gamma)]\ni t \to \gamma(t)$ issues from one of  
the components of $S_h$  
then $[0,L(\gamma)]\ni t \to \gamma(L(\gamma)-t)$ issues from the other  
component. In particular, the primary  length of the closed broken  
geodesics issuing from $S_h$ is constant and we denote it by $l(h)$.   
The same is true for $h\in (0,f(1/4))$ since $B$ interchanges the  
connected components of $S_h$ in this case.    
   
  
  
We consider now the closed  broken geodesics issuing from  $S_h$ for   
$h$ close to $0$ and to $f(1/4)$. Denote by   
${\cal L}_b(X,g)$ the set of  lengths $L(\gamma)$   
of all closed generalized geodesics $\gamma$ having at least one common point  
with the boundary. By definition the set $L(\gamma)\, {\bf N}^\ast$ is  
contained also in ${\cal L}_b(X,g)$.   
\begin{Lemma}\hspace{-2.mm}.\label{Lengths2}  
There exist neighborhoods $U_0$ and $U_1$ in ${\bf R}$ of $0$ and $f(1/4)$  
respectively such that for each $T>0$ there are at most  finitely many  
$L \in {\cal L}_b(X,g)\cap (0,T)$ such that $L$ is the length of  
a closed broken geodesic issuing from $S_h$ with $h\in  
U_0 \cup U_1$.   
\end{Lemma}  
  
\noindent{\em Proof. }  
We shall first prove the Lemma for  
$h$ in a neighborhood of 0. In view of Lemma \ref{singularlevel} we can exclude  
the singular level $S_0$ from our consideration.  
     
  
    
Suppose that there is an infinite sequence $\{x_j\}_{j=1}^\infty$  with   
$\displaystyle \lim_{j\to\infty}x_j =0$ such that the geodesics issuing  
from the level sets $S_{x_j}= \{{\cal I}(x,p_1)=x_j\} \subset  
B^\ast \Gamma$ are all  closed ($\rho^{\pm}(x_j) \in {\bf Q}$)   
and $\displaystyle \lim_{j\to\infty}l(x_j) = T_0$. Denote be    
 $n_j$  the number of  vertices of  the   primitive closed geodesic   
issuing from  $\nu \in S_{x_j}$, i.e.   
the smallest positive integer such that  
$B^{n_j}(\nu) = (\nu)$, which is independent  
of $\nu$.   
Since the set of lengths $l(x_j)$, $j\in {\bf N}$,  is bounded,  
using Lemma 4.9 b),  we obtain that     
the set $\{n_j\}_{j\in {\bf N}}$,  
is bounded as well. Hence, we can suppose that $n_j=n$ does not depend  
on $j\ge 1$.  
  
   
  
Choose sufficiently small     
neighborhoods $W_0\subset W$ in $B^\ast \Gamma$   
of  one of the critical points, say $A_1$,  
such that $B^n(W_0) \subset W$.    
Then $B^n$ is  
analytic in $W_0$ and $B^n(q,p)=(q,p)$ for each $(q,p)\in S_{x_j}\cap W_0$.  
On the other hand, by  Proposition \ref{Morse}, $A_1$ is   
a non-degenerate critical point of ${\cal I}(q,p)$ of index $1$   
(hyperbolic), and  we can  provide $W$ with analytic local coordinates  
$(q,p)\in {\bf R}^2$ such that $(q,p)(A_1)=(0,0)$ and ${\cal I}(q,p) = q^2 -  
p^2$ in $W$.  Then    
there is an open cone $\Gamma_0$ with a vertex at $(0,0)$ such that  
any ray in $\Gamma_0$ starting from the origin intersects infinitely  
many level curves $S_{x_j}\cap W_0$. This implies $B^{n}(q,p)=(q,p)$  
in $W_0$. Since $B$ is analytic in the connected component $U$ of $A_1$  
in the complement of $\cup_{j=1}^l S_{h_j}$, $h_j\in \Xi(q)$,    
in $B^\ast \Gamma$, we obtain   
$B^{n}(\nu)=\nu$ for each $\nu \in U$. Moreover,  the length of the  
closed geodesic corresponding to the periodic orbit $\{\nu,  
B(\nu),\ldots, B^{n-1}(\nu)\}$ of $B$ does not depend on $\nu \in U$,  
and  we denote it by  $T$.     
Taking into account  Lemma \ref{Lengths}, a), we obtain that  
$\Xi(q)=\emptyset$.  Hence,  $B^{n}$  
is the identity mapping in $B^\ast \Gamma$.   
This contradicts to the strict geodesic  
convexity of $\Gamma$.    
The same is true in the case when $h$ is close to $f(1/4)$ since  
the corresponding critical point is non-degenerate elliptic.  \finishproof  
  
  
We investigate  now the lengths of the closed broken   
geodesics approximating $\Gamma$ with winding number $m=1$.  
Given a value $h$ of the integral ${\cal I}(x,p_1)$, close to $q(N)$, and such  
that $\rho^-(h)$ is rational, we denote by $l(h)$ the length of the   
primitive closed broken geodesics issuing from $S_h$.   
Consider a sequence $\{p_k\}_{k=1}^{\infty}$   
tending to $q(N)$ and such that  
$\rho^{-}(p_k)=\frac{1}{k}$, $k\ge M_0$,   
where $M_0 \gg 1$ is a fixed natural number.  
\begin{Lemma}\hspace{-2.mm}.\label{Popov}  
There is  $M_0 \gg 1$ such that for each $k\ge M_0$,   
\[  
l(p_k)<l(p_{k+1})<L(\Ga) \quad {\rm and} \quad   
l(p_k)\to L(\Ga) \ {\rm as} \  
k\to\infty\, .  
\]  
Moreover, if $\rho^{-}(h)=\frac{m}{n}<1/M_0$, $(m,n)=1$, and  
$m\ge 2$, then $l(h)> L(\Ga)$.  
\end{Lemma}  
{\em Proof.}   
Set $t_0 = L(\Gamma)/2\pi$ and   
denote by $(r, \varphi)\in [t_0 -\delta_0,t_0]\times ({\bf R}/2\pi {\bf Z})$   
the ``action-angle'' coordinates corresponding to  
the smooth foliation of invariant ``circles'' $S_h$ for $h \ge q(N)$ and  
close to $q(N)$. The billiard ball map is given by   
$B(r,\varphi) = (r, \varphi + \mu(r))$ for   
$r \in [t_0 -\delta_0,t_0]$, where $\mu$ is   
smooth in the open interval and continuous in the closed one.   
Taking into account  that  $\Gamma$ is strictly    
geodesically convex, we are going to  
show  that  $\mu$ is given by the derivative $\mu(r)= \tau'(r)$,  
where $\tau(r)=- Q(r)^{3/2}$ for $r\in [t_0-\delta_0,t_0]$,  $Q$ is  
smooth in a neighborhood of $r=t_0$ and $Q(t_0)=0$, $Q'(t_0)<0$.  
To this end we    
make use of  an approximate interpolating Hamiltonian $\zeta$ of  
$B$ \cite{kn:MM}. The function $\zeta$ is smooth in a neighborhood of  
$t_0$,  $\zeta(t_0,\varphi)=0$, $\partial\zeta/\partial  
r (t_0,\varphi)<0$ and $R(r,\varphi)=B(r,\varphi) - \exp  
(H_{-\zeta^{3/2}})(r,\varphi)$ can be extended as a  smooth function  
across $t_0$ and it  vanishes  to infinite order at  
$r=t_0$. Here   
\[ 
\exp(H_{-\zeta^{3/2}})(r,\varphi)=  
\left(r,\varphi - {3\over 2}\zeta(r,\varphi)^{1/2}{\partial  
\zeta \over{\partial r}} (r,\varphi) + O((t_0-r))\right) 
\] 
stands for the  
time-one-flow of the Hamiltonian $-\zeta^{3/2}$. We are going to prove 
 that $\zeta(r,\varphi) = \zeta(r,0)(1 + R_0(r,\varphi))$, where  
$R_0$ is smooth and vanishes  to infinite order at  
$r=t_0$. Obviously, we have  
$\mu(r)=- {3\over 2}\zeta(r,\varphi)^{1/2}{\partial  
\zeta \over{\partial r}} (r,\varphi)(1  + O((t_0-r)^{1/2})$.  On the 
other hand,   
$B^\ast \zeta = \zeta + O(\zeta^\infty)$, and  we get   
\[  
\zeta (r, \varphi + \mu(r))  
= \zeta(r,\varphi)  
+ O((t_0-r)^\infty) \, .  
\]  
Expanding the smooth function $\zeta$ in Fourier  series   
$\zeta(r,\varphi) = 
\sum_{k\in {\bf Z}} f_k(r)e^{ik\varphi}$, we obtain  
\[ 
f_k(r)(e^{ik\mu(r)} -1)= O_k((t_0-r)^\infty) 
\]  
which implies 
$f_k(r)= O_k((t_0-r)^\infty)$ for $k\neq 0$. Since $f_k$ is smooth we get 
$(d^jf_k/dr^j)(t_0)=0$ for each $j\ge 0$,  $k\neq 0$. Hence    
$\zeta(r,\varphi) - \zeta(r,0)$  vanishes to 
infinite order at $r=t_0$ and  
$ Q(r) = \zeta(r,0) + O((t_0-r)^\infty)$ is smooth in  
$[t_0 -\delta_0,t_0]$.  
  
  
Given an invariant circle $S_h$ of $B$ with a rational rotation number  
$\rho^-(h) =\rho = m/n$, where $m,n\in {\bf N}$, $(m,n)=1$, and  $\rho$ is  
close to $0$, we consider the common primitive length $\ell(\rho) 
=l(h)$ of the  
 family of broken geodesics issuing from $S_h$.   
Using a simple argument about the  
symplectic invariance of the   
length spectrum  (see \cite{Po}, Sect. 4, (4.2)-(4.6), with $Q_0=0$),      
we obtain  that   
\[  
\frac{\ell(\rho)}{2\pi m}\ =\  \frac{{\cal S}(\rho)}{\rho}\, ,  
\]   
where   
${\cal S}(\rho)$ is the Legendre  transformation of $\tau(r)/2\pi$.  
In other words,    
\[  
{\cal S}(\rho) \ =\   
 r(\rho)\rho - \tau(r(\rho))/2\pi\, ,  
\]  
where   
$[0,\varepsilon_0]\ni\rho \rightarrow r(\rho)\in  
[t_0-\delta_0,t_0]$, $r(0)=t_0$,  is the inverse function to the   
``frequency map'' $r \rightarrow \tau'(r)/2\pi$.   
It is easy to see that   
\[  
{d\over{d\rho}} \left ({{\cal S}(\rho)\over \rho} \right ) =       
{\tau(r(\rho))\over {2\pi \rho^2}}\ < \ 0 \,  ,    
\]  
which implies the firs statement.  
To prove the second one, we observe that   
\[ 
r(\rho) = t_0 - C\rho^2(1 + O(\rho))\ ,\ \tau(r(\rho))/\rho =   
- C_0\rho^2 (1 + O(\rho))\, ,\ C, C_0>0,\   \mbox{as}\  \rho \to +0 . 
\] 
Then  $\ell(\rho) = 2\pi m \,  {\cal S}(\rho)/\rho =  
m(L(\Gamma) - \rho^2 s(\rho))$, where $s$ is  
smooth in a neighborhood of $0$ and  independent of $m$,    
which completes the  
proof of the Lemma.  \finishproof  
  
  
  
\section{Proof of the main theorem}  
\setcounter{equation}{0}  
First we recall some facts about the singularities of the distribution  
$Z_K$ given by (\ref{trace}).   
It is known  that the singular support of $Z_K(t),\ t>0$, is contained  
in the length spectrum ${\cal L}(X,g)$ of the corresponding billiard  
table. We denote also by $Z_0(t)$ the corresponding distribution for the  
Laplace-Beltrami operator with Neumann boundary conditions. As in   
\cite{GM} and \cite{GM1} we consider the distribution  
$\sigma(t)=Z_K(t) - Z_0(t)$, $t>0$.   
We observe that the singular support of $\sigma$ on ${\bf R}_+$ is  
contained in the set ${\cal L}_b(X,g)$ which consists of the lengths  
of all closed generalized geodesics having at least one common point  
with the boundary. More precisely, we prove that the contribution in  
$\sigma$  of  
any closed geodesic $\gamma$ lying entirely in   
$\displaystyle\mathop X^\circ$ is  
$C^\infty$.   
Indeed, let $B$ be a  
pseudo-differential operator of order $0$ the wave-front of which is  
contained in a small conic neighborhood of $\gamma$ in $T^\ast   
\displaystyle\mathop X^\circ \setminus \{0\}$   
and let $\zeta\in C_0^\infty ({\bf R})$  have   
support in a small neighborhood of $L (\gamma)$.   
Consider the distribution   
\[  
\sigma_B(t) = {\rm tr}\, (\cos(t\sqrt {\Delta_K})\, B) -   
{\rm tr}\, (\cos(t\sqrt {\Delta_D})\, B)\, ,  
\]  
where  $\Delta_D$ is the Laplace-Beltrami operator in $X$ with Dirichlet  
boundary conditions. Since the parametrices of $\cos(t\sqrt  
{\Delta_K})B$ and $\cos(t\sqrt {\Delta_D})B$ differ by a smooth function  
for $t\in [-T,T]$, $T>L (\gamma)$, choosing the  
wave-front set of $B$ sufficiently small, we obtain that $\sigma_B$ is  
a smooth function in that interval.   
  
  
  
  
  
Consider now a regular level $S_h$ of the integral ${\cal I}$,   
where $h\notin\Xi(q)$ is  
a fixed real number in the interval $q(N)<h<f(1/4)$.  
It is clear that if the rotation number $\rho^{-}(h)$ is rational,  
say $\rho^{-}(h)=\frac{m}{n}$, $0 \le m <  n$,  
then all the geodesics issuing from $S_h$ are closed. The  
corresponding primitive geodesics have  $n$ vertices,  
their winding number is $m$, and they have  
the same primary length $l(h)$. Moreover,   
if $h$ is sufficiently close to $q(N)$, then   
$d\rho^{-}/dh>0$ in a neighborhood of $h=q(N)$  
(see Proposition \ref{RotBihav1}), which implies  that $S_h$ is a  
{\it clean submanifold} for the iterated billiard ball map $B^n$.  
If there are no other broken geodesics of  length $l(h)$,  
using Theorem 4.2, \cite{GM}, we recover from the leading term of  
the asymptotic expansion of $\sigma(t)$ at $t=l(h)$ the integral   
\[  
M(h)\eqdef\int\limits_{S_h}\frac{K}{\cos\phi}d\mu_{S_h}\, .   
\]  
In the last formula  $\phi$ is the angle between the initial vector of the  
corresponding  geodesic issuing form $S_h$   
and the inward normal to the boundary of  
the billiard at the initial point of the geodesic. The measure  
$\mu_{S_h}$ on $S_h$ coincides (up to multiplication with a constant) with  
the Leray form $\la_h$ defined in Sect. 4.1.  
It is easy to see that $\cos\phi=\sqrt{\frac{h-q(N)}{f(x)-q(N)}}$.  
Therefore,  
\begin{equation}\label{invariant}  
M(h)=\frac{1}{\sqrt{h-q(N)}}  
\int\limits_0^1\frac{K(x)}{\sqrt{f(x)-h}}\sqrt{f(x)-q(N)}\;dx  
\end{equation}  
is a spectral invariant.  
\begin{Lemma}\hspace{-2.mm}.\label{sequence}  
There exists a strictly monotone sequence $\{p_k\}_{k=1}^{\infty}$   
such that  
\begin{itemize}  
\item[a)] $p_k\to q(N)+0$ as $k\to\infty$;  
\item[b)] the  geodesics issuing from $S_{p_k}$ are closed and  
$l(p_k)\ne L(\Gamma)$;  
\item[c)] the primitive closed geodesics issuing  
from $S_{p_k}$ are the only closed broken geodesics  
in  $X$ with length $l(p_k)$.  
\end{itemize}  
\end{Lemma}  
\noindent{\em Proof.}  
The lemma follows   from the properties of the  
Liouville billiard tables  proved in the previous sections.  
  
Given  $h>q(N)$  close to $q(N)$  and such that $\rho^-(h)$ is 
rational, we  
consider the closed broken geodesics  $\gamma$ issuing from $S_{h}$  
and denote by $L(\gamma)$ the length of  $\gamma$  
(then $h=I(\gamma(t),\dot\gamma(t)))$.  
Using   Lemma \ref{Popov} we choose $M_0$ such that  
the geodesics $\gamma$ issuing from $S_{p_k}$, $p_k = 1/k$,   $k\ge 
M_0$,  
are the only closed geodesics  satisfying  
\[ 
L(\gamma) \in [l(p^0), L(\Ga))\ ,\   
h=I(\gamma(t),\dot\gamma(t)) \in (q(N),p^0]\ ,\ p^0 =p_{M_0}. 
\] 
Obviously  the sequence $\{p_k\}_{k\ge M_0}$ satisfies items $a)$ and  
$b)$ of Lemma \ref{sequence}.  
In view  of Lemma \ref{Lengths}  
there exist  $\de_i>0$ $(i=1,...,l)$ such that for each   
$h\in (h_i-\de_i,h_i+\de_i)$ the length $l(h)>L(\Ga)$.  
Denote by $K$ the set $U_0\cup U_1\cup_j(h_j-\de_j,h_j+\de_j)$  
where $U_0$ and $U_1$ are the neighborhoods from Lemma \ref{Lengths2}.  
To prove $c)$ it is sufficient to show that there are only  
finitely many $k_1,...,k_r\in (p^0,f(1/4)]\setminus K$ such that  
$\rho^\pm(k_j)\in{\bf Q}$ and $l(k_j)\in [l(p^0),L(\Gamma))$.  
Taking $\ep$ such that $0<\ep\le p^0<h_1$ and applying   
the second statement of Lemma \ref{Lengths},  we find $J_0>0$ such  
that if the number of the vertices of a closed geodesic $\gamma$   
is greater than   
$J_0$ and   $h=I(\gamma(t),\dot\gamma(t)) \in (p^0,f(1/4)]$,   
then $L(\gamma)>L(\Ga)$.  
Therefore, if $h\in(p^0,f(1/4)]\setminus K$,    
$l(h)<L(\Ga)$, and $\rho^{\pm}(h)$ is rational,   
then  
\begin{equation}\label{RotFuncEq}  
\rho^{\pm}(h)=\frac{u}{v}\;(\mod 1)\ {\rm with}\  v<J_0.  
\end{equation}  
Relation (\ref{RotFuncEq}) gives a finite number of equations on $h$.  
The rotation function $\rho^{\pm}(h)$ is analytic on  
$(p^0,f(1/4)]\setminus K$ and  
we obtain that each equation in  (\ref{RotFuncEq})  
has only finitely many solutions $h$ in that set.  
Therefore, there is only a finite number of values  
$k_1,...,k_r\in(p^0,f(1/4)]$ such that  
$l(k_i)\in[l(p^0), L(\Ga))$. Choosing $M_0$ sufficiently big we 
complete the proof of Lemma \ref{sequence}.    
\finishproof  
  
  
Suppose that ${\rm Spec}\, (\Delta_{K_1})={\rm Spec}\,  
(\Delta_{K_2})$,   
where $K_1$ and $K_2$ are  
smooth functions on $\partial X$ invariant under the action of the group $G$.  
It follows from (\ref{invariant}) that  
\[  
\int\limits_0^1\frac{K(x)}{\sqrt{f(x)-p_k}}\sqrt{f(x)-q(N)}\;dx\equiv 0,  
\]  
where $K =  K_1-K_2$.  
Denote by $\Omega$ the complement of $[0,f(1/4)]\cup \{z\in {\bf C}:\,  
{\rm arg}\, z=3\pi/2\}$ in ${\bf C}$ and consider the analytic function  
\[  
V(h)\eqdef\int\limits_0^1\frac{K(x)}{\sqrt{f(x)-h}}\sqrt{f(x)-q(N)}\;dx  
\, , \quad h\in \Omega.  
\]  
Then $V(h)\equiv 0$, and we obtain   
\[  
V_1(h) \eqdef \int\limits_0^{1/4}\frac{K(x)}{\sqrt{f(x)-h}}  
\sqrt{f(x)-q(N)}\;dx \equiv 0\, .  
\]  
We are going to show that  $K$ has zeros  
at $x=0,\frac{1}{4},\frac{1}{2}$ and $x=\frac{3}{4}$. Let $K(x)= K_0 +  
O(x)$ as $x\to 0$. Since $f(x) = f_1x^2 + O(x^3)$, $f_1>0$, using the 
identity  $V_1(h) \equiv 0$,  
we get for $h<0$   
\[  
0\ =\  K_0\, \int_0^1 \,   
\frac{ds}{\sqrt{s^2  
-h}}\, + \, O(1)\ =\ - \frac{K_0}{2}\, \log(-h) + O(1)\, ,  
\]  
which implies $K_0=0$. In the same way we prove that $K(1/4)=0$.  
Then we have  
\[  
V_1(h) = \int\limits_0^{f(1/4)}  
\frac{{\widetilde K}(t)}{\sqrt{t-h}}\;dt\equiv 0  
\]  
where $t=f(x)$ and ${\widetilde K}(t)=K(x)\sqrt{f(x)-q(N)}/f^{'}(x)$  
 is continuous on the interval $[0,f(1/4)]$.  
Finally, the arguments used in \cite{GM} show that $K\equiv 0$.  
Indeed, differentiating the function $V_1(h)$ with respect to $h$ at  
the point $h=q(N)$ we obtain  
\[  
\int\limits_0^{f(1/4)}\frac{{\widetilde K}(t)}{\sqrt{t-q(N)}}  
(t-q(N))^{-k}\;dt=0,  
\]  
for $k=0,1,...$.  
The last equality and the Stone-Weierstrass theorem show that  
${\widetilde K}\equiv 0$.  
This proves Theorem \ref{MainTh}.  
  
  
  
  
  
  
\begin{thebibliography}{99}  
\bibitem{Birkhoff} {\sc G. Birkhoff}:   
{\em Dynamical systems}, Amer. Math. Soc. Colloq. Publ. $\bf 9$,  
AMS, New York, 1927  
  
\bibitem{Bolotin} {\sc S. Bolotin}:  
{\em Integrable billiards on surfaces of constant curvature},  
Math. Notes $\bf 51$(1992), no. 1-2, 117-123  
  
\bibitem{B-F} {\sc V. Bolsinov, A.Fomenko}:  
{\em Integrable geodesic flows on two dimensional surfaces},  
Monographs in Contemporary Mathematics, Consultants Bureau, New York,  
2000  
  
\bibitem{C-S} {\sc S. Chang, K. Shi}:   
{\em Billiard systems on quadric surfaces and the Poncelet theorem},   
J. Math. Phys $\bf 30$(4),  
1989  
  
\bibitem{Drizzi} {\sc B. Drizzi, B. Grammaticos, A. Kalliterakis}:  
Integrable curvilinear billiards, Phys. Lett. A., $\bf 115$(1986),  
25-28   
  
\bibitem{GM} {\sc V.Guillemin and R.Melrose}:   
{\em An inverse spectral result for  
elliptical regions in ${\bf R}^2$},   
Advances in Mathematics, $\bf 32$, 128-148, 1979.  
  
\bibitem{GM1} {\sc V.Guillemin and R.Melrose}:   
{\em The Poisson summation formula for manifolds with boundary},   
Advances in Mathematics, $\bf 32$, 204-232, 1979.  
  
\bibitem{Hor} {\sc L. H\"{o}rmander}:    
{\em The Analysis  of  Linear  Partial   
Differential Operators}, Vol. III,  Berlin -    
Heidelberg  -  New  York:  Springer,  1985.  
  
\bibitem{Kiyo1} {\sc K.Kiyohara}:   
{\em Compact Liouville surfaces},  
J. Math. Soc. Japan $\bf 43$(1991), 555-591  
  
\bibitem{Kiyo2} {\sc K.Kiyohara}:   
{\em Noncompact Liouville surfaces},  
J. Math. Soc. Japan $\bf 45$(1993), 459-479  
  
\bibitem{Kiyo3} {\sc K.Kiyohara}:   
{\em Two Classes of Riemannian   
Manifolds Whose Geodesic Flows Are Integrable},   
Memoirs of the AMS, Vol.$\bf 130$, Number 619(1997).  
  
\bibitem{Kolokol} {\sc V. Kolokol'tsov}:  
{\em Geodesic flows on two-dimensional manifolds with an  
additional first integral that is polynomial with   
respect to velocities} (Russian)   
Izv. Akad. Nauk SSSR Ser. Mat. $\bf 46$(1982), no. 5,   
994-1010, 1135  
  
\bibitem{kn:MM} {\sc Sh. Marvizi,  R. Melrose}:   
{\em Spectral invariants of convex planar regions},   
J. Differ. Geom. {\bf 17}, 475-502 (1982).  
  
\bibitem{M-T} {\sc  V. Matveev, P. Topalov}:   
{\em Geodesic equivalence of metrics on surfaces as integrability},   
Dokl. Akad. Nauk, $\bf 367$(1999), no. 6.  
English translation to appear in Russian Acad. Sci. Dokl. Math.  
  
\bibitem{Po}  {\sc G. Popov}:  
{\em Invariants of the length spectrum and spectral invariants for convex   
planar domains}, Commun.  Math. Phys. {\bf 161}, 335-364 (1994).  
  

  
\bibitem{Poritsky}  {\sc Poritsky, H}:   
{\em The billiard ball problem on a table with convex boundary -  
an illustrative dynamical problem},  
Annals of Mathematics, $\bf 51$, no.2, 1950  
  
\bibitem{Tabach} {\sc Tabachnikov, S}:  
{\em Billiards}, Panoramas et Syntheses,  
Societe Mathematique de France, 1995  
  
\bibitem{Veselov} {\sc A. Veselov}: Confocal surfaces and integrable  
billiards on the sphere and in the Lobachevsky space, J. Geom. Phys.,  
$\bf 7$(1990), 81-107  
  
\bibitem{Whittaker} {\sc E.Whittaker}:   
{\em A treatise on the Analytical Dynamics  
of particles and rigid bodies}, Cambridge University Press, 1964.   
\end{thebibliography}  
  
\vspace{0.5cm}  
\noindent  
G. P.:  
Universit\'e de Nantes, D\'epartement de Math\'ematiques,     
UMR 6629 du CNRS, 2, rue de la Houssini\`ere, BP 92208,  44072 Nantes    
Cedex 03, France  
  
\vspace{0.5cm}  
\noindent  
P.T.:  
Institute of Mathematics, BAS,  
Acad.G.Bonchev Str., bl.8,  
Sofia 1113,  
Bulgaria  
  
  
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