Content-Type: multipart/mixed; boundary="-------------9901070306253"

This is a multi-part message in MIME format.

---------------9901070306253
Content-Type: text/plain; name="99-4.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="99-4.keywords"

globally coupled maps, exchangeable distributions,  
violation of the law of large numbers
---------------9901070306253
Content-Type: application/x-tex; name="gcml.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="gcml.tex"

\documentclass[10pt]{article}				%%% 30.11.98
\usepackage{amsfonts,amssymb,latexsym,a4wide,xspace}
\usepackage{amsmath,delarray}
\newcommand{\mypar}{\medskip\newline}
\newcommand{\myitem}{\item[$\triangleright$]}
 
\begin{document}
%
{
\newcommand{\nz}{{\mathbb N}}
\newcommand{\rz}{{\mathbb R}}
\newcommand{\tz}{{\mathbb T}}
\newcommand{\eps}{\epsilon}
\newcommand{\ph}{\varphi}
\newcommand{\limn}{\lim_{n\to\infty}}
\newcommand{\liminfn}{\liminf_{n\to\infty}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\T}{{\cal T}}
\newcommand{\M}{{\cal M}}
\newcommand{\A}{{\cal A}}
\newcommand{\D}{{\cal D}}
\newcommand{\F}{{\cal F}}
\renewcommand{\L}{{\cal L}}
\renewcommand{\P}{{\cal P}}
\newcommand{\Q}{{\cal Q}}
\renewcommand{\S}{{\cal S}}
\newcommand{\var}{\operatorname{var}}
\newcommand{\nn}{\nonumber}
\newcommand{\qed}{\hfill$\Box$\bigskip\\\noindent}
\newcommand{\To}{\Rightarrow}
\newcommand{\const}{\operatorname{const}} 

\newcommand{\dbv}{\D}
\newcommand{\f}{f} 
\newcommand{\mf}{\bar f}
\newcommand{\Mf}{\bar F}
\newcommand{\eg}{{e.g.}\xspace} 
\newcommand{\ie}{{i.e.}\xspace}
\newcommand{\etc}{{etc.}\xspace} 
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega} 
\newcommand{\Omreg}{\Om_\nz^{\text{reg}}}
\newcommand{\bp}{Borel probability\xspace}
\newcommand{\bps}{Borel probabilities\xspace}
\newcommand{\Mex}{\M^{\mathit{ex}}}
\newcommand{\Ftail}{\F^{\mathit{tail}}}
\newcommand{\limi}{\lim_{i\to\infty}} 
\newcommand{\Perm}{\Sigma}
\newcommand{\perm}{\sigma} 
\renewcommand{\ae}{a.e.\xspace}
\renewcommand{\aa}{a.a.\xspace} 
\newcommand{\as}{a.s.\xspace}
\newcommand{\1}[1]{1_{#1}} 
\newcommand{\bvnorm}[1]{\|#1\|_{BV}}
\newcommand{\BV}{\mathit{BV}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} 
\renewcommand{\div}{\operatorname{div}}
\newcommand{\linf}{l^\infty}
\newcommand{\lI}{\linf(I)}
\newcommand{\inn}[1]{\overset{\circ}{#1}}
\newcommand{\Der}{{\rm D}}
%
\newtheorem {defi}{Definition} 
\newtheorem {lemma}{Lemma} 
\newtheorem{theo}{Theorem} 
\newtheorem {bemerkung}{Remark} 
\newtheorem{prop}{Proposition} 
\newtheorem {koro}{Corollary} 
\newtheorem{beispiel}{Example}
\newenvironment{remark} {\begin{bemerkung} \rm }{\end{bemerkung}}
\newenvironment{example} {\begin{beispiel} \rm }{\end{beispiel}}
\newenvironment{proof}[1]{{\em Proof{}#1:} }{\qed}
%
\title{
An ergodic theoretic approach to mean field coupled maps}
%
\author{
  Gerhard Keller\thanks{Research partially supported by the Deutsche Forschungsgemeinschaft (DFG).},
  Universit{\"a}t Erlangen}
\date{December 2, 1998}
%
\maketitle
\begin{center}
\textbf{Abstract}\medskip\\
\begin{minipage}{13cm}{{\small
We study infinite systems of globally coupled maps with permutation invariant interaction as limits of large finite-dimensional systems. Because of the symmetry of the interaction the interesting invariant measures are the exchangeable ones. For infinite systems this means in view of de Finetti's theorem that
we must look for time invariant measures within the class of 
mixtures of spatial i.i.d. processes. If we consider only those invariant measures in that class as physically relevant which are weak limits of SRB-measures
of the finite-dimensional approximations, we find for systems of piecewise expanding interval maps that the limit measures are in fact mixtures of absolutely continuous measures on the interval which have densities of uniformly bounded variation.\par
The law of large numbers is violated (in the sense of Kaneko) if a nontrivial mixture of i.i.d. processes can occur as a weak limit of finite-dimensional SRB-measures. 
We prove that this does neither happen for $C^3$-expanding maps of the circle (extending slightly a result of J\"arvenp\"a\"a) nor for mixing tent maps for which the critical orbit finally hits a fixed point (making rigorous a result of
Chawanya and Morita).
}}
\end{minipage}
\end{center}
%
\section{Introduction}
%
The purpose of these notes is to provide a precise mathematical
framework within which it is possible to describe the dynamics of
finite and infinite systems of {\em mean field coupled maps} as well
as the limiting transition from large finite to infinite systems.
Mean field coupled maps (also called {\em globally coupled maps} in
the literature) have the potential to show a surprising dynamical
behaviour which is sometimes referred to as {\em violation of the law
  of large numbers}, a term which conveys only insufficiently the
rich structure of these dynamics.  The following
observation is reported in the literature: If the action of an
infinite (for practical purposes: extremely large finite) system of
very simple chaotic dynamical units, \eg of identical tent maps, is
perturbed by its own mean field, then, in many cases, the mean field
itself shows a highly complex and structured dynamical behaviour,
although it is just the average state of a family of chaotic units
which, given the value of the mean field, act independently of each
other. In particular, there exists a ``component'' of the dynamics
which cannot be located, not even partially, in any finite
subcollection of the family of chaotic units.  This effect is
investigated, both numerically \cite{Kaneko2,Kaneko1,ErPo1,ChaMo,NaKo1,NaKo2} and
theoretically \cite{Just,ErPo1,ChaMo}, but on a mathematical level
much remains to be done in order to achieve a deeper understanding.
\par
I would like to advocate the following point of view: Since an infinite
system of mean field coupled maps is completely invariant under
arbitrary permutations of its components, its statistical properties
should be described by {\em exchangeable distributions} (in the sense
of probability theory). From the general theory of these distributions
\cite{Aldous} it follows at once that the mean field fluctuations of
the system reflect that part of the dynamics which can be described in
terms of the tail field of the infinite collection of units. So the
term {\em violation of the $0$-$1$-law} describes the situation much
better than {\em violation of the law of large numbers}. The second
aspect I want to underline is the importance to understand how this
violation of the $0$-$1$-law shines up in the behaviour of large
finite systems. In fact, it is known that for moderate coupling
strengths the dynamics of an arbitrary finite number of mixing coupled
tent maps is governed by a unique mixing absolutely continuous
invariant measure \cite{Keller-wien}, and only a good knowledge of the
infinite system can reveal something about the important dynamical
fine structure of this measure, \ie structure below the very coarse
concepts of ergodicity, mixing, \etc Needless to say that such ideas
are common knowledge in the stochastic theory of many particle
systems.
\par
At the end of this introduction I want to stress that some of the
ideas in these notes appear, although often only implicitly and without mathematical rigour, in the
existing literature, most noteworthy in \cite{ErPo1} and \cite{ChaMo}.
%
\section{Systems with continuous units}
\label{cont-units-sect}
%
In this section (and, in fact, in the whole paper) 
we assume that the individual maps are continuously acting on a compact metric space and that also the coupling mechanism is continuous.
This allows us to study exchangeable distributions as the natural class of measures that is left invariant by the action of the system and to describe the transition from finite to infinite systems in two complementary ways.
We finish this section with a precise definition of what we mean by 
\emph{violation of the $0$-$1$-law}.
%
\subsection{The uncoupled system}
\label{basic-sect}
%
$X$
is a compact metric space with metric $d$ and $\tau:X\to X$ is
continuous. The pair $(X,\tau)$ is our basic dynamical unit. For a
finite or countable index set $I$ let $\Om:=X^I$ and denote by
$T:\Om\to\Om$ the map defined by
\[
(T\om)_i=\tau\om_i\quad(i\in I)\ .
\]
$T$ describes the uncoupled dynamics of a family of copies of
$(X,\tau)$ indexed over $I$. If necessary we write $\Om_I$ and $T_I$
instead of $\Om$ and $T$. In 
order to avoid double subscripts $i_j$, we assume from now
on that $I=I_n=\{1,\ldots,n\}$ if $I$ is finite and that $I=\nz$ if
$I$ is infinite.
\par
Before we describe a general form of a mean field interaction which allows
us to define coupled maps $T_\eps:\Om\to\Om$, where $\eps$ is a parameter
that reflects the coupling strength, we discuss two particular examples
which motivated this paper.
%
\subsection{Two examples of mean field couplings}
%
Let $I=\{1,\dots,n\}$.
\begin{example}{\bf [Tent maps]}\label{tent-example}\\
$X=[0,1]$, $\tau$ is piecewise monotone, \eg $\tau$ might be a tent map, and 
\begin{equation}\label{DLap-coupling}
(T_\eps\om)_i=(1-\eps)\tau\om_i+\frac\eps n\sum_{j=1}^n\tau\om_j\quad
\mbox{for all $i\in I_n$.}
\end{equation}
Note that the coupling term
$\eps\,(-\tau\om_i+\frac 1n\sum_{j}\tau\om_j)$ at position $i$
can be written as a continuous function of $\tau\om_i$ and of
the empirical distribution $p_\om:=\frac 1n\sum_{j=1}^n\delta_{\om_j}$,
namely as $\eps\,(-\tau\om_i+\int_X\tau(y)\,dp_\omega(y))$. Since this term can be rewritten as $\eps\,(-\tau\om_i+\int_X y\,dp_{T\omega}(y))$ in this example (and in the next one as well), $T_\eps$ is the composition of two maps, namely
\begin{displaymath}
T_\eps=\Phi_\eps\circ T\quad\text{where}\quad
(\Phi_\eps\omega)_i=\omega_i+\eps\,(-\omega_i+\int_X y\,dp_\omega(y))\ .
\end{displaymath}
This is the kind of systems studied in
\cite{Just,ErPo1,ChaMo,NaKo1,NaKo2} for which the phenomenon of violation of the law of large numbers is reported. Observe that 
$\Phi_\eps:\Omega\to\Omega$ is not a diffeomorphism (it is not onto).
This, so far, prevents all attempts to study these examples using techniques from statistical physics.
\end{example}
%
\begin{example}{\bf[Expanding circle maps]}\label{circle-example}\\
$X=\tz^1$ (where $\tz=[0,2\pi)$ 
with addition$\mod 2\pi$), $\tau:X\to X$ of class $C^2$ (not only piecewise $C^2$), and 
\begin{equation}\label{circle-coupling}
(T_\eps\om)_i=\tau\om_i+\frac\eps n\sum_{j=1}^n \sin(\tau\om_i-\tau\om_j)\quad
\mbox{for all $i\in I_n$.}
\end{equation}
As $\sum_j \sin(\tau\om_i-\tau\om_j)=\sin\tau\om_i\sum_j\cos\tau\om_j-\cos\tau\om_i\sum_j\sin\tau\om_j$, the coupling term at position $i$ is again a 
continuous function of $\tau\om_i$ and of
the empirical distribution $p_{T\omega}$.
In this case the coupling transformation $\Phi_\eps:\Om\to\Om$, 
$(\Phi_\eps\om)_i=\om_i+\frac\eps n\sum_j\sin(\om_i-\om_j)$
is an analytic diffeomorphism of the torus $\tz^n$.
This fact plays a crucial role in \cite{Jarvenpaa} where, among many other results, it is proved that no violation of the law of large numbers occurs provided
that $\tau$ extends analytically to a complex annulus around $\tz^1$.
That proof relies heavily on techniques from statistical physics.
\footnote{The author studies more general analytic couplings, indeed.}
For related work see also \cite{Just}.
\end{example}
%
\subsection{A general model for mean field couplings}
\label{sec:general-coupling}
Next we describe a general form of a mean field interaction
motivated by the preceding examples.
The only requirement is that for finite $I$
the coupling term at position $i$ is a 
continuous function of $\om_i$ and of
the empirical distribution $p(\om):=\frac 1n\sum_{j=1}^n\delta_{\om_j}$.
\par
More precisely, denote by $\P$ the set of all \bps on $X$. $\P$ is a compact
metrizable space with respect to the topology of weak convergence.
Then each continuous map $S_\eps:X\times\P\to X$ defines a coupled
map
\begin{displaymath}
  T_\eps:\Om\to\Om,\quad(T_\eps\om)_i:=S_\eps(\tau\om_i,p(\om))\ .
\end{displaymath}
Sometimes it is convenient to consider the map $x\mapsto S_\eps(x,\nu)$ for some fixed $\nu\in\P$. Then we write $S_{\eps,\nu}(x):=S_\eps(x,\nu)$.
In Example \ref{tent-example} we used 
\begin{displaymath}
S_\eps(x,\nu)=x+\eps\cdot\left(-x+\int_X \tau(y)\,d\nu(y)\right)  \ ,
\end{displaymath}
in Example \ref{circle-example} it was
\begin{displaymath}
S_\eps(x,\nu)=x+\eps\cdot
\left(\sin(x)\,\int_X\cos(\tau y)\,d\nu(y)-\cos(x)\,\int_X\sin(\tau y)\,d\nu(y)\right)
\end{displaymath}
%
\begin{remark}
Note that in typical examples the family of maps $(S_\eps)_\eps$ will be defined only for a certain parameter interval including $\eps=0$, that in these cases $S_{0,\nu}=\Id_X$ for all $\nu\in\P$ and that the maps $S_\eps$ typically depend continuously on the parameter $\eps$. Most of our further analysis does not depend on these properties, however.
\end{remark}
%
The above chosen set-up extends naturally to the case of infinite $I$.
If $I=\nz$, $p:\Om_\nz\to\P$ can be partially defined by
\begin{displaymath}
  p(\om):=p_\om:=\limn\frac 1n\sum_{i=1}^n\delta_{\om_i}
\quad\mbox{if this limit exists in the weak topology.}  
\end{displaymath}
Denote the set of $\om$ where $p$ is defined by $\Omreg$. Then $\Omreg$ has full measure for each shift invariant Borel probability on $\Om_\nz$ and hence
{\em a fortiori} also for each exchangeable Borel probability on $\Om_\nz$. Exchangeability will be further discussed in Section \ref{exchangeable-sect}.
\par
In order to describe the essential symmetry property of
$T_\eps$ denote by $\Perm(I)$ the set of all permutations $\perm$ of $I$
such that $\perm(i)=i$ for all but finitely many $i\in I$. For
$\perm\in \Perm(I)$ define $\perm^*:\Om_I\to\Om_I$ by
$(\perm^*\om)_i:=\om_{\perm(i)}$ $(i\in I)$.  It is easily checked
that
\begin{equation*}
T_\eps\circ\perm^*=\perm^*\circ T_\eps\ .
\end{equation*}
%
\begin{remark}
  If $I$ is finite, then $T_\eps:\Om_I\to\Om_I$ is continuous with
  respect to the usual Euclidean metric on $\Om_I$. If $I=\nz$, we
  equip $\Om_I$ with the metric
  $d_I(\om,\om'):=\sum_{i=1}^\infty2^{-i}d(\om_{i},\om_{i}')$.  With
  this metric $T_\eps$ is not continuous, but it is Borel measurable.
\end{remark}
%
\subsection{The action of $T_\eps$ on probability measures on $X$}
\label{Teps-sect}
%
The action of $T$ and of $T_\eps$ can be transferred completely to $\P$.
Observe first that for finite and infinite $I$ holds
\begin{itemize}
\myitem
 $p(T\om)=\tau p_\om$ for all $\om\in\Om$,
\end{itemize}
and if $I=I_n$ is finite also the following is true:
%
\begin{itemize}
\myitem $p:\Om_I\to\P$ is continuous; so all $\P_n:=p(\Om_{I_n})\subseteq\P$ are
  compact.
\myitem $p\circ\perm^*=p$ for all $\perm\in \Perm(I)$.
\myitem If $\mu$ is a \bp on $\Om_I$, then $p\mu$ is a \bp on $\P$
supported by $\P_n$.\\
\myitem 
\quad\vspace*{-6.5ex}
  \begin{displaymath}
p(T_\eps\om)=\frac 1n\sum_{i=1}^n\delta_{S_{\eps,p(\om)}(\tau\om_i)}
=\frac 1n\sum_{i=1}^n (S_{\eps,p_\om}\circ\tau)\delta_{\om_i}
=(S_{\eps,p_\om}\circ\tau)p_\om
  \end{displaymath}
\end{itemize}
%
This motivates to define $\T_\eps:\P\to\P$ by
\[
\T_\eps(\nu):=(S_{\eps,\nu}\circ\tau)\nu
\ .
\]
As $\tau$ and $S_\eps$ are continuous, also
$\T_\eps$ is continuous. 
\par
For the subsequent discussion it is important to be aware of the following notational subtlety: If we write $\T_\eps(\nu)$, then $\nu\in\P$, \ie $\nu$ is a probability measure on $X$ and so is $\T_\eps(\nu)$. If we write $\T_\eps P$, \ie if the argument is not put into brackets, then $P$ is a 
probability measure on $\P$, and $\T_\eps P$ is the measure on $\P$ defined by $(\T_\eps P)(\A)=P(\T_\eps^{-1} \A)$ for measurable $\A\subseteq\P$.
Analogous conventions are used for other maps.
Observe now that for finite $I$ holds:
\begin{itemize}
\myitem $p({T_\eps\om})=\T_\eps(p_\om)$ for all $\om\in\Om$.
  As $\bigcup_n\P_n$ is dense in $\P$, this means that $\T_\eps$
  is the unique continuous extension of the action induced by $T_\eps$ on
  $\bigcup_n\P_n$ to all of $\P$.  In a slightly different notation
  this is the same as $p\circ T_\eps=\T_\eps\circ p$.
\myitem If $T_\eps\mu=\mu$ for a \bp $\mu$ on $\Om_I$, then
  $\T_\eps\, p\mu=(\T_\eps\circ p)\mu=(p\circ T_\eps)\mu=p\mu$.
\end{itemize}
Finally we note that for each $\nu\in\P$ and $T_\eps:\Om\to\Om$ holds:
\begin{equation}
  \label{eq1}
  T_\eps \nu^\nz=\left((S_{\eps,\nu}\circ\tau)\nu\right)^\nz
  =\left(\T_\eps(\nu)\right)^\nz\ .
\end{equation}
%
\subsection{The action of $T_\eps$ on exchangeable distributions on $\Om$}
\label{exchangeable-sect}
%
Denote by $\M=\M_I$ the set of all \bps on $\Om=\Om_I$ and let
\[
\Mex=\Mex_I:=\{\mu\in\M: \perm^*\mu=\mu\ \forall\perm\in \Perm(I)\}
\]
be the set of all {\em exchangeable} distributions on $\Om$.  For
$I=\nz$, a classical theorem of de Finetti asserts, that each
exchangeable distribution is a mixture of product distributions in the
following sense:
%
\begin{theo}[de Finetti, first version; see \cite{Aldous}]\label{deFinetti1}
  To each $\mu\in\Mex_\nz$ there exists a unique \bp $P$ on $\P$ such
  that
\begin{equation}\label{de-Finetti-eqn}
\mu(A)=\int_\P\nu^\nz(A)\,dP(\nu)\quad\mbox{for all measurable $A\subseteq \Om$.}
\end{equation}
In short-hand: $\mu=\int\nu^\nz\,dP(\nu)$. Here $\nu^\nz$ denotes the
infinite product measure on $\Om$ with all factors equal to $\nu$.
(Vice versa, each $\mu\in\M$ that can be represented as in
(\ref{de-Finetti-eqn}) belongs to $\Mex_\nz$.)
\end{theo}
%
In particular there is a one-to-one correspondence given by
(\ref{de-Finetti-eqn}) between measures $\mu\in\Mex_\nz$ and \bps $P$ on
$\P$.
\par
There is another way to represent $\mu\in\Mex_\nz$, which conveys more of
the probabilistic structure of the coordinate process on $\Om$ under
the distribution $\mu$. Denote by $\F$ the Borel $\sigma$-algebra and
by $\Ftail\subseteq\F$ the tail-field of $\Om$.
%
\begin{theo}[de Finetti, second version; see \cite{Aldous}]\label{deFinetti2}
  To each $\mu\in\Mex_\nz$ there exists a unique $\Ftail$-measurable
  stochastic kernel $Q:\Om\times\F\to[0,1]$ such that
\begin{enumerate}
\item $\mu(A)=\int_\Om Q(\om,A)\,d\mu(\om)$ for each $A\in\F$ and
\item for each $\om\in\Om$ there is some $\nu_\om\in\P$ such that
  $Q(\om,\,.\,)=\nu_\om^\nz$.
\end{enumerate}
\end{theo}
%
\begin{remark}\label{Finetti-remark}
Given $\mu=\int\nu_\om^\nz\,d\mu(\om)$, define $\psi:\Om\to\P$ by $\psi(\om):=\nu_\om$.
Then the two versions of the de Finetti theorem are related by the identity
$P=\psi\mu$.
\end{remark}
%
\begin{remark}
  If $\mu\in\Mex_\nz$, then each finite-dimensional marginal
  distribution of $\mu$ on $\Om_{I_n}$ is exchangeable, but not each
  $\mu\in\Mex_{I_n}$ is a marginal distribution of some measure from
  $\Mex_\nz$, see \cite{Aldous}.
\end{remark}
%
Fix any $\mu=\int\nu^\nz\,dP(\nu)\in\Mex_\nz$. Then, because of (\ref{eq1}),
\[
T_\eps\mu = \int T_\eps\nu^\nz\,dP(\nu) = \int
(\T_\eps(\nu))^\nz\,dP(\nu) = \int \nu^\nz\,d(\T_\eps P)(\nu)\ .
\]
This shows that $T_\eps\mu$ is represented by the distribution
$\T_\eps P$ on $\P$.  In particular,
\[
T_\eps\mu=\mu\quad\mbox{if and only if}\quad\T_\eps P=P\ .
\]
%
\subsection{Transition to the limit as $n\to\infty$}
%
Consider \bps $\mu_n$ on $\Om_{I_n}$ such that $T_\eps\mu_n=\mu_n$ for
all $n$, and fix a subsequence $\mu_{n_i}$ such that the sequence
$p\mu_{n_i}$ converges weakly to some \bp $P$ on $\P$.  Then
\begin{equation}
\label{limit-invariance-eqn}
\T_\eps P=\limi (\T_\eps\circ p)\mu_{n_i} =\limi (p\circ T_\eps)\mu_{n_i}=\limi
p\mu_{n_i}=P\ .
\end{equation}
Suppose now that the $\mu_n$ are exchangeable and that the subsequence
$\mu_{n_i}$ converges weakly to some $\mu\in\M_\nz$ in the sense that
$\limi\mu_{n_i}(\ph)=\mu(\ph)$ for each continuous $\ph:\Om_\nz\to\rz$
which depends only on a finite number of coordinates (so that
$\mu_n(\ph)$ is well defined for large enough $n$). Then
$\mu\in\Mex_\nz$ as is easily seen, and our goal is to show that
$\mu=\int\nu^\nz\,dP(\nu)$ for the limiting distribution $P=\limi
p\mu_{n_i}$.
\par
Denote by $C(\P)$ the algebra of all continuous functions
$F:\P\to\rz$.  Particularly simple elements from $C(\P)$ are the
functions $F_\ph:\nu\mapsto\nu(\ph)$ with $\ph\in C(X)$. Denote
$A_0(\P):=\{F_{\ph_1}\cdots F_{\ph_n}:n\in\nz,\ph_i\in C(X)\}$ and let
$A(\P)$ be the linear space generated by $A_0(\P)$. Then $A(\P)$ is a
subalgebra of $C(\P)$ which contains the constant functions and
separates points in $\P$ so that, by the theorem of Stone and
Weierstrass, $A(\P)$ is dense in $C(\P)$. Hence we have
%
\begin{lemma}
  A sequence $(P_k)_{k\in\nz}$ of \bps on $\P$ converges weakly to
  $P$, if and only if
\[
\lim_{k\to\infty}\int\nu(\ph_1)\cdots\nu(\ph_n)\,dP_k(\nu) =
\int\nu(\ph_1)\cdots\nu(\ph_n)\,dP(\nu)
\]
for all $n\in\nz$ and all $\ph_1,\ldots,\ph_n\in C(X)$.
\end{lemma}
%
Similarly, let $A_0(\Om_\nz)$ be the family of all functions
$\om\mapsto\ph_1(\om_1)\cdots\ph_n(\om_n)$ from $\Om_\nz$ to $\rz$
with $n\in\nz$ and $\ph_i\in C(X)$. Denote by $A(\Om_\nz)$ the linear
space generated by $A_0(\Om_\nz)$.  Then $A(\Om_\nz)$ is a subalgebra
of $C(\Om_\nz)$ which contains the constant functions and separates
points in $\Om_\nz$ so that, by the theorem of Stone and Weierstrass
again, $A(\Om_\nz)$ is dense in $C(\Om_\nz)$. Hence we have
%
\begin{lemma}
  A sequence $(\mu_{n_i})_{i\in\nz}$ of \bps on $\Om_{n_i}$ converges
  weakly to $\mu\in\M_\nz$, if and only if
\[
\limi\int\ph_1(\om_1)\cdots\ph_n(\om_{n})\,d\mu_{n_i} =
\int\ph_1(\om_1)\cdots\ph_n(\om_{n})\,d\mu
\]
for all $n\in\nz$ and all $\ph_1,\ldots,\ph_n\in C(X)$.
\end{lemma}
%
Now we are ready to prove
%
\begin{prop}\label{weak-convergence-prop}
  Consider a sequence $\mu_n\in\Mex_{I_n}$. A subsequence
  $(\mu_{n_i})_{i\in\nz}$ converges weakly to a measure
  $\mu\in\Mex_\nz$ if and only if the sequence
  $(p\mu_{n_i})_{i\in\nz}$ converges weakly to a \bp $P$ on $\P$. In
  this case, $\mu=\int\nu^\nz\,dP(\nu)$.
\end{prop}
%
\begin{proof}{}
  Suppose first that $\mu_{n_i}\to\mu$. Then $\mu\in\Mex_\nz$ and
  hence $\mu=\int\nu^\nz\,dP(\nu)$ for some \bp $P$ on $\P$.  Let
  $F_{\ph_1}\cdots F_{\ph_n}\in A_0(\P)$. Then, as the $\mu_{n_i}$ are
  exchangeable,
\begin{eqnarray}\label{rewrite-eqn}
\int F_{\ph_1}(\nu)\dots F_{\ph_n}(\nu)\,d(p\mu_{n_i})(\nu)
&=&
\int\nu(\ph_1)\cdots\nu(\ph_n)\,d(p\mu_{n_i})(\nu)\\
&=&
\int\prod_{k=1}^n\left(\frac 1{n_i}\sum_{j=1}^{n_i}\ph_k(\om_j)\right)
\,d\mu_{n_i}(\om)\nn\\
&=&
\frac 1{n_i^n}\sum_{j_1=1}^{n_i}\cdots\sum_{j_n=1}^{n_i}
\int\ph_1(\om_{j_1})\cdots\ph_n(\om_{j_n})\,d\mu_{n_i}(\om)\nn\\
&=&
\int\ph_1(\om_{1})\cdots\ph_n(\om_{n})\,d\mu_{n_i}(\om)+O(n_i^{-1})\nn\\
&\to&
\int\left[\int\ph_1(\om_{1})\cdots\ph_n(\om_{n})\,d\nu^\nz(\om)\right]\,
dP(\nu)\nn\\
&=&
\int F_{\ph_1}(\nu)\dots F_{\ph_n}(\nu)\,dP(\nu)
\end{eqnarray}
as $i\to\infty$. Hence $p\mu_{n_i}\to P$ weakly.
\par
Suppose conversely that $p\mu_{n_i}\to P$ for some \bp $P$ on $\P$.
Let $\ph_1,\ldots,\ph_n\in C(X)$. Then
\begin{eqnarray*}
\int\ph_1(\om_{1})\cdots\ph_n(\om_{n})\,d\mu_{n_i}(\om)
&=&
\int F_{\ph_1}(\nu)\cdots F_{\ph_n}(\nu)\, d(p\mu_{n_i})(\nu)
+O(n_i^{-1})\quad\mbox{as in (\ref{rewrite-eqn})}\\
&\to&
\int F_{\ph_1}(\nu)\dots F_{\ph_n}(\nu)\,dP(\nu)\\
&=&
\int\left[\int\ph_1(\om_{1})\cdots\ph_n(\om_{n})\,d\nu^\nz\right]\,
dP(\nu)\ .  
\end{eqnarray*}
Hence $\mu_{n_i}\to\int\nu^\nz\,dP(\nu)$.
\end{proof}
%
\begin{koro}
  Consider a sequence $\mu_n\in\Mex_{I_n}$ and a probability measure
  $\nu\in\P$. For any subsequence
  $(\mu_{n_i})_{i\in\nz}$ the following are equivalent:
  \begin{itemize}
  \item[1)]
  $(\mu_{n_i})_{i\in\nz}$ converges weakly to the product measure $\nu^\nz$,
  \item[2)]
  $(p\mu_{n_i})_{i\in\nz}$ converges weakly to the point mass $\delta_\nu$,
  \item[3)]
  $\lim_{i\to\infty}\int \ph(\om_1)\psi(\om_2)\,d\mu_{n_i}(\om)
  =\nu(\ph)\nu(\psi)$ for all $\ph,\psi\in C(X,\rz)$,
  \end{itemize}
\end{koro}
%
\begin{proof}{}
1)$\Leftrightarrow$2) follows from the previous proposition,
1)$\To$3) is  trivial. We turn to
3)$\To$2): Because of the weak compactness of $\P$ it suffices to show 
that if $\lim_{i\to\infty}\mu_{n_i}=\mu=\int\tilde\nu^\nz\,dP(\tilde\nu)$, then
$P=\delta_\nu$. But, if $\lim_{i\to\infty}\mu_{n_i}=\mu$, then 
$\int\ph(\om_1)\psi(\om_2)\,d\mu_{n_i}(\om)
\to\int\tilde\nu(\ph)\tilde\nu(\psi)\,dP(\tilde\nu)$ 
for all $\ph,\psi\in C(X)$, and it follows from 3) that for each $\ph\in C(X)$
\begin{eqnarray*}
\nu(\ph)^2
&=&
\limi \int\ph(\om_1)\ph(\om_2)\,d\mu_{n_i}(\om)
=
\int\tilde\nu(\ph)^2\,dP(\tilde\nu)\\
&\ge&
\left(\int\tilde\nu(\ph)\,dP(\tilde\nu)\right)^2
=
\left(\limi \int\ph(\om_1)\,d\mu_{n_i}(\om) \right)^2
=
\nu(\ph)^2\ .
\end{eqnarray*}
Hence $\tilde\nu(\ph)=\nu(\ph)$ for $P$-\ae $\tilde\nu\in\P$,
and as $C(X)$ has a countable dense subset, it follows that
$\tilde\nu=\nu$ for $P$-\ae $\tilde\nu\in\P$, \ie $P=\delta_\nu$.
\end{proof}

%
As a consequence of Proposition \ref{weak-convergence-prop}, a suitable dynamical model for the infinite globally coupled map lattice is the topological dynamical system $(\T_\eps,\P)$. Invariant measures $P$ for $\T_\eps$ should 
be considered to be dynamically relevant (in the sense of an SRB-measure), 
if they are weak limits of sequences 
$(p\mu_{n_j})_j$ where the $\mu_n$ are SRB-measures for 
$T_\eps:\Om_{I_n}\to\Om_{I_n}$. This suggests the following precise definition:
%
\begin{defi}\label{violation-defi}
A globally coupled system {\em violates the $0$-$1$-law}, if 
there exists a subsequence $(\mu_{n_i})_i$
of unique SRB-measures
for $T_\eps:\Om_{I_{n_i}}\to\Om_{I_{n_i}}$
such that the $p\mu_{n_i}$ converge weakly to a
probability distribution $P$ on $\P$ 
that is not just a point mass
in a fixed point of $\T_\eps$.
\end{defi}
%

%
\section{Systems with piecewise differentiable units}
%
In Section \ref{pwexpanding-sect}
we shall see that if the individual map $\tau$ is a 
piecewise $C^2$ and expanding mixing interval map (like the tent map),
then for each $n$ there is a unique SRB-measure $\mu_n$, 
which is absolutely continuous and has a density of bounded variation. 
Beforehand we show in this section that weak limits $P$ of such measures 
$\mu_n$ are concentrated on a relatively small subset $\dbv$ of $\P$, namely on the set of those measures $\nu$ on $X$ that have densities (with respect to Lebesgue measure) of bounded variation.
%
\subsection{Functions of bounded variation}
%
A Lebesgue-integrable
function $f:\rz^n\to\rz$ has bounded variation, if its derivative in the
distributional sense is a finite $\rz^n$-valued measure. A rather
complete treatment of the basic properties of these functions is given
in Chapter 5 of \cite{EvGa}.  
As for our purposes a slight variation of the notations introduced there is preferable, we start with some basic definitions.
\par
For $f\in L^1(\rz^n)$ let
\begin{align*}
  \var_i(f)& = \sup\left\{\int f\,\pd{\ph}{x_i}\,dx:
  \ph\in C_c^1(\rz^n;\rz),|\ph|\le 1\right\}  \\
\intertext{and}
\var(f) & = \sum_{i=1}^n\var_i(f)\ .
\end{align*}
If $\Om\subset\rz^n$ is open and bounded and if $f:\Om\to\rz$ is integrable, then we define
$\var_i(f):=\var_i(f\,\1{\Om})$. Therefore, $\var(f)$ is the sum of the variation of $f$ inside $\Om$ plus the contributions caused by the jumps of $f$ to zero on the boundary of $\Om$.
Denote by $\BV=\BV(\Om)$ the space of 
functions of bounded variation on $\Om$
\footnote{More precisely, $\BV$ is the space of all equivalence classes 
(modulo Lebesgue measure zero) of
such functions}.
It is a linear space which, equipped with the norm 
$\bvnorm{f}:=\var(f)+\|f\|_1$, becomes a Banach space.
This fact, as well as the compact embedding of $\BV(\Om)$ into
$L^1(\Om)$ and many further results follow directly from the corresponding statements in \cite[Chapter 5]{EvGa} if one observes that
$\frac 1n\var(f)\le\|Df\|(\rz^n)\le\var(f)$ where $\|Df\|$ denotes the total variation measure of the distributional derivative of $f$ which can be interpreted as a vector-valued Radon measure. In particular, 
$\bvnorm{.}$ and the variation norm defined in \cite[p.170]{EvGa}
are equivalent norms on $\BV(\Om)$.
\footnote{As $\Omega$ is open and bounded, the Sobolev inequality
\cite[Theorem 5.6.1-1]{EvGa} implies that there is a constant $C>0$ such that
$\|f\|_1\le C\var(f)$ for all $f\in\BV(\Omega)$. Hence $\bvnorm{f}\le(1+C)\var(f)\le\bvnorm{f}$, so $\var(.)$ is itself a norm equivalent to $\bvnorm{.}$.}
\par
If $\mu$ is a finite signed measure on $\rz^n$, we define similarly
\begin{align*}
  \var_i(d\mu)& = \sup\left\{\int \pd{\ph}{x_i}\,d\mu(x):
  \ph\in C_c^1(\rz^n;\rz),|\ph|\le 1\right\}  \\
\intertext{and}
\var(d\mu) & = \sum_{i=1}^n\var_i(d\mu)\ .  
\end{align*}
Then $\var_i(d\mu)=\var_i(h)$, 
if $\mu$ has density $h$ with respect to Lebesgue measure on $\rz^n$.
\footnote{So $\var(d\mu)$ is not related to the total variation (= total absolute mass) of $\mu$.}
%
\begin{lemma}\label{abs-cont-lemma}
If $\mu$ is a finite signed measure on $\rz^n$ and if $\var(d\mu)<\infty$,
then $\mu$ has a density $h$ with respect to Lebesgue measure and
$\var_i(d\mu)=\var_i(h)$ for $i=1,\dots,n$.
\end{lemma}
%
\begin{proof}{}
In view of the preceding remark, only the absolute continuity of $\mu$ remains to be proved. Consider $\ph\in C^1_c(\rz^n;\rz)$ and
let $\eta_\eps(x)=\frac 1{\eps^n}\eta(\frac x\eps)$ $(\eps>0)$
be a standard mollifier (see \cite[4.2.1]{EvGa}). As $\pd{\ph}{x_i}$
is continuous with compact support,
$\lim_{\eps\to 0}\pd{\ph}{x_i}\ast\eta_\eps=\pd{\ph}{x_i}$ uniformly on
$\rz^n$.
\par
The measures $\mu\ast\eta_\eps$ have densities, say $h_{\eps}$,
w.r.t.\ Lebesgue measure on $\rz^n$. 
Therefore, if $|\ph|\le 1$, then
\begin{displaymath}
  \int\pd{\ph}{x_i}\,h_{\eps}\,dx
=\int\pd{\ph}{x_i}\,d(\mu\ast\eta_\eps)
= \int\left(\pd{\ph}{x_i}\ast\eta_\eps\right)\,d\mu
=\int\pd{}{x_i}(\ph\ast\eta_\eps)\,d\mu
\le\var_i(d\mu)
\end{displaymath}
so that $\var_i(h_\eps)\le\var_i(d\mu)$ for all $\eps>0$.
Hence there exist a sequence
$\eps_k\to 0$ and a probability density $h$ such that
$\lim_{k\to\infty}\|h_{\eps_k}-h\|_1=0$.
Therefore, for each $\psi\in C_c(\rz^n;\rz)$,
\begin{displaymath}
 \int\psi\,d\mu
=\lim_{k\to\infty}\int\psi\ast\eta_{\eps_k}\,d\mu
=\lim_{k\to\infty}\int\psi h_{\eps_k}\,dx
= \int\psi h\,dx
\end{displaymath}
so that $d\mu=h\,dx$.
\end{proof}
\par
The most important property of functions of bounded variation we make use 
of in this section is the following compactness property: 
If $\Om\subset\rz^n$ is bounded and if
$f_j:\rz^n\to\rz$ are of bounded variation $(j\in\nz)$ with
$\var_i(f)\le C_i<\infty$ 
for all $i=1,\ldots,n$ and all $j\in\nz$, then 
there is a subsequence $f_{j_k}$ converging in $L^1$ to some integrable function $f$ on $\Om$ and 
$\var_i(f)\le C_i$ for all $i=1,\ldots,n$.
%
\subsection{Measures with densities of bounded variation}
%
In this section we make no dynamical assumptions.  Suppose that
$X=[0,1]$ (or any other interval or the unit circle line 
with normalized Lebesgue measure $m$).
Let $\Lambda_0\subset\Lambda_1\subset\Lambda_2\subset\dots\subset
\nz$ be a filtration of $\nz$ by finite sets. We use the following
notations:
\begin{eqnarray*}
&&
\pi_J:\Om_\nz\to\Om_J\mbox{ for $J\subseteq \nz$ 
is the canonical projection, }\pi_n:=\pi_{\Lambda_n}\\
&&
\F_{J}\mbox{ for $J\subseteq\nz$ is the Borel $\sigma$-algebra of $\Om_J$,
$\hat\F_J:=\pi_J^{-1}\F_J$,}
\\
&&
\hat A:=\pi_{\Lambda_J}^{-1}(A)\mbox{ for $J\subseteq\nz$ and $A\in\F_J$.}
\end{eqnarray*}
Fix a probability measure $\mu$ on $\Om_\nz$.
We define conditional probability distributions $K_n$ and $K_\infty$
on $\F_{\Lambda_0}$ given $\hat\F_{\Lambda_n\setminus\Lambda_0}$
(resp.\ $\hat\F_{\nz\setminus\Lambda_0}$) by
\[
K_n(\om,A):=E_\mu[\hat
A\mid\hat\F_{\Lambda_n\setminus\Lambda_0}](\om)\ , \quad
K_\infty(\om,A):=E_\mu[\hat A\mid\hat\F_{\nz\setminus\Lambda_0}](\om)\ 
.
\]
As $\om\mapsto K_n(\om,A)$ is
$\hat\F_{\Lambda_n\setminus\Lambda_0}$-measurable, we may write as
well $K_n((\om_i)_{i\in{\Lambda_n\setminus\Lambda_0}},A)$.
%
\begin{lemma}\label{density-lemma1}
  Suppose that all $\pi_n\mu$ are absolutely continuous with respect
  to Lebesgue measure on $\Om_{\Lambda_n}$ with densities $h_n$ and
  that these densities are of bounded variation. In particular,
  assume that there are constants $C_i$ $(i\in\Lambda_0)$ such that
  $\sup_n\var_i(h_n)\le C_i$ for all
  $i\in\Lambda_0$.
\par
Then, for $\mu$-\ae $\om$, $K_\infty(\om,\,.\,)$ is absolutely
continuous with respect to Lebesgue measure on $\Om_{\Lambda_0}$ with
density $k_\infty(\om,x)$ of bounded variation and
$\int_{\Om_{\nz}} 
\var_i(x\mapsto k_\infty(\om,x))\,d\mu(\om)\le C_i$ 
for all
$i\in\Lambda_0$.
\end{lemma}
%
\begin{proof}{}
  For $\ph\in C_c(\rz^{\Lambda_0};\rz)$ we have
  $E_\mu[\ph\circ\pi_0\mid\hat\F_{\Lambda_n\setminus\Lambda_0}](\om)
  =\int_{\Om_{\Lambda_0}}\ph(x)\,K_n(\om,dx)$, and the same holds for 
  $n=\infty$. Hence,
  by the martingale convergence theorem, 
  $\int_{\Om_{\Lambda_0}}\ph(x)\,K_\infty(\om,dx)
  =\limn\int_{\Om_{\Lambda_0}}\ph(x)\,K_n(\om,dx)$
  for $\mu$-\ae $\om$ and fixed $\ph$. As $C_c(\rz^{\Lambda_0};\rz)$ is
  separable,
  $K_n(\om,\,.\,)$ converges weakly to $K_\infty(\om,\,.\,)$ for
  $\mu$-\ae $\om$.  Let $A_0\in\F_{\Lambda_0}$ and
  $A\in\F_{\Lambda_n\setminus\Lambda_0}$. Then 
\begin{eqnarray*}
\int_{A_0\times A}h_n(x,y)\,d(x,y) 
&=&
(\pi_n\mu)(A_0\times A)\\
&=& 
E_\mu\left[\1{\hat A}
\cdot E_\mu[\1{\hat A_0}\mid\hat\F_{\Lambda_n\setminus\Lambda_0}]\right]\\
&=&
E_\mu\left[\1{\hat A}\cdot K_n(\,.\,,A_0)\right]\\
&=& 
\int_{\Om_{\Lambda_n\setminus\Lambda_0}} \int_{\Om_{\Lambda_0}}
  \1{A}(y)\,K_n(y,A_0)\,h_n(x,y)\,dx\,dy\\
&=& 
\int_{A} K_n(y,A_0)\,\bar h_n(y)\,dy 
\end{eqnarray*}
where $\bar h_n(y):=\int_{\Om_{\Lambda_0}} h_n(x,y)\,dx$.
  Hence $K_n(y,A_0)\cdot\bar h_n(y)=\int_{A_0}h_n(x,y)\,dx$ for
  $\pi_{\Lambda_n\setminus\Lambda_0}\mu$-\ae $y$ so that for such $y$ the
  measure $K_n(y,\,.\,)$ is absolutely continuous with respect to
  Lebesgue measure.
\par
We estimate the variation of the densities $x\mapsto k_n(y,x)$ of the
measures $K_n(y,\,.\,)$. Observe first that $k_n(y,x)=\bar
h_n(y)^{-1}\,h_n(x,y)$.  Hence 
\begin{eqnarray}
&&
\int_{\Om_{\Lambda_n\setminus\Lambda_0}} \var_i(x\mapsto k_n(y,x))
\,d(\pi_{\Lambda_n\setminus\Lambda_0}\mu)(y)\nn\\
&=&
\int_{\Om_{\Lambda_n\setminus\Lambda_0}} \var_i(x\mapsto h_n(x,y))
\,\bar h_n(y)^{-1}\,\bar h_n(y)\,dy\nn\\
&=&
\var_i((x,y)\mapsto h_n(x,y))
\le C_i \label{eq4}
\end{eqnarray}
The last equality is trivial for differentiable
$h_n$, and it caries over to general $h_n$ by approximation with
smoothened functions.
\par
Define
$k_n(\om,x):=k_n((\om_i)_{i\in\Lambda_n\setminus\Lambda_0},x)$.
As $K_n(\omega,.)\to K_\infty(\omega,.)$ weakly for $\mu$-\ae $\omega$,
it follows that for $\mu$-\ae $\omega$ and each 
$\ph\in C^1_c(\rz^{\Lambda_0},\rz)$
with $|\ph|\leq 1$ 
\begin{align*}
  \int \pd{\ph}{x_i}\,K_\infty(\omega,dx)
&= \limn \int \pd{\ph}{x_i}\,K_n(\omega,dx)
= \limn \int \pd{\ph}{x_i}k_n(\omega,x)\,dx\\
&\le \liminfn \var_i(x\mapsto k_n(\omega,x))
\end{align*}
for each $i\in\Lambda_0$.
Therefore $\var_i(dK_\infty(\omega,\,.\,))\le\liminfn\var_i(x\mapsto k_n(\omega,x))$.
for $\mu$-\ae $\omega$ so that
\begin{align*}
\int \var_i(dK_\infty(\omega,\,.\,))\,d\mu(\omega)
&\le \liminfn\int \var_i(x\mapsto k_n(\omega,x))\,d\mu(\omega)\\
&= \liminfn \int_{\Om_{\Lambda_n\setminus\Lambda_0}} \var_i(x\mapsto k_n(y,x))
\,d(\pi_{\Lambda_n\setminus\Lambda_0}\mu)(y)\\
&\leq C_i\ .
\end{align*}
In particular, $\var_i(dK_\infty(\omega,.))<\infty$ for $\mu$-\ae $\omega$ and each $i$ such that $K_\infty(\omega,.)$ has density $k_\infty(\omega,x)$
with $\var_i(x\mapsto k_\infty(\omega,x))=\var_i(dK_\infty(\omega,\,.\,))$
by Lemma \ref{abs-cont-lemma}.
It follows that $\int_{\Om_{\nz}} 
\var_i(x\mapsto k_\infty(\om,x))\,d\mu(\om)\le C_i$ 
for all
$i\in\Lambda_0$.
\end{proof}
%
Next we define conditional probability distributions $K^n$ and
$K^\infty$ on $\F_{\Lambda_0}$ given $\hat\F_{\nz\setminus\Lambda_n}$
(resp.\ $\Ftail$) by
\[
K^n(\om,A):=E_\mu[\hat A\mid\hat\F_{\nz\setminus\Lambda_n}](\om)\ ,\quad
K^\infty(\om,A):=E_\mu[\hat A\mid\Ftail](\om)\ .
\]
Then $(K^n(\om,A),\hat\F_{\nz\setminus\Lambda_n})_n$ is a 
uniformly bounded reverse martingale
so that $\limn K^n(\om,A)=K^\infty(\om,A)$ $\mu$-a.s.\ and in $L^1_\mu$
and $E_\mu[K^0(\,.\,,A)\mid\Ftail]=K^\infty(\,.\,,A)$. 
As in the proof of the previous lemma it follows
that $\limn K^n(\om,\,.\,)=K^\infty(\om,\,.\,)$ weakly for $\mu$-\ae
$\om$.
%
\begin{lemma}\label{density-lemma2}
  Under the assumptions of Lemma \ref{density-lemma1}, let
  $\mu=\int\mu_\om\,d\mu(\om)$ be the $\Ftail$-decomposition of
  $\mu$, \ie $(\om,A)\mapsto\mu_\om(A)$ is a conditional probability
  distribution on $\Om$ given $\Ftail$.
\par
Then, for $\mu$-\ae $\om$, $\pi_{\Lambda_0}\mu_\om=K^\infty(\om,\,.\,)$ 
and $K^\infty(\om,\,.\,)$ is
absolutely continuous with respect to Lebesgue measure on
$\Om_{\Lambda_0}$ with density
$k^\infty(\om,x)=\int_{\Om_\nz}k_\infty(\tilde\om,x)\,d\mu_\om(\tilde\om)$
of bounded variation and 
\begin{equation}
\int_{\Om_{\nz}} \var_i(x\mapsto k^\infty(\om,\,.\,))\,d\mu(\om)\le C_i  
\end{equation}
for all
$i\in\Lambda_0$.
\end{lemma}
%
\begin{proof}{}
$\pi_{\Lambda_0}\mu_\om=K^\infty(\om,\,.\,)$ for $\mu$-\ae $\om$ is just the definition of
conditional distributions.
  As $K^0(\om,A)=K_\infty(\om,A)$, we have 
\begin{eqnarray*}   
  K^\infty(\om,A) 
  &=&
  E_\mu[K^0(\,.\,,A)\mid\Ftail](\om)\\
  &=&
  E_\mu[K_\infty(\,.\,,A)\mid\Ftail](\om)\\
  &=&
  E_\mu\left[\int_A k_\infty(\,.\,,x)\,dx\mid\Ftail\right](\om)\\
  &=&
  \int_{\Om_\nz}\int_A k_\infty(\tilde\om,x)\,dx\,d\mu_\om(\tilde\om)\\
  &=&
  \int_A\left[\int_{\Om_\nz}k_\infty(\tilde\om,x)\,d\mu_\om(\tilde\om)\right]\,dx
\end{eqnarray*}
 so that $K^\infty(\om,\,.\,)$ has density $x\mapsto
  \int_{\Om_\nz}k_\infty(\tilde\om,x)\,d\mu_\om(\tilde\om)$.  The
  average variation bound for this density follows at once from that
  for $k_\infty(\om,x)$.
\end{proof}
%
\subsection{Exchangeable densities of bounded variation}
%
In the special case of exchangeable distributions $\mu$ on $\Om_\nz$
the previous lemma yields at once:
%
\begin{prop}\label{abs-cont-decomp-prop}
  Let $\mu$ be an exchangeable distribution on $\Om_\nz$ all
  of whose finite-dimensional mar\-ginals are absolutely continuous with
  respect to Lebesgue measure and have densities of bounded variation,
  namely $\var_1(h_n)\le C$. Let
  $\mu=\int\nu_\om^\nz\,d\mu(\om)$ be the de Finetti-decomposition of
  $\mu$ (see Theorem \ref{deFinetti2}).  Then $\mu$-\aa measures
  $\nu_\om$ of this decomposition are absolutely continuous with
  densities $k^\infty(\om,x)$ of bounded variation and 
  $\int\var_1(x\mapsto k^\infty(\omega,x))\,d\mu(\om)\le C$.
\end{prop}
%
\begin{proof}{}
  Apply Lemma \ref{density-lemma2} to $\mu\in\Mex_\nz$ with
  $\Lambda_0=\{1\}$.  As the de Finetti-decomposition coincides with
  the $\Ftail$-decomposition from that lemma, $\mu_\om=\nu_\om^\nz$ so
  that $\pi_{\Lambda_0}\mu_\om=\nu_\om$.
\end{proof}
%
In the dynamical setting discussed in Section \ref{cont-units-sect}
this proposition leads to the following theorem:
%
\begin{theo}\label{abs-cont-decomp-theo}
  Suppose that for some fixed $\eps$ each $T_\eps:\Om_{I_n}\to\Om_{I_n}$ $(n>0)$
  has an absolutely continuous invariant probability measure 
  $\mu_n$ with density $h_n$ w.r.t.\ {}Lebesgue measure and that
  these densities satisfy the following variation bound:
\[
\exists C>0\ \forall i\in\nz\ \forall n\ge i:\
\var_i(h_n)\le C\ .
\]
Let $P$ be a weak accumulation point of the measures $p\mu_n$ on $\P$
(these are the Borel probability measures on $X=[0,1]$).
Denote by 
$\dbv:=\{\nu\in\P: d\nu(x)=h_\nu(x)\,dx\mbox{ and }
\var_1(h_\nu)<\infty\}$.
Then $P$ is $\T_\eps$-invariant (see (\ref{limit-invariance-eqn})),
\begin{equation}\label{limit-claim}
P(\P\setminus\dbv)=0\quad\mbox{ and }\quad
\int\var_1(h_\nu)\,dP(\nu)\le C\ .
\end{equation}
\end{theo}
%
\begin{proof}{}
Suppose that $P=\lim_{j\to\infty}p\mu_{n_j}$. In view of Proposition
\ref{weak-convergence-prop} and Remark \ref{Finetti-remark},
\begin{equation}\label{Finetti-eq}
\lim_{j\to\infty}\mu_{n_j}=\mu:=\int\nu^\nz\,dP(\nu)=\int\nu_\om^\nz\,d\mu(\om)\ .
\end{equation}
Fix $k\in\nz$ and consider $\pi_k\mu_{n_j}$ for $n_j\ge k$.
By assumption, $\pi_k\mu_{n_j}$ has density 
\[
h_{n_j}^{(k)}(x):=\int_{\Om_{\{k+1,\ldots,n_j\}}}h_{n_j}(x,y)\,dy
\quad\mbox{with}\quad \var_i(h_{n_j}^{(k)})\le C\quad(i=1,\ldots,k)\ .
\]
As $\lim_{j\to\infty}\mu_{n_j}=\mu$ in the weak sense, it follows from the relative compactness of the family of densities $(h_{n_j}^{(k)})_j$ that $\pi_k\mu$ is absolutely continuous w.r.t.\ {}Lebesgue measure with some density $h^{(k)}$ and that
$\lim_{j\to\infty}\|h^{(k)}-h_{n_j}^{(k)}\|_1=0$. In particular,
$\var_i(h^{(k)})\le C$ for all $i=1\ldots,k$.
Now Proposition \ref{abs-cont-decomp-prop} implies that 
$\mu$-\ae $\nu_\om$ is absolutely continuous with density
$k^\infty(\om,x)$ of bounded variation and 
$\int\var_1(x\mapsto k^\infty(\omega,x))\,d\mu(\om)\le C$.
In view of (\ref{Finetti-eq}) this is equivalent to (\ref{limit-claim}).
\end{proof}
%
In the next section we show that for piecewise expanding interval maps 
$\tau$ the assumptions of Theorem \ref{abs-cont-decomp-theo}
are satisfied and that, even more, there is some $K>0$ such that
$P(\P\setminus\dbv_K)=0$ where 
$\dbv_K:=\{\nu\in\dbv: \var_1(h_\nu)\le K\}$.
%
\section{Systems with piecewise expanding units}
\label{pwexpanding-sect}
%
In this section we specialize our setting. Now $X$ is either a compact interval (for definitiveness we let $X=[0,1]$ in this case), or $X$ is the unit 
circle line. In both cases we denote the normalized Lebesgue measure on $X$ by $m$.
The map $\tau:X\to X$ is continuous, piecewise $C^2$ and piecewise expanding 
in the following sense: There are points $0=a_0<a_1<\dots<a_N=1$
such that
\begin{enumerate}
\item 
$\tau_{|(a_{i-1},a_i)}$ is monotone and $C^2$ for all $i=1,\ldots,N$,
\item
there are $\kappa>2$ and $k\ge 1$ such that
\begin{enumerate}
\item 
$\tau^ja_i\not\in\{a_1,\ldots,a_{N-1}\}$ for $j=0,\ldots,k-1$ and $i=0,\ldots,N$ and
\item
$|(\tau^k)'(x)|\ge\kappa$ for all $x\in X$ for which the derivative is well defined,
\end{enumerate}
\item
$\sup_x|\tau''(x)/\tau'(x)^2|<\infty$ where the supremum extends over all 
$x\not\in\{a_0,\ldots,a_N\}$.
\end{enumerate}
These are traditional assumptions guaranteeing that $\tau$ has an absolutely continuous invariant measure with density of bounded variation. (There are weaker assumptions under which the same conclusion holds.) For a detailed review of 
this type of results see \cite{BoGobook}.
\par
The two main results of this section show that expanding circle maps with a coupling as in (\ref{circle-coupling}) do not violate the $0$-$1$-law 
(Theorem \ref{circle-map-theo}) and that the same holds for tent maps $\tau$ with $|\tau'|>1.65$ and such that the critical trajectory of $\tau$ ends in a fixed point (Theorem \ref{pwl-map-theo} and Remark \ref{pwl-map-remark}).
%
%
\subsection{A specific class of couplings}
%
In order to be able to use some results from the literature
we consider only couplings of the following type:
\begin{equation}\label{coupling-ass}
S_\eps(x,\nu)=x+\eps\cdot\gamma(x,\bar\nu)\quad
\text{where}\quad\bar\nu=\int_X F\circ\tau\,d\nu\ (\in\rz^s)
\end{equation}
for some fixed $F\in C^2(X,\rz^s)$ and $\gamma\in C^2(X\times\rz^s,\rz)$.
Denoting, as in Examples \ref{tent-example} and \ref{circle-example} 
(coupled tent maps and circle maps),
\begin{displaymath}
\Phi_\eps:\Omega\to\Omega,\quad
(\Phi_\eps\omega)_i=\omega_i+\eps\cdot\gamma(\omega_i,\int_X F\,dp_\omega)
\end{displaymath}
we can write $T_\eps=\Phi_\eps\circ T$. 
Therefore the 
first model fits directly into the framework of \cite{KeKu} and of the thesis
\cite{kuenzle}, the results of which are also summarized in \cite{keller-amst}.
In the second model, the structure of the coupling map is not exactly the one studied in these references, but a glance at the proofs given there shows that
indeed any coupling as described in (\ref{coupling-ass}) allows the same conclusions. 
All one has to check is that there are constants $A_0,A_1,A_2>0$, which are independent of the lattice size $n$, such that
for all $\om\in\Om$
\begin{displaymath}
|\Phi_\eps(\om)-\om|_\infty\le A_0\eps,\quad
\|D\Phi_\eps(\om)-E\|_\infty\le A_1\eps,\quad
\|D_iD\Phi_\eps(\om)\|_\infty\le A_2\eps\mbox{ for all $i$}\ ,
\end{displaymath}
where $D$ denotes the total derivative, $D_i$ the partial derivative with respect to $\om_i$, $E$ the identity matrix,
$|\,.\,|_\infty$ the maximum norm for vectors, and $\|\,.\,\|_\infty$
the corresponding operator norm for matrices (maximal row sum).
\footnote{These are the essential properties of the more special couplings studied in \cite{kuenzle}. They are listed in Proposition 2.15 of that thesis, and all further proofs use only these properties.}
It is not hard to check that these inequalities are satisfied for twice continuously differentiable $F$ and $\gamma$.
Therefore the dynamics of $T_\eps$ can be studied as in \cite{KeKu,kuenzle,keller-amst} by means of its associated Perron-Frobenius operator
$\L_\eps:L^1_\Om\to L^1_\Om$. This operator is uniquely defined by the property
\begin{displaymath}
\int_\Om\ph(T_\eps\om)\psi(\om)\,dm(\om)=\int_\Om\ph(\om)(\L_\eps\psi)(\om)\,dm(\om)
\quad\mbox{for all $\ph\in L^\infty_\Om,\psi\in L^1_\Om$.}
\end{displaymath}
It is proved in these references that there are constants $C_1,C_2>0$ 
and $\eps_0>0$ depending
only on $\tau$ and $A_0,A_1,A_2$ and not on the dimension $n$ 
of $\Omega_{I_n}$ such that for all $\eps\in[0,\eps_0]$ and all 
$f\in L^1_\Om$ of bounded variation holds:
\begin{equation}\label{LY-inequ}
\var(\L^k_\eps f)\le \left(\frac 2\kappa+C_1\eps\right)\cdot\var(f)+C_2\,n\cdot\|f\|_1
\end{equation}
where $k\in\nz$ and $\kappa>2$ is the lower expansion bound of $\tau^k$
as introduced above.
Hence $\L_\eps$ leaves the space $\BV(\Om)$ of functions of bounded variation
on $\Om$
invariant and acts on it as a quasicompact operator such that 
$T_\eps:\Om_{I_n}\to\Om_{I_n}$ has an absolutely continuous (w.r.t.\ {}Lebesgue measure) invariant probability measure with density $h_{\eps,n}$ of bounded variation.
Let
$\alpha\in(\frac 2\kappa,1)$. Then $\eps_0$ can be chosen such that 
$(\frac 2\kappa+C_1\eps)<\alpha$ for all $\eps\in[0,\eps_0]$.
Therefore 
$\var(h_{\eps,n})\le \frac{C_2}{1-\alpha}n$.
If $T_\eps$ is ergodic, then it has a unique
invariant density. In any case we can assume that the density is exchangeable, because the invariance of $h_{\eps,n}$ implies immediately that also
$\bar h_{\eps,n}:=|\Perm(I_n)|^{-1}\sum_{\perm\in\Perm(I_n)}h_{\eps,n}\circ\perm^*$ is invariant (see Section \ref{basic-sect}). This implies that 
$\var_i(h_{\eps,n})=\frac 1n\var(h_{\eps,n})\le\frac{C_2}{1-\alpha}$ 
for all $i\in I_n$.
\par
For coupled tent maps with slope larger than $\sqrt{2}$ and for a coupling
as in (\ref{DLap-coupling}), it was proved in
\cite{Keller-wien} that there is indeed $\eps_2\in(0,\eps_1]$ independent of $n$ such that
$T_\eps$ is even mixing for $\eps\in[0,\eps_2]$. This result carries over to arbitrary mixing $\tau$ and also to more general couplings.
If $\tau$ is a $C^r$-endomorphism of the circle ($r\in\{2,3,\ldots,\infty\}\cup\{\om\}$)
and if the coupling is the one from (\ref{circle-coupling}), then it is well known
that $T_\eps$ is mixing and
that the unique density $h_\eps$ is of class $C^{r-1}$, see \cite{krzyzewski}.
\par
In any case we are now in a position to apply Theorem \ref{abs-cont-decomp-theo}
which tells us that all weak accumulation points of the sequence $(p\mu_{\eps,n})_n$
(where $d\mu_{\eps,n}(\om)=h_{\eps,n}(\om)d\om$) are measures on $\P$ concentrated on the set of probability measures on $X$ with densities
of bounded variation. Even more is true. Recall the notations
$\dbv=\{\nu\in\P: d\nu(x)=h_\nu(x)\,dx\mbox{ and }\var_1(h_\nu)<\infty\}$
and $\dbv_K=\{\nu\in\dbv: \var_1(h_\nu)\le K\}$.
Denote also by $\L$ the Perron Frobenius operator acting on $L^1_m(X)$ which 
is associated with $\tau$ and by $\S_{\eps,\nu}$ the one associated
with $S_{\eps,\nu}:X\to X$. 
%
\begin{prop}
\label{P-on-compact-prop}
For maps $\tau$ and couplings $S_{\eps,\nu}$ as introduced above let
$K_0:=\frac{C_2}{1-\alpha}$. Then each weak accumulation point
$P$ of the sequence $(p\mu_{\eps,n})_n$ satisfies $P(\P\setminus\dbv_{K_0})=0$.
\end{prop}
%
\begin{proof}{}
We know already that $P(\P\setminus\dbv)=0$.
As $P$ is invariant under $\T_\eps$, it suffices to show that 
for each $\nu\in\dbv$ all weak accumulation points of the sequence 
$(\T_\eps^i(\nu))_i$ belong to $\dbv_{K_0}$.
If $\nu\in\dbv$ has density $h$, then 
$\T_\eps(\nu)=(S_{\eps,\nu}\circ\tau)\nu$ has density
$\S_{\eps,\nu}(\L h)$.
A simple calculation, similar to the one leading to (\ref{LY-inequ}) in the case $n=1$, shows that for all $h\in\BV(\Omega)$
\begin{equation}
\label{LY-inequ-1D}
\var_1((\S_{\eps,\nu}\L)^k h)
\le \left(\frac 2\kappa+C_1\eps\right)\cdot\var_1(h)+C_2
\end{equation}
with $k\in\nz$ and $\kappa>2$ as introduced above.
Hence, iterating this estimate shows that the variation 
per variable of the densities
of the $\T_\eps^k(\nu)$ tend to be at most $\frac{C_2}{1-\alpha}$ as $k\to\infty$, thus proving the claim.
\end{proof}
%
Denote  by $\tilde \L_\eps$
the operator which associates with a density $h$ (corresponding to the measure $\nu=hm$) the density
$\S_{\eps,\nu}\L h$. This operator is often called the
{\em self consistent Perron Frobenius operator}.
The proof shows that the nonlinear operator $\tilde\L_\eps:L^1_m(X)\to L^1_m(X)$ and the
single site Perron-Frobenius operator $\L$ are 
closely related. In general, however, $\tilde\L_\eps$ is not a small perturbation of $\L$ neither in the sense of the $L^1$-operator norm nor 
of the BV-operator norm. We shall see that this fact is responsible for the
{\em violation of the $0$-$1$-law} mentioned in the introduction.
On the other hand, there are cases, where $\tilde\L_\eps$ is a small perturbation of $\L$, namely, when $\tau$ and $S_{\eps,\nu}$ satisfy suitable
analyticity assumptions (see \cite{Jarvenpaa,Just}).
We shall have a closer look at systems not violating the $0$-$1$-law in 
Section \ref{circle-map-sect}.
%
\subsection{Fixed points of $\T_\eps$}
%
Proposition \ref{P-on-compact-prop} leaves us with the task to study
$\T_\eps$-invariant measures on $\dbv_{K_0}$. This is particularly simple,
if $\tilde\L_\eps$ is a $\bvnorm{.}$-contraction on $\dbv_{K_0}$ with a unique fixed point
$h_\eps$, because in this case $P=\delta_{\nu_\eps}$ with $\nu_\eps=h_\eps m$
is the only $\T_\eps$-invariant measure. So we start with a simple characterization of fixed points for $\T_\eps$.
\par
With the notation and assumptions of the previous section denote 
$\tau_{\eps,\nu}:=S_{\eps,\nu}\circ\tau$.
Suppose that $\tau_{0,\nu}=\tau$ is ergodic and therefore 
has a unique absolutely continuous invariant measure. Then, for
$\eps$ sufficiently close to $0$, 
also $\tau_{\eps,\nu}$ has a unique invariant measure
$\mu_{\eps,\nu}\in\dbv_{K_0}$
with density $h_{\eps,\nu}$ such that 
$\S_{\eps,\nu}\L h_{\eps,\nu}=h_{\eps,\nu}$. Furthermore,
$G_\eps:\nu\mapsto h_{\eps,\nu}$
is continuous as a function from $\P$ to $L^1_m(X)$,
see \cite[Corollary 14]{keller-mh}. 
In particular, the map $G_{\eps|\D_{K_0}}:\D_{K_0}\to \D_{K_0}$, 
$fm\mapsto h_{\eps,fm}$, is continuous w.r.t.\ the $\|.\|_1$-norm.
Since $\D_{K_0}$ is compact convex, we have
\begin{lemma}
$fm\mapsto G_\eps(f)m$ has at least one fixed point in $\D_{K_0}$.
\end{lemma}
The following observation is well known \cite{ErPo1,Kaneko1},
its proof is straightforward:
%
\begin{lemma}\label{fixed-point-lemma}
  $\nu\in\P$ is a fixed point of $\T_\eps$ if and only if 
  $\nu$ has density $f=G_\eps(fm)$ for some $f$.
  In this case $f=h_{\eps,fm}$.
\end{lemma}
%
Therefore, if $G_{\eps|G_\eps(\P)}$ is a contraction, then $\T_\eps$ has a unique fixed point.
Now $G_{\eps|G_\eps(\P)}$ is certainly a contraction for small $\eps$
if it is Lipschitz continuous 
with a Lipschitz constant $l_\eps$ that can be made arbitrarily small as $\eps\to0$. Unfortunately this is not true in general. The best known general result seems to be
\begin{equation}
  \label{inv-density-dep}
  \|G_\eps(fm)-G_\eps(\tilde fm)\|_1
=O\left(\eps\|f-\tilde f\|_1\cdot|\log(\eps\|f-\tilde f\|_1)|\right)
\quad\mbox{as $\eps\to0$.}
  \end{equation}
for a coupling satisfying (\ref{coupling-ass}), see \cite{keller-mh}.
Under more restrictive assumptions, however,
such a unique fixed point may exist (see Example \ref{tent-example-a} below),
and it may even be an attractive fixed point for the map $\T_\eps$, as we shall see in the next section.
%
\begin{example}
\label{tent-example-a}
{\bf [Tent maps]}
A particularly simple case is that of a tent map
$\tau:[0,1]\to[0,1], x\mapsto a(1-|2x-1|)$ with $\frac 12<a<1$. Choosing a simple coupling as in (\ref{DLap-coupling}) we find that 
$\tau_{\eps,\nu}(x)=\tau_{\eps,0}(x)+\eps\bar\nu$ where $\bar\nu=\int\tau(y)\,d\nu(y)$.
So $\tau_{\eps,\nu}$ is a tent map on the interval 
$[-\frac {\eps\bar\nu}{2a(1-\eps)-1},1+\frac {\eps\bar\nu}{2a(1-\eps)-1}]$ 
with the same slope 
$a(1-\eps)$ as $\tau_{\eps,0}$ and it is
therefore linearly conjugated to $\tau_{\eps,0}$. 
Recall that $h_{\eps,\nu}$ denotes the invariant density of $\tau_{\eps,\nu}$. 
An elementary computation shows that 
\begin{displaymath}
h_{\eps,\nu}(x)=\lambda_{\eps,\bar\nu}^{-1}\cdot h_{\eps,0}
\left(\frac 12+\lambda_{\eps,\bar\nu}^{-1}(x-\frac 12)\right)
\quad\mbox{where}\quad
\lambda_{\eps,t}:=1+\frac{2\eps t}{2a(1-\eps)-1}\ .
\end{displaymath}
Therefore, if $\nu_1,\nu_2\in\P$, then $h_{\eps,\nu_1}=h_{\eps,\nu_2}$
if and only if $\bar\nu_1=\bar\nu_2$. So we may write
$h_{\eps,\bar\nu}$ instead and study
$\bar G:\rz\to\rz,\ t\mapsto \int\tau(x)h_{\eps,t}(x)\,dx=\eps A_{\tau,\eps}t+B_{\tau,\eps}$ for suitable constants $A_{\tau,\eps},B_{\tau,\eps}$ that are bounded uniformly for $\eps$ ranging in a neighbourhood of $0$.
As $G_\eps(\nu)=\nu$ if and only if $\bar G_\eps(\bar\nu)=\bar\nu$, it follows that $G_\eps$ has a unique fixed point $\nu_\eps$ which is 
also the unique fixed point of $\T_\eps$ (see Lemma~\ref{fixed-point-lemma}).
\par
The reader should be aware that this does not imply 
that tent maps do not violate the $0$-$1$-law, since
the $\T_\eps$-invariant measure $\delta_{\nu_\eps}$ need not
be a weak accumulation point of 
unique SRB-measures $\mu_n$
for $T_\eps:\Om_{I_n}\to\Om_{I_n}$, see Definition \ref{violation-defi}. 
Indeed, there is both, numerical and theoretical evidence (not to be confused with a rigorous mathematical proof, however), that for many parameters tent maps violate the $0$-$1$-law
\cite{ErPo1,NaKo1,ChaMo}, while \eg for tent maps with eventually fixed critical trajectory there are convincing arguments (close to a mathematical proof)
that the $0$-$1$-law is not broken \cite{ChaMo}. Below we prove a result that covers also this latter case.
\end{example}
%
\subsection{Globally coupled $C^3$ circle maps}
\label{circle-map-sect}
%
In this section we prove that globally coupled expanding circle maps where both the maps and the coupling are of class $C^3$ do not violate the $0$-$1$-law.
For the case where maps and coupling are analytic this was proved in \cite{Jarvenpaa}.
%
\begin{theo}
  \label{circle-map-theo}
Let $\tau:\tz^1\to\tz^1$ be $C^3$ and strictly expanding, 
and suppose that the coupling is of the type described in (\ref{coupling-ass})
and satisfies also $\sup_{(x,\bar\nu)}\left|\pd{}{\bar\nu}\pd{^2\gamma}{x^2}\right|<\infty$.
Then there is $\eps_0>0$ such that for all $\eps\in[0,\eps_0]$
holds: 
Let $h_\eps$ be a fixed point of $G_\eps$ (uniqueness is not assumed {\em a priori}), $\nu_\eps:=h_\eps m$.
Then
\begin{displaymath}
\T_\eps\nu_{\eps}=\nu_{\eps}\quad\text{and}\quad
\lim_{i\to\infty}\T_\eps^i\nu=\nu_{\eps}\quad\text{exponentially for all } 
\nu\in\dbv
\end{displaymath}
(in the sense that there are $C>0$ and $\gamma\in(0,1)$ 
independent of $\nu$ and
such that for the density $u$ of $\nu$ we have 
$\|\tilde\L_\eps^i u-h_{\eps}\|_1\leq C\gamma^i\,\bvnorm{u}$).
\end{theo}
%
We give two different proofs for this theorem. The first one makes use of the following perturbation property for invariant densities of $C^3$ circle maps,
see \cite[Theorem 1]{BaYo} or, for a slightly weaker assertion \cite[Theorem 1]{krzyzewski}:
\begin{equation}
\label{Lip-dep-ass}
\text{
There are $l_\eps>0$ with $\lim_{\eps\to0}l_\eps=0$ such that
$\bvnorm{h_{\eps,\nu}-h_{\eps,\nu'}}\le l_\eps\,|\bar\nu-\bar\nu'|$ for all
$\nu,\nu'\in\P$.}
\end{equation}
The second one uses simply the $C^2$-smoothness of invariant densities of $C^3$ circle maps, see \eg \cite[Theorem 1]{krzyzewski}.
For both proofs we need the following lemma:
%
\begin{lemma}
\label{L1-contraction-lemma}
Under the assumptions of Theorem \ref{circle-map-theo} there are
$\eps_1>0$, $C>0$ and $\beta\in(0,1)$ 
such that for all $\eps\in[0,\eps_1]$ and for all $\nu_1,\ldots,\nu_n\in\P$ holds
\begin{displaymath}
  \bvnorm{\S_{\eps,\nu_n}\L\cdots\S_{\eps,\nu_1}\L(u)}
  \le C\,\beta^n\bvnorm{u}\quad\mbox{for all $u\in \BV$ with $\int u(x)\,dx=0$.}
\end{displaymath}
\end{lemma}
%
\begin{proof}{}
As the map $\tau:\tz^1\to\tz^1$ is strictly expanding 
and of class $C^2$, 
it is mixing and satisfies the Lasota-Yorke type inequality
(\ref{LY-inequ}) with $n=1$, $\eps=0$ and with a factor 
$\frac 1\kappa$ instead of $\frac 2\kappa$. 
Fix $\alpha\in(\frac 1\kappa,1)$. By continuity reasons 
there are $\eps_0>0$ and $C_3>0$ 
such that each $\tau_{\eps,\nu}$ with $\eps\in[0,\eps_0]$ and $\nu\in\P$ has a minimal expansion rate
$\alpha$ and satisfies the estimate $|\tau_{\eps,\nu}''|/|\tau_{\eps,\nu}'|^2\le C_3$. Therefore $\tau_{\eps,\nu}$ is mixing and
\begin{equation}\label{LY-1D-eqn}
\bvnorm{\S_{\eps,\nu}\L u}\le\alpha\bvnorm{u}+(C_3+1)\,\|u\|_1
\end{equation}
for all $\eps\in[0,\eps_0]$, $\nu\in\P$ and $u\in\BV$. Let
$C_4:=\frac {C_3+1}{1-\alpha}$ and consider any sequence
$\nu_1,\nu_2,\ldots\in\P$. Denote
$\Q_{\eps,n}:=\S_{\eps,\nu_n}\L\cdots\S_{\eps,\nu_1}\L$. 
Repeated application of (\ref{LY-1D-eqn}) leads to
\begin{displaymath}
  \bvnorm{\Q_{\eps,n}u}\le\alpha^n\bvnorm{u}+C_4\,\|u\|_1
\end{displaymath}
for all $\eps\in[0,\eps_0]$, $n\in\nz$, $\nu\in\P$ and $u\in\BV$.
As $\|f\|_1\le\bvnorm{f}$ by definition, it follows also that
$\bvnorm{\Q_{n,\eps}}\le C_4+1$.
\par
A well-known first consequence of estimate (\ref{LY-1D-eqn}) is that
the spectrum of
$\S_{\eps,\nu_\eps}\L$ consists of $1$ and a part strictly contained in the unit disc. It follows that
there is $N\in\nz$ such that
\begin{displaymath}
  \|(\S_{\eps,\nu_\eps}\L)^N(u)\|_1
  \le\frac 1{8C_4}\bvnorm{u}\quad\mbox{for all $u\in \BV_0$}
\end{displaymath}
where $\BV_0:=\{u\in\BV:\int u(x)\,dx=0\}$.
Choosing $N$ even larger if necessary we can also make sure that
\begin{displaymath}
  \alpha^N(C_4+1)<\frac 14\ .
\end{displaymath}
Since
$\|(\S_{\eps,\nu_j}-\S_{\eps,\nu_\eps})(u)\|_1\le\const\cdot\eps
\cdot|\bar\nu_j-\bar\nu_\eps|\cdot\bvnorm{u}$ for each $u\in\BV$ (see \cite[Lemma 13]{keller-mh}), we have
\begin{eqnarray*}
&&
\left|\|\Q_{\eps,N}(u)\|_1
-\|(\S_{\eps,\nu_\eps}\L)^N(u)\|_1\right|\\
&\le&
\sum_{j=1}^N\|\S_{\eps,\nu_N}\L\cdots\S_{\eps,\nu_{j+1}}\L
\,(\S_{\eps,\nu_j}-\S_{\eps,\nu_\eps})\L\,
(\S_{\eps,\nu_\eps}\L)^{j-1}(u)\|_1\\
&\le&
\const_N\cdot\eps\cdot\bvnorm{u}\cdot\|F\|_\infty
\end{eqnarray*}
for all $\eps\in[0,\eps_0]$, $\nu_1,\dots,\nu_N\in\P$ 
and $u\in\BV_0$ with $F$ from (\ref{coupling-ass}).
Together with the previous estimates it follows that there is some 
$\eps_1=\eps_1(N)\in(0,\eps_0]$ such that for all $\eps\in[0,\eps_1]$,
$\nu_1,\dots,\nu_N\in\P$ and $u\in\BV_0$
\begin{eqnarray*}
\bvnorm{\Q_{\eps,2N}(u)}
&\le&
\alpha^N\bvnorm{\Q_{\eps,N}(u)}+C_4\,\|\Q_{\eps,N}(u)\|_1\\
&\le&
\alpha^N(C_4+1)\bvnorm{u}+C_4\left(\frac1{8C_4}+\const_N\eps\,\|F\|_\infty
  \right)\,\bvnorm{u}\\
&\le&
\frac 12\bvnorm{u}
\end{eqnarray*}
Now the lemma follows if one observes that 
$\Q_{\eps,n}(\BV_0)\subseteq\BV_0$ and
$\bvnorm{\Q_{\eps,n}}\le C_4+1$ for all $n$.
\end{proof}
%
\begin{proof}{ of Theorem \ref{circle-map-theo}}
In most quantities occurring in this proof we skip the subscript $\eps$.
For $\nu\in\dbv$ let $\nu_i:=\T^i(\nu)$.
Denote by $u_i$ the density of $\nu_i$ with respect to Lebesgue measure. Then
$u_i=\tilde\L^i u_0$ and in particular 
$u_{i+1}=\tilde\L u_i=\S_{\eps,\bar\nu_i}\L u_i$.
\medskip\\
{\em First proof} (using (\ref{Lip-dep-ass})):\quad
Observe first that $G$ has only one fixed point so that also $\T$ has a unique fixed point: 
If $G(fm)=f$ and $G(\tilde fm)=\tilde f$, then $f=h_{\eps,fm}$,
$\tilde f=h_{\eps,\tilde f m}$, and
\begin{displaymath}
\|f-\tilde f\|_1=\|h_{fm}-h_{\tilde fm}\|_1
\le
l_\eps\cdot \|F\|_\infty\,\|f-\tilde f\|_1
\end{displaymath}
by assumption (\ref{Lip-dep-ass}), which proves the uniqueness for sufficiently small $\eps>0$.
\par
As $h_{\nu_i}=S_{\nu_i}\L h_{\nu_i}$,
\begin{eqnarray*}
u_{i+1}-h_{\nu_{i+1}}
&=&
\S_{\nu_i}\L(u_i-h_{\nu_i})+(h_{\nu_i}-h_{\nu_{i+1}})\\
&\vdots&\\
&=&
\S_{\nu_i}\L\dots\S_{\nu_0}\L(u_0-h_{\nu_0})
+\sum_{k=0}^i\S_{\nu_i}\L\dots\S_{\nu_{k+1}}\L(h_{\nu_k}-h_{\nu_{k+1}})\ .
\end{eqnarray*}
Hence,
\begin{eqnarray*}
&&
u_{i+1}-u_i\\
&=&
\S_{\nu_i}\L(u_i-h_{\nu_i})+(h_{\nu_i}-u_i)\\
&=&
(\S_{\nu_i}\L-\Id)
\left(\S_{\nu_{i-1}}\L\dots\S_{\nu_0}\L(u_0-h_{\nu_0})
+\sum_{k=0}^{i-1}\S_{\nu_{i-1}}\L\dots\S_{\nu_{k+1}}\L
(h_{\nu_k}-h_{\nu_{k+1}})\right)\ ,
\end{eqnarray*}
and it follows from Lemma \ref{L1-contraction-lemma} that
\begin{eqnarray*}
\bvnorm{u_{i+1}-u_i}
&\le&
(C_4+2)\,C\,\left(\beta^i\bvnorm{u_0-h_{\nu_0}}
+\sum_{k=0}^{i-1}\beta^{i-k-1}\bvnorm{h_{\nu_k}-h_{\nu_{k+1}}}
\right)\ .
\end{eqnarray*}
As
\begin{eqnarray*}
\bvnorm{h_{\nu_k}-h_{\nu_{k+1}}}
&\le&
l_\eps\cdot|\bar\nu_k-\bar\nu_{k+1}|
=
l_\eps\cdot|\nu_k(F\circ\tau)-\nu_{k+1}(F\circ\tau)|\nn\\
&\le&
l_\eps\cdot\|F\|_\infty\,\|u_k-u_{k+1}\|_1\ ,
\label{h-u-comparison-eqn}
\end{eqnarray*}
we can conclude that
\begin{eqnarray*}
\bvnorm{u_{i+1}-u_i}
&\le&
(C_4+2)\,C\,\left(\beta^i\bvnorm{u_0-h_{\nu_0}}
+\sum_{k=0}^{i-1}\beta^{i-k-1}\,l_\eps\cdot\|F\|_\infty\,\|u_{k+1}-u_k\|_1
\right)\ .
\end{eqnarray*}
Now a simple induction argument shows that for $\gamma\in(\beta,1)$ 
and sufficiently small $\eps>0$ there is $\Gamma>0$ such that 
$\bvnorm{u_{i+1}-u_i}\le \Gamma\,\gamma^i$ for all $i\in\nz$. It follows that the sequence
$(u_i)$ converges to some $\tilde\L_\eps$-invariant density $u_\infty$, and 
as $\T_\eps$ has only one fixed point, this density must coincide with $h_{\eps,\nu_\eps}$.
\medskip\\
{\em Second proof} (using that the invariant density is twice differentiable):\quad
Denote 
$h_*:=h_{\eps,\nu_\eps}$, $\nu_*:=\nu_\eps$. Then
\begin{eqnarray}
\label{decomposition-eqn}
  u_{i+1}-h_*
&=&
\S_{\nu_i}\L(u_i-h_*)+(\S_{\nu_i}-\S_{\nu_*})\L h_*\nn\\
&\vdots&\\
&=&
\S_{\nu_i}\L\dots\S_{\nu_0}\L(u_0-h_*)
+\sum_{k=0}^i\S_{\nu_i}\L\dots\S_{\nu_{k+1}}\L
(\S_{\nu_k}-\S_{\nu_*})\L h_*\nn
\end{eqnarray}
since $h_*=\S_{\nu_*}\L h_*$. 
It follows from Lemma \ref{L1-contraction-lemma} that
\begin{eqnarray*}
\bvnorm{u_{i+1}-h_*}
&\le&
C\,\beta^{i+1}\bvnorm{u_0-h_*}
+C\,\sum_{k=0}^{i}\beta^{i-k}
\bvnorm{(\S_{\nu_k}-\S_{\nu_*})\L h_*}\ .
\end{eqnarray*}
A straightforward calculation based on (\ref{coupling-ass}) 
and on the assumption 
$\sup_{(x,\bar\nu)}\left|\pd{}{\bar\nu}\pd{^2\gamma}{x^2}\right|<\infty$ 
shows that there is a constant 
$K>0$ depending only on the coupling and on $\L$ such that
\begin{displaymath}
  \sup_{x\in X}
\left|\left((\S_{\nu_{k}}-\S_{\nu_*})\L h_*\right)'(x)\right|
\le
K\,\eps\,\left(\sup|h_*|+\sup|h_*'|+\sup|h_*''|\right)
\,|\bar\nu_{k+1}-\bar\nu_*|\ .
\end{displaymath}
In a similar way as in part 1 it follows that for $\gamma\in(\beta,1)$ 
and sufficiently small $\eps>0$ there is $C>0$ such that 
$\bvnorm{u_{i+1}-h_*}\le C\,\gamma^i$ for all $i\in\nz$. 
This proves at the same time the uniqueness of the invariant density $h_*=h_{\eps,\nu_\eps}$
of $\tilde\L_\eps$.
\end{proof}
%

\subsection{Globally coupled piecewise linear maps}
\label{sec:piecewise-linear-sect}
%
In this section we consider piecewise linear, expanding $\tau:[0,1]\to[0,1]$
and very simple couplings described 
by the function $\gamma(x,\bar\nu)=\bar\nu$,
see (\ref{coupling-ass}).
\par
There are two references which study such systems by mathematical methods.
In \cite{ErPo1} the authors argue that if $|\tau'|>2$, then each fixed point of $\tilde\L_\eps$ is unstable so that one might expect a violation of the $0$-$1$-law to happen. 
In \cite{ChaMo} the authors mostly concentrate on post-critically finite tent maps and prove among others that the fixed point of $\tilde\L_\eps$ is stable and hence no violation of the $0$-$1$-law happens if some iterate of the critical point is a fixed point of $\tau$. Both papers are mathematically not really rigorous, because they rely on linear response theory. 
\par
Below we prove a result along the lines of reasoning of \cite{ChaMo} which
confirms the observation of \cite{ChaMo} and shows at the same time that
the assumption $|\tau'|>2$ made in \cite{ErPo1} is not sufficient to guarantee
instability of the fixed point(s) of $\tilde\L_\eps$. Nevertheless I believe that ``generically'' these fixed points are unstable.
%
\begin{theo}
\label{pwl-map-theo}
Let $\tau:[0,1]\to[0,1]$ be piecewise linear and expanding
and suppose that the coupling is of the type described in (\ref{coupling-ass})
with a function $\gamma(x,\bar\nu)=\gamma(\bar\nu)$.
Assume that the conclusions of Lemma \ref{L1-contraction-lemma}
are satisfied.
Then there is $\eps_0>0$ such that for all $\eps\in[0,\eps_0]$
holds: 
\par
Let $h_\eps$ be a fixed point of $G_\eps$ (uniqueness is not assumed {\em a priori}), $\nu_\eps:=h_\eps m$.
Assume that $\tau_{\eps,\nu_\eps}$ has a fixed point $z$, different from all its turning points,
and that there exists an integer $N$ such that 
$\tau_{\eps,\nu_\eps}^N(c)=z$ for all turning points $c$ of 
$\tau_{\eps,\nu_\eps}$.
Then
\begin{displaymath}
\T_\eps\nu_{\eps}=\nu_{\eps}\quad\text{and}\quad
\lim_{i\to\infty}\T_\eps^i\nu=\nu_{\eps}\quad\text{for all }
\nu\in\dbv
\end{displaymath}
(in the sense that there are $C>0$ and $\gamma\in(0,1)$ 
independent of $\nu$ and
such that for the density $u$ of $\nu$ we have 
$\|\tilde\L_\eps^i u-h_{\eps}\|_1\leq C\gamma^{\sqrt i}
\,\bvnorm{u}$).
\end{theo}
%
\begin{remark}\label{pwl-map-remark}
  For a broad class of piecewise linear maps it was proved in
\cite{Keller-barcelona} 
that the conclusions of Lemma \ref{L1-contraction-lemma}
are satisfied. This is in particular true for tent maps $\tau$ with slope
$|\tau'|>1.65$ and for each $\tau$ which is an iterate of such a tent map. 
\par
In the pure tent map case all maps $\tau_{\eps,\nu}$ are conjugated to the basic map $\tau$, see Example \ref{tent-example-a}. Therefore the extra assumption on the turning point of $\tau_{\eps,\nu_\eps}$ is satisfied if and only if it is satisfied for $\tau$ itself.
\end{remark}
%
\begin{proof}{ of Theorem \ref{pwl-map-theo}}
As in the proof of Theorem \ref{circle-map-theo} consider $\nu\in\D$
and let $\nu_i=\T^i\nu$ with density $u_i$ so that 
$u_{i+1}=\tilde\L u_i=\S_{\nu_i}\L u_i$. (Again we mostly
skip the subscript $\eps$.)
Fix $\nu_*\in\D$ with density $h_*$ such that $\T\nu_*=\nu_*$, \ie
$\tilde\L h_*=h_*$.
Recall that $\bar\nu_i=\int F(\tau x)u_i(x)\,dx$ and let
\begin{displaymath}
  \delta_i:=|\bar\nu_i-\bar\nu_*|\ .
\end{displaymath}
Denote $s:=|\tau_{\eps,\nu_*}'(z)|>1$ where $z$ is the fixed point of 
$\tau_{\eps,\nu_*}$ with $\tau_{\eps,\nu_*}^N(C)=\{z\}$,
$C$ being the set of turning points of $\tau_{\eps,\nu_*}$.
We are going to prove that there is $Q>0$ such that 
$\delta_i\leq Q\cdot s^{-\sqrt{i}}\cdot\bvnorm{u_0}$ for all $i\in\nz_0$.
The proof proceeds by induction, and $Q$ will be chosen in the course 
of the proof. Note that all restrictions on $Q$ will be satisfied if 
$Q$ is sufficiently large and that $Q$ depends neither on $\eps$ nor 
on the initial density $u_0$. Other constants that neither depend 
on $\eps$ and $u_0$ nor on $Q$ are denoted by ``$\const$''.
\par
Recall from (\ref{decomposition-eqn}) that
\begin{displaymath}
    u_{i+1}-h_*
=
\S_{\nu_i}\L\dots\S_{\nu_0}\L(u_0-h_*)
+\sum_{k=0}^i\S_{\nu_i}\L\dots\S_{\nu_{k+1}}\L
(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*
\end{displaymath}
where we used that $(\S_{\nu_k}-\S_{\nu_*})\L h_*=
(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*$.
Denote
\begin{displaymath}
  H_{k,i}:=\S_{\nu_i}\L\dots\S_{\nu_{k+1}}\L(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*
\end{displaymath}
and let $p=p(i):=[\sqrt i]$.
Then
\begin{align}
  \label{2star-eqn}
\delta_{i+1}
&=
\left|\int(u_{i+1}-h_*)(x) F(\tau x)\,dx\right|\\
&\leq
\|F\|_\infty\cdot\left(
\|\S_{\nu_i}\L\dots\S_{\nu_0}\L(u_0-h_*)\|_1
+\sum_{k=0}^{i-p}\|H_{k,i}\|_1\right)
+\sum_{k=i-p+1}^i\left|\int H_{k,i}(x) F(\tau x)\,dx\right|\nn
\end{align}
Since the conclusions of Lemma \ref{L1-contraction-lemma} are assumed to hold true
and since $\bvnorm{\S_{\nu_k}\S_{\nu_*}^{-1}h_*}=\bvnorm{h_*}$, 
the first term of the r.h.s. of this inequality can be estimated by
\begin{equation}
  \label{3star-eqn}
\const\cdot\left(\beta^{i+1}\bvnorm{u_0-h_*}
+\sum_{k=0}^{i-[Ap]}\beta^{i-k}2\bvnorm{h_*}
+\sum_{k=i-[Ap]+1}^{i-p}\beta^{i-p-k}\bvnorm{H_{k,k+p}}\right)  
\end{equation}
where $A:=\frac{\log s}{-\log\beta}$.
\par
In order to further estimate this term and also the second one on the r.h.s. of
(\ref{2star-eqn}), we have a closer look at $(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*$.
Observe first that $h_*$, as an invariant density of a post-critically finite
piecewise linear map, is piecewise constant with all discontinuities in the finite set $C$.
Since $S_\nu(x)=x+\eps\gamma(\bar\nu)$, we have
\begin{displaymath}
\left|S_{\nu_k}S_{\nu_*}^{-1}(x)-x\right|
=
\left|\eps\,(\gamma(\bar\nu_k)-\gamma(\bar\nu_*))\right|
\leq
\eps\cdot\|\gamma'\|_\infty\cdot\delta_k\ .  
\end{displaymath}
The function $(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*$ 
is therefore supported in a neighbourhood of the set $C$ with radius $\eps\cdot\|\gamma'\|_\infty\cdot\delta_k$.
As $\tau_{\eps,\nu_*}^N(C)=\{z\}$,
the function $H_{k,k+N}$ is thus supported in a neighbourhood of $z$ with radius $\eps\cdot\const\cdot\sum_{j=0}^N\delta_{k+j}$.
Since, by assumption, $z$ is not a turning point of $\tau_{\eps,\nu_*}$, there
is $\rho>0$ such that
$\tau_{\eps,\nu_*}'(x)=\tau_{\eps,\nu_*}'(z)=\pm s$ 
for all $x\in(z-\rho,z+\rho)$. It follows that 
for $\ell=k+N,\dots,i$ the function 
$H_{k,\ell}$ is supported in a neighbourhood of $z$ with radius
\begin{align*}
r_{k,\ell}
&:=
\eps\cdot\const\cdot\left(\dots\left.\left.\left(
\sum_{j=0}^N\delta_{k+j}\cdot s+\delta_{k+N+1}\right)\cdot s+\delta_{k+N+2}\right)\cdot s+\ldots\right)\cdot s+\delta_\ell\right)\\
&=
\eps\cdot\const\cdot\left(s^{\ell-k-N}\sum_{j=0}^N\delta_{k+j}
+\sum_{j=1}^{\ell-k-N}s^{\ell-k-N-j}\delta_{k+N+j}\right)\ .
\end{align*}
\par
Now we proceed to the inductive estimate. Choose $Q>0$ such that 
$\delta_j\leq Q\cdot s^{-\sqrt j}$ for $j=0,\dots,N^2$. 
Let $i\geq N^2$ and suppose
by induction that $\delta_j\leq Q\cdot s^{-\sqrt j}\cdot\bvnorm{u_0}$
for $j=0,\dots,i$. (Since $\bvnorm{u_0}\geq\|u_0\|_1=1$, this assumption
is satisfied for $i=N^2$.)
Then
\begin{equation}
\begin{split}
r_{k,\ell}
&\leq
\eps\cdot\const\cdot Q\cdot\bvnorm{u_0}\left(s^{\ell-k-N} s^{-\sqrt k} +
 s^{-\sqrt{k+N+1}}\cdot\frac{s^{\ell-k-N}-1}{s-1}\right)\\
&\leq
\eps Q\cdot\bvnorm{u_0}\cdot\const\cdot s^{\ell-k-\sqrt k}
\label{rki-est}
\end{split}
\end{equation}
for $\ell=k+N,\dots,i$.
\par
If $\max\{i-[Ap],\ell-p\}\leq k\leq \ell\leq i$,
this yields in view of $p=[\sqrt i]$:
\begin{displaymath}
  r_{k,\ell}
\leq
\eps Q\cdot\bvnorm{u_0}\cdot\const\cdot s^{p-\sqrt{i-[Ap]}}
\leq \eps Q\cdot\bvnorm{u_0}\cdot\const\ .
\end{displaymath}
Therefore, if $\eps<\frac\rho{Q\cdot\const}$,
then $r_{k,\ell}<\rho$ for these $k$ and $\ell$
so that the function $H_{k,\ell}$ is supported in the interval
$(z-\rho,z+\rho)$.
\par
Since all $S_\nu$ and also $\tau$ are affine on $(z-\rho,z+\rho)$, it follows at once that
\begin{displaymath}
  \var(H_{k,\ell})=s^{-(\ell-k-N)}\cdot\var(H_{k,k+N})
\end{displaymath}
for $\max\{i-[Ap],\ell-p\}\leq k\leq \min\{\ell,i-N\}$, and as
$\int H_{k,\ell}(x)\,dx=0$, it follows that
\begin{displaymath}
  \bvnorm{H_{k,\ell}}
\leq
\const\cdot s^{-(\ell-k)}\cdot\bvnorm{H_{k,k+N}}
\leq
\const\cdot s^{-(\ell-k)}
\end{displaymath}
where we used the Lasota-Yorke type inequality
(\ref{LY-inequ-1D}) in the last step and included $\bvnorm{h_*}$ in the constant.
We apply this estimate to $k=i-[Ap],\dots,i-p$ and $\ell=k+p$
(observe that $p=[\sqrt i]\geq N$):
\begin{displaymath}
  \bvnorm{H_{k,k+p}}
\leq
\const\cdot s^{-p}
=
\const\cdot \beta^{[Ap]}
\ .
\end{displaymath}
Combined with (\ref{2star-eqn}) and (\ref{3star-eqn}) this yields
\begin{equation}
\label{5star-est}
  \delta_{i+1}
\leq
\const\cdot\left(\beta^{i+1}\bvnorm{u_0-h_*}
+\beta^{[Ap]}+\beta^{[Ap]}\right)
+\sum_{k=i-p+1}^i\left|\int H_{k,i}(x) F(\tau x)\,dx\right|
\end{equation}
Observe next that $\int H_{k,i}(x)\,dx=0$ for all $k$. Therefore
we can apply (\ref{rki-est})
with $k=i-p,\dots,i$ and $\ell=i$ and obtain
\begin{align*}
  \left|\int H_{k,i}(x) F(\tau x)\,dx\right|
&\leq
\text{Lip}(F\circ\tau)\cdot r_{k,i}\cdot\|H_{k,i}\|_1\\
&\leq
\eps Q\cdot\const\cdot s^{i-k-\sqrt{k}}\cdot\left\|(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*\right\|_1\ .
\end{align*}
As $|S_{\nu_k}S_{\nu_*}^{-1}(x)-x|\leq\eps\cdot\|\gamma'\|_\infty\cdot\delta_k$,
it follows from \cite[Lemma 13]{keller-mh} that
\begin{equation}
\label{Hki-est}
\left\|(\S_{\nu_k}\S_{\nu_*}^{-1}-\Id)h_*\right\|_1
\leq\eps\cdot\const\cdot\delta_k
\end{equation}
so that
\begin{displaymath}
  \left|\int H_{k,i}(x) F(\tau x)\,dx\right|
\leq
\eps^2Q^2\cdot\const\cdot s^{i-k-2\sqrt k}
\end{displaymath}
for $k=i-p,\dots,i$.
Inserting this estimate into (\ref{5star-est}) and observing that
$1=\|u_0\|_1\leq\bvnorm{u_0}$ we arrive at
\begin{displaymath}
    \delta_{i+1}
\leq
\const\cdot\bvnorm{u_0}\cdot\left(\beta^{i+1}+\beta^{[Ap]}
+\eps^2Q^2\sum_{k=i-p+1}^i s^{i-k-2\sqrt k}
\right)
\end{displaymath}
We estimate the sum in this expression: As
$\sqrt i-\sqrt{i-j}\leq 1$ for $j=0,\dots,p=[\sqrt i]$, we have
\begin{displaymath}
  s^{\sqrt i}\cdot\sum_{k=i-p+1}^i s^{i-k-2\sqrt k}
=
  \sum_{j=0}^{p-1} s^{j-{\sqrt i}+2({\sqrt i}-\sqrt{i-j})}
\leq
\sum_{j=0}^{p-1}s^{j-p+2}
\leq
\frac{s^2}{s-1}\ .
\end{displaymath}
Therefore there is a constant $K>0$ independent of $Q$, $\eps$ and $u_0$ 
such that
\begin{displaymath}
  \frac{\delta_{i+1}}{Q\cdot s^{-\sqrt i}\cdot\bvnorm{u_0}}
\leq
K\cdot\left(\frac 1Q\beta^{i+1}s^{\sqrt{i+1}}+\frac 1Q +\eps^2Q\right)\ .
\end{displaymath}
Choosing $Q$ large enough we may assume that 
$Q\geq 2K\cdot(1+\max_{i\ge0}\beta^{i+1}s^{\sqrt{i+1}})$. Then, 
if $\eps<\sqrt{\frac 1{2KQ}}$ (and $\eps<\frac\rho{\const\cdot Q}$
from above), we have
\begin{displaymath}
  \delta_{i+1}\leq Q\cdot s^{-\sqrt{i+1}}\cdot\bvnorm{u_0}
\end{displaymath}
which finishes the induction.
\par
Finally we use these bounds to estimate $\|u_{i+1}-h_*\|_1$: With a decomposition as in (\ref{2star-eqn}) and using (\ref{Hki-est}) and the conclusions of Lemma \ref{L1-contraction-lemma} we have
\begin{align*}
\|u_{i+1}-h_*\|_1
&\leq
\const\cdot\left(\beta^{i+1}+\sum_{k=0}^{i-[Ap]}\beta^{i-k}\right)\bvnorm{u_0}
+\sum_{k=1-[Ap]+1}^{i}\|H_{k,i}\|_1\\
&\leq
\const\cdot\left(\beta^{[Ap]}\bvnorm{u_0}+\sum_{k=i-[Ap]+1}^i\delta_k\right)
\leq    
\const\cdot\sqrt{i}\cdot s^{-\sqrt i}\bvnorm{u_0}
\end{align*}
which gives the assertion of the theorem.
\end{proof}

\begin{thebibliography}{123}
%\footnotesize
\bibitem{Aldous} 
D.J.\ Aldous, {\em Exchangeability and related
topics}, Springer Lecture Notes in Mathematics {\bf 1117} (1985),
1-198.  
\bibitem{BaYo}
V.\ Baladi, L.-S.\ Young,
{\em On the spectra of randomly perturbed expanding maps,}
Commun.\ Math.\ Phys. {\bf 156} (1993), 355-385.
\bibitem{BoGobook}
A.\ Boyarsky, P.\ G\'ora,
{\em Laws of Chaos - Invariant Measures and Dynamical Systems in One Dimension,} Birkh\"auser (1997).
\bibitem{ChaMo} T.\ Chawanya, S.\ Morita, {\em On the
   bifurcation structure of the mean-field fluctuation in the globally
   coupled tent map systems},    
   Physica D {\bf 116} (1998), 44-70.
\bibitem{ErPo1} S.V.\ Ershov,
  A.B.\ Potapov, {\em On mean field fluctuations in globally coupled
    maps}, Physica D {\bf 86} (1995), 523-558.  
\bibitem{EvGa} L.C.\ 
  Evans, R.F.\ Gariepy, Measure Theory and Fine Properties of
  Functions, CRC Press (1992), 
\bibitem{Jarvenpaa}
  E.\ J\"arvenp\"a\"a,
{\em An SRB-measure for globally coupled analytic expanding circle maps,}
Nonlinearity {\bf 10} (1997), 1435-1469.
\bibitem{Just} W.\ Just, {\em Globally
    coupled maps: phase transitions and synchronization}, Physica D
  {\bf 81} (1995), 317-340.  
\bibitem{Kaneko2} K.\ Kaneko, {\em Globally coupled chaos violates the law of large numbers but not the central limit theorem},
Phys.\ Rev.\ Letters {\bf 65} (1990), 1391-1394.
\bibitem{Kaneko1} K.\ Kaneko, {\em Remarks
    on the mean field dynamics of networks of chaotic elements},
  Physica D {\bf 86} (1995), 158-170.  
\bibitem{keller-mh}
G.\ Keller, {\em Stochastic stability in some chaotic dynamical systems,} 
Monatshefte Math. 
{\bf 94} (1982), 313-333.
\bibitem{keller-amst}
G.\ Keller,
{\em Coupled map lattices via transfer
operators on functions of bounded
variation,}
Stochastic and spatial structures of dynamical systems (Ed.: S.J.\ van 
Strien, S.M.\ Verduyn Lunel)
Kon.\ Nederl.\ Akad.\ Wetensch.\ Verhandelingen, Afd. Natuurkunde,
Eerste Reeks, Vol. {\bf 45} (1996), 71-80. 
\bibitem{Keller-wien} G.\ 
  Keller, {\em Mixing for finite systems of coupled tent maps},
  Proc.\ Steklov Inst.\ Math. {\bf 216} (1997), 315-321.
\bibitem{Keller-barcelona}
  G.\ Keller, {\em Interval maps with strictly contracting Perron-Frobenius operators,}
Preprint (1997).
\bibitem{KeKu} G.\ 
  Keller, M.\ K\"unzle, {\em Transfer operators for coupled map
    lattices}, Ergod.\ Th.\& Dynam.\ Sys. {\bf 12} (1992), 297-318.
\bibitem{kuenzle}
M.\ K\"unzle,
  {\em Invariante Ma\ss{}e f\"ur gekoppelte Abbildungsgitter,}
Thesis, Erlangen (1993).
\bibitem{krzyzewski}
K.\ Krzyzewski,
{\em Some results on expanding mappings,}
Ast\'erisque {\bf 50} (1977), 205-218.
\bibitem{NaKo1} N.\ Nakagawa, T.S.\ Komatsu, {\em Dominant collective
    motion in globally coupled tent maps}, Preprint (1996).
\bibitem{NaKo2} N.\ Nakagawa, T.S.\ Komatsu, 
{\em Confined chaotic behavior in collective motion for populations of globally coupled chaotic elements}, Preprint (1998).
\end{thebibliography}
}
\end{document}
---------------9901070306253--
