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\title{Topological entropy of Standard type monotone twist maps}
\author{Oliver Knill\thanks{Division of Physics, Mathematics and Astronomy,
Caltech, 91125 Pasadena, CA, USA}}
\date{}
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\begin{document}
\maketitle
\begin{abstract}
We study invariant measures of families of monotone twist maps
$S_{\gamma}(q,p)$ $=$ $(2q-p+ \gamma \cdot V'(q),q)$
with periodic Morse potential $V$. We prove
that there exists a constant $C=C(V)$
such that the topologlical entropy satisfies
$h_{top}(S_{\gamma}) \geq \log(C \cdot \gamma)/3$. In particular,
$h_{top}(S_{\gamma}) \to \infty$ for $|\gamma| \to \infty$. We show also that
there exists arbitrary large $\gamma$ such that $S_{\gamma}$ has nonuniformly
hyperbolic invariant measures $\mu_{\gamma}$
with positive metric entropy. For larger $\gamma$, the measures
$\mu_{\gamma}$ become hyperbolic and the Lyapunov exponent
of the map $S$ with invariant measure $\mu_{\gamma}$
grows monotonically with $\gamma$.
\end{abstract}
\vspace{1mm}
\begin{center}{\bf Mathematics subject classification: } \end{center}
\begin{center} 28D10,28D20,58F11,58E30,58F15 \end{center}
\vspace{1mm}
\begin{center} {\bf Keywords:} \end{center}
Ergodic theory, monotone twist maps, topological entropy,
hyperbolic sets, Lyapunov exponents, invariant measures.
\pagestyle{myheadings}
\thispagestyle{plain}
\section{Introduction}
For a one-parameter family of $C^{\infty}$-diffeomorphisms $S_{\gamma}$
on a two-dimensional compact manifold $M$, the topological entropy
$h_{top}(S_{\gamma})$ depends continuously on $\gamma$ and there exists
an invariant ergodic measure of maximal entropy \cite{New}. Moreover, according
to Katok \cite{Kat} (using terminology of \cite{You})
the map $S_{\gamma}$ is a limit of
hyperbolic subshifts of finite type: if $h_{top}(S_{\gamma})>0$, there exist
$S_{\gamma}$-invariant
hyperbolic sets having entropy arbitrarily close to $h_{top}(S_{\gamma})$.
It follows that a change of the topological entropy is accompanied by
creations or destructions of hyperbolic sets.
Any of these sets $K_{\gamma}$ is the support of an invariant
measure $\mu_{\gamma}$ having metric entropy equal to
the topological entropy of $S_{\gamma}$ restricted to $K_{\gamma}$.
The invariant measure $\mu_{\gamma}$
depends continuously on the parameter $\gamma$.
The abstract dynamical system, $(Y,S_{\gamma},\mu_{\gamma})$ is an
ergodic subshift of finite type and we will say,
it is embedded in the topological
dynamical system $(Y,S_{\gamma})$. At the boundary of a
maximal interval for which this Markov chain is embedded,
there still exists
an invariant measure $\mu_{\gamma}$ but $(S_{\gamma},\mu_{\gamma})$
is in general only a measure-theoretical factor of $(S,T,m)$. We will call
this an immersion.
%The knowledge of the topological entropy function is in some sense equivalent
%with the knowledge about embeddings of abstract dynamical systems because
%it tells about the existence of hyperbolic stable embeddings of Markov chains.\\
We will see that the immersions of a dynamical system are compact in a suitable
topology. This will imply that for parameter values, for which a hyperbolic
invariant measure loses hyperbolicity, a nonuniform hyperbolic invariant
measure exists.
In this paper, we study embeddings and immersions of dynamical systems in
generalized Standard maps
$S_{\gamma}: (q,p) \mapsto (q',p')= (2q-p+ \gamma \cdot V'(q),q)$ on the
torus $Y=\TT^2$.
Such maps are discrete Hamiltonian systems and have a natural parametrization
by the coupling constant $\gamma$. The twist property allows to
describe $S_{\gamma}$ by variational methods. Immersions of
a given dynamical system $(X,T,m)$ are in one to one correspondence with
critical points of a class of variational functionals. For large $|\gamma|$,
critical points can be found with the implicit function theorem. This idea
is due to Aubry and Abramovici \cite{Aub} and is called the
"anti-integrable limit".
Together with some ergodic theory this will show that
$h_{top}(S_{\gamma}) \to \infty$ for $|\gamma| \to \infty$. We want to stress
that we get {\em quantitative results} about the topological entropy.
The existence of homoclinic points or other arguments like \cite{Ang1,Ang2}
give qualitative results. \\
For a fixed Standard map $S_{\gamma}$, all the possible variational functionals
on $L^{\infty}(X)$ are labeled by functions $N \in L^{\infty}(X,2\pi \ZZ)$. They
fall into equivalence classes and the quotient space is a cohomology
space $H(T)$ attached to $(X,T,m)$. The number $\int N \; dm$, the
averaged force, is a function on $H(T)$.
To each embedded system belongs an element of the cohomology space. It follows
that if two embedded systems belong to different elements $H(T)$, then they
are different. The numerical invariant $\int N \; dm$ will be used to get
multiplicity results, i.e. uncountably many different critical points.
There are other invariants of the embedding like the integrated density of
states of the Hessian at a critical point. This invariant is up to a constant
the Morse index in the case when the variational problem is finite dimensional,
i.e. if we embed a periodic orbit.
With an implicit function theorem approach similar with the one
to find critical points for large $|\gamma|$, one can compute
the stable and unstable direction-fields on the hyperbolic set.
These direction-fields are related to the Titchmarch-Weyl functions of
the Hessian, a discrete ergodic Schr\"odinger operator. By computing the
motion of the critical point belonging to a hyperbolic invariant measure
$\mu_{\gamma}$ for large $|\gamma|$, we will show that
these stable and unstable direction fields move monotonically in dependence of
$\gamma$. It follows that the Lyapunov exponent of $S_{\gamma}$ is
growing monotonically for large $|\gamma|$. This will imply with a theorem
of Fathi that the Hausdorff dimension of the hyperbolic sets is decreasing
monotonically as $|\gamma| \rightarrow \infty$. \\
Using the compactness of the immersions, we will show also that for
arbitrary large $\gamma$, there exist invariant measures with positive
Lyapunov exponents but which do not have a hyperbolic set as support.
It is an open question if such sets form a set of positive
Lebesgue measure for large $\gamma$. \\
In the end of the article, we show how some of the ideas can be
carried over to higher dimensional symplectic maps, real or complex
H\'enon like maps. We will also treat noninvertible systems like
one-dimensional circle maps, complex polynomial
or meromorphic maps.
The Aubry-Abramovici perturbation idea applies also to discrete partial
difference equations like coupled map lattices,
where it can give an easy prove of the existence of embedded two-dimensional
shifts.
\section{Invariant measures of monotone twist maps}
\subsection{Immersions and embeddings of dynamical systems}
A measure theoretical dynamical system $(X,T,m)$ is {\em embedded}
in a topological dynamical system $(Y,S)$, if there
exists an $S$-invariant probability measure $\mu$ on $Y$, such that
$(X,T,m)$ and $(Y,S,\mu)$ are isomorphic as measure theoretical
dynamical systems.
If the system $(Y,S,\mu)$ is only a factor of $(X,T,m)$ we speak of
an {\em immersion}. Two embeddings or immersions
are called {\em different}, if the measures
are different. \\
A basic question is to decide whether and how often
a given abstract dynamical system $(X,T,m)$ can be embedded or immersed.
Given two embeddings of the same dynamical system, one can not
use purely ergodic theoretical invariants to distinguish the embeddings because
both embedded systems are isomorphic.
Other invariants should allow to distinguish different
embeddings and hopefully label
the set $\Ecal_T(S)$ of embeddings and the set $\Ical_T(S)$ of immersions. \\
According to Goodman's variational theorem,
the metric entropy $h_m(T)$ of
an embedded dynamical system must be bounded from above by the topological
entropy $h_{top}(S)$. This
simple necessary condition for an embedding is in general not
sufficient: $h_m(T) \leq h_{top}(S)$ does not allow in general
an embedding. An example is an uniquely ergodic topological
dynamical system $(Y,S)$ with $h_{top}(S)>0$. No system $(X,T,m)$
with metric entropy smaller than $h_{top}(S)$ can be embedded. \\
%The problem of embedding dynamical systems can be compared with the
%analoguous problem in differential topology
%to embed or immerse manifolds in other manifolds. Invariants of immersions
%or embeddings are of cohomological nature and indices or degrees are numerical
%invariants which allow to distinguish homotopically not equivalent embeddings.
%The category of topological dynamical systems is complicated.
%Topological dynamical systems are in general not structurally stable: they
%do not allow general homotopy deformations.
%Along a one-parameter family $(Y,S_\gamma)$ of systems, the embedded systems
%$\Ecal_T(S_\gamma)$ as well as the immersed systems changes drastically. In the
%case, when $X$ is a finite set, such changes are described by bifurcations of
%periodic orbits.
%A single embedded system can however be structurally stable. In the case, when
%$(Y,S)$ is a smooth dynamical system, there can exist
%hyperbolic invariant measures, measures which have as their support a
%hyperbolic set. \\
\subsection{The ergodic Percival functional}
We study the embedding problem in the case, when the
topological dynamical system is a monotone twist map.
Embedding an ergodic dynamical system $(X,T,m)$ is then a variational problem.
We consider the special case of {\em generalized Standard maps}
$$ S: (q,p) \mapsto (q',p')= (2q-p+ V'(q),q) \; , $$
with $V \in C^2(\TT^1,\RR)$. The map $S$
acts also on the torus $Y=\TT^2$ because $S(q,p+2\pi)=(q',p'+2\pi)$ and the
topological entropy $h_{top}(S)$ is defined.
Consider for each $N \in L^{\infty}(X,2 \pi \ZZ)$
the {\em ergodic Percival functional}
$$ q \mapsto \Lcal_N(q)=\int_X h(q(x),q(Tx)) - N(x) q(x)\; dm(x) $$
on the Banach space $L^{\infty}(X,\RR)$, where
$ h(q,q')= -(q'-q)^2/2 -V(q)$
is the {\em generating function} of the twist map. These functionals $\Lcal_N$
are bounded and have the same smoothness as $V$.
Define $\Delta(q)=q(T)-2q+q(T^{-1})$.
\begin{lemma}
If there exists $q \in L^{\infty}(X)$ satisfying the {\em Euler equations}
$$ \delta \Lcal_N(q)= \Delta(q) - V'(q) - N = 0 \; ,$$
then $(X,T,m)$ is immersed in the twist map $(Y,S)$.
\end{lemma}
\begin{proof} The homomorphism
between $(X,T,m)$ and the embedded system $(Y,S,\mu)$ is given by the map
$\phi: X \rightarrow Y$,
$$ x \mapsto (q(x),q(T^{-1}x)) \; {\rm mod} \; 2\pi \ZZ \; . $$
The condition $S \circ \Phi = \Phi \circ T$ follows immediately from
$\delta \Lcal_N(q)=0$.
The push-forward measure $\mu$ on $Y$
defined by $\mu(Z)=m(\phi^{-1}(Z))$ is $S-$ invariant.
\end{proof}
If $q$ is not constant, then the immersed factor is nontrivial in the sense
that the immersed system is not isomorphic to the identity map.
If $\phi$ is injective almost everywhere on $X$,
the system $(X,T,m)$ itself is embedded. \\
{\bf Remarks}. \\
1) The fact that we look at {\em bounded} measurable functions $q$
forces the introduction of a family of variational functionals. The case
$N=0$ gives only a few embeddings. \\
2) Most of the literature
about monotone twist maps deals with a
variational problem in a space of sequences. \\
3) Other versions of the functional $\Lcal$ have been introduced earlier.
For the embedding of irrational rotations, we refer to
\cite{Per}, \cite{Mat2}, \cite{Laz}. The ergodic functional $\Lcal$ for
general dynamical systems $(X,T,m)$ appeared in \cite{Kni} in the case $N=0$.
{\bf Examples of embeddings} \\
1) Every embedded system corresponds to a critical point of a variational
problem: given an $S$-invariant probability measure $m$ on $Y$,
defining the dynamical system
$({\rm supp}(m),S,m)$, where $T$ is the restriction of $S$ to the support of $m$.
Choose a function $K \in L^{\infty}(X,2\pi \ZZ)$.
Denote with $\pi_1: \TT \times \RR \rightarrow \TT$ the projection onto the
first coordinate.
The function $q(x)=\pi_1(x) + K(x)$,
is a critical point of the functional $\Lcal_N(q)$ with
$N=\Delta(q)-V'(q)$.
The choice of the function $K$ determines the chosen lift
from $q \in L^{\infty}(X,\TT^1)$ to $L^{\infty}(X,\RR)$. \\
2) Embedded finite ergodic
dynamical system $T: x_i \mapsto x_{i+1}$
corresponds to periodic orbits. According to the Poincar\'e-Birkhoff theorem,
there exist many periodic orbits of each period. \\
3) The embedding of irrational rotations $x \mapsto x+\alpha$
of the circle is well
investigated. Functions $q(x)=x+v$ with smooth $v$ correspond to {\em KAM tori},
(see for example \cite{Mos}, \cite{Her1}).
If $v$ is discontinuous then $q$ belongs to an invariant Cantor set like
{\em Aubry-Mather sets} (see \cite{Mat2,Kat1,Ban}). \\
4) Immersions of subshifts of finite type
are possible, if there exists homoclinic points (see for example \cite{Fon}).
Embeddings of Bernoulli shifts can be obtained
by constructing horse-shoes. If a Bernoulli shift $(I^{\ZZ},T,m)$
is embedded, Krieger's
theorem (see for example \cite{Den})
assures that every ergodic dynamical system with metric entropy
$\leq \log(|I|)$ can be embedded.
\section{Invariants of embeddings}
The Hessian at a critical point $q$ of $\Lcal_N$ is the
Fr\'echet derivative of
$$ \delta \Lcal: L^{\infty}(X) \rightarrow L^{\infty}(X), \;
q \mapsto \Delta(q) - V'(q) $$
and is a discrete Schr\"odinger operator on $L^{\infty}(X)$ defined by
$ Lu =\Delta u - V''(q) u $.
One can extend this operator to a bounded linear operator
on $L^2(X)$. For $m$-almost all $x \in X$, we get a family of ergodic
Jacobi matrices $L(x) : l^2(\ZZ) \rightarrow l^2(\ZZ) $
defined by
$$ (L(x) u)_n = u_{n+1} -2 u_n + u_{n-1} + V''(q(x)) \; u_n \; . $$
Attached to such a measurable family of operators
is the {\em density of states} $dk$ (see for example \cite{Cyc}).
This is a measure on the real line having as the support
the spectrum $\sigma(L(x))$ for almost all $x \in X$. \\
A critical point $q$ is called {\em hyperbolic}, if its Hessian $L$ is
invertible. This is equivalent with $0 \notin \sigma(L(x))$ for almost
all $x \in X$. The embedded system of a hyperbolic
critical point is a hyperbolic set (see \cite{Aub1}).
Therefore, we call an invariant measure defined by a hyperbolic critical
point also {\em hyperbolic} and say also the embedding is
{\em hyperbolic}, if $\mu$ is hyperbolic.
If $\mu$ is a hyperbolic invariant measure, then it defines
by the implicit function theorem
a family of measures $S' \mapsto \mu_{S'}$, for $S'$ in a neighborhood $S$. \\
A simple example of a hyperbolic set
is a hyperbolic critical point of a finite
dynamical system which is a hyperbolic periodic orbit.
An {\em invariant} of an embedding
is a map $\psi$ from the set $\Ecal_T(S)$ into
some topological space so that for any
hyperbolic embedding $\mu$,
$S' \mapsto \psi(\mu_{S'})$ is constant for $S'$ in a neighborhood of $S$.
An example of an invariant is the {\em topological
index} of a periodic orbit. It is useful for classifying and counting
periodic orbits. Another example of an invariant
is the {\em Morse index} of a periodic orbit, if a
periodic orbit can be characterized as a critical point of some functional. \\
One motivation to consider invariants of embeddings
is that it allows to distinguish different invariant measures
$\mu,\nu \in \Ecal_T(S)$
which are not distinguishable from the ergodic point of view.
The density of states $dk(\mu)$,
the spectrum $\sigma(\mu)$ of $L$ and the
integrated density of states are
functions on $\Mcal_S$ but not invariants of the embedding since
a general perturbation of $S$ can change them even if $\mu$ is hyperbolic.
\subsection{The generalized Morse index}
A {\em generalised Morse index} $k$
of an embedding is defined as
$ k(\mu):=\int_{-\infty}^0 \; dk(E)$.
\begin{propo}
The generalized Morse index is an invariant
of the embedding.
\end{propo}
\begin{proof}
If $\mu$ is hyperbolic, the
Hessian is invertible and $0$ is not in the spectrum. This
stays true for $S'$ in a neighborhood of $S$.
The value of $k$ therefore does not change under perturbations of
$S$.
\end{proof}
For a hyperbolic
periodic orbit, the classical Morse index $\tilde{k}$
is related with the generalised Morse index by $k=\tilde{k}/Q$, where $Q$
is the period of the orbit \cite{Mat3}.
\subsection{A cohomology set}
Call two functions $N_1,N_2 \in L^{\infty}(X,2 \pi \ZZ)$
{\em T-cohomologuous}, $N_1 \sim_T N_2$,
if there exists $K \in L^{\infty}(X,2\pi \ZZ)$
and an automorphism $U$ commuting with $T$ such that
$N_1-N_2(U)=\Delta K$. Define the cohomology set
$$ H(T)= L^{\infty}(X,2\pi \ZZ)/ \sim_T \; . $$
Denote with $\Mcal_S$ the set of $S$-invariant probability measures
and let $\Ecal_S(T) \subset \Mcal_S$ be the set of $S$-invariant
probability measures which are embeddings of $(X,T,m)$.
\begin{lemma}
The function $\Psi(\mu):=(\Delta(q)-V'(q))/\sim_T$ maps
$\Ecal_S(T)$ into $H(T)$ and is independent of
the critical point $q$ belonging to $\mu \in \Ecal_S(T)$.
\end{lemma}
\begin{proof}
Let $q_1,q_2$ be two critical points with the same measure $\mu$ and
$$ \phi_i : (X,T,m) \rightarrow (Y,S,\mu), x \mapsto (q_i(x),q_i(T^{-1}x)) $$
the corresponding conjugations. The automorphism
$ U: X \rightarrow X, x \mapsto \phi_1^{-1} \phi_2(x) $
satisfies $TU=UT$ and $\phi_1(U)=\phi_2$. From the first coordinate of the
equation $\phi_1(U)=\phi_2$, we read off
$q_1(U)-q_2=K \in L^{\infty}(X,2\pi \ZZ)$ which means
$$ N_2=\Delta(q_2)-V'(q_2)
= \Delta(q_1(U))-V'(q_1(U)) + \Delta K = N_1(U) + \Delta K $$
and $N_2 \sim_T N_1$.
\end{proof}
\begin{propo}
\label{coho}
$\psi$ is an invariant of the embedding.
\end{propo}
\begin{proof}
Attached to every $\mu \in \Ical_S(T)$ is a critical point of
the functional $\Lcal_N$ and $\psi(\mu)$ is given by
$\psi(\mu)=N/\sim_T$. Assume $\mu$ is hyperbolic.
Let $q_S$ be the critical point corresponding
to $\mu_S$. By the implicit function theorem, there exists a critical point
$q_{S'}$ of $\Lcal_N$ for $S'$ near $S$ and
$\psi(\mu_S)=\psi(\mu)$, as long as $q_S$ is a critical point of $\Lcal_N$.
\end{proof}
\subsection{The averaged force}
The trace
$\rho: L^{\infty}(X,2\pi \ZZ) \rightarrow \RR, N \mapsto \int N \; dm$
projects to a function $\rho: H(T) \rightarrow \RR$ which we call the
{\it averaged force}. Define $\rho$ on $\Ecal_S(T)$ by
$$ \rho(\mu) = \rho \circ \Psi(\mu) \; . $$
We can rewrite $\rho$ with the Euler equation
$\Delta(q)=V'(q)+N$ as
$$ \rho(\mu)= -\int V'(q(x)) \; dm(x) \; , $$
where $q$ is any critical point belonging to the measure $\mu$. This
explains the name averaged force.
\begin{coro}
The averaged force $\rho$ is an invariant of the embedding.
\end{coro}
\begin{proof}
This follows from Proposition~\ref{coho}.
\end{proof}
Because it is difficult to
decide whether two functions $N,N'$ are $T$-cohomologous or not, the
simpler invariant $\rho$ is useful and will lead to multiplicity results.
%{\bf Remarks}. \\
%1) If the twist map has a homotopically nontrivial invariant circle,
%then $N=\psi(\mu)=0$ and the averaged force is zero for any invariant measure.\\
\section{Hyperbolic invariant measures and the topological entropy}
Denote by $\Sigma \subset \TT$ the set
of nondegenerate critical points of a Morse function $V \in C^2(\TT)$.
We consider a one-parameter family of
twist maps $S_{\gamma}$ on $Y=\TT^2$ given by
$$ S_{\gamma}: (q,p) \mapsto (2q-p+ \gamma \cdot V'(q),q) \; . $$
\begin{thm}
Fix $N \in L^{\infty}(X,2\pi \ZZ)$ and $q_0 \in L^{\infty}(X,\Sigma+2\pi \ZZ)$
and an ergodic dynamical system $(X,T,m)$ with
metric entropy $h_m(T) \leq \log(|q_0(X)|)$. \\
There exists $\gamma_0=\gamma(N,q_0)>0$ such that
for $|\gamma|>\gamma_0$,
there exist hyperbolic measures in $\Ecal_{S_{\gamma}}(T)$
with metric entropy $\log(|q_0(X)|)$. \\
\end{thm}
\begin{proof}
By Krieger's theorem, it is enough to embed a Bernoulli shift
$(X,T,m)=(q_0(X)^{\ZZ},T,m)$ with entropy $\log|q_0(X)|$.
In this case, any system $(X,T,m)$ with smaller entropy can be embedded. \\
Define $\epsilon=1/\gamma$.
The variational functional $ \epsilon \cdot \Lcal_N(q)$
has the Euler equations
$$ \epsilon \cdot \Delta(q) = V'(q) + \epsilon N q \; . $$
For $\epsilon=0$, every $q \in L^{\infty}(X,\Sigma)$ is a
critical point. Because $V$ is a Morse function, the Hessian $L=-V''(q)$
of such a critical point is invertible and
the implicit function theorem implies the existence of the critical
point also for small $\epsilon < \epsilon_N$.
The map $\phi(x)=(q(x),q(T^{-1}))$ conjugates $(X,T,m)$ with $(Y,S)$
and $(X,T,m)$ is an immersion. The map $\phi$ is injective: if
$\phi(x)=\phi(y)$, then $\phi(T^nx)=\phi(T^ny)$ and comparing the
first coordinates of $\phi(T^nx)=\phi(T^ny)$ gives $x_n=y_n$ and so $x=y$. \\
The twist map
$$ S: (q,p) \mapsto (2q-p+\gamma \cdot V'(q),q) $$
on $Y=\RR^2/(2\pi \ZZ)^2$ is conjugated by $(q,p) \mapsto (q/k,p/k)$
to the twist map
$$ S_k: (q,p) \mapsto (2q-p+\frac{\gamma}{k} \cdot V'(k \cdot q),q) $$
on the torus $Y_k=\RR^2/((2\pi/k) \cdot \ZZ)^2$.
The torus $Y$ is a $k^2-$ sheeted
cover of $Y_k$ and we can lift $S_k$ to a map $\tilde{S}_k$ on $Y$.
The potential $\tilde{V}(x)=V(k x)$ has $k \cdot |\Sigma|$
different critical points. For large $\gamma$, we can embed a
shift $(X,T,m)$ with $k \cdot |\Sigma|$ symbols by a map $\tilde{\phi}$.
Denote with $\tilde{\mu}$ the invariant measure given by this embedding.
$(Y,\tilde{S}_k,\tilde{\mu})$
has metric entropy $\log(k \cdot |\Sigma|)$. The projection
$\pi: Y \mapsto Y_k$ satisfies $\pi \circ \tilde{S}_k=S_k \circ \pi$.
Call $\mu=\pi^* \tilde{\mu}$.
Because $(Y,\tilde{S}_k,\tilde{\mu})$
is a finite skew-product extension of $(Y,S_k,\mu)$, we know
from the Abramov-Rokhlin formula (see for example \cite{Pet}) that
the metric entropy of $(Y_k,S_k,\mu)$ is the same as the metric entropy
of $(Y,\tilde{S}_k,\tilde{\mu})$.
Using Ornstein's results that a
factor of a Bernoulli shift is again a Bernoulli shift and the isomorphy
classes of Bernoulli shifts is determined by the metric entropy, we have
embedded the Bernoulli shift and not only a factor.
\end{proof}
{\bf Remarks}. \\
1) The idea to go to the limit $|\gamma| \rightarrow \infty$ is due to
Aubry \cite{Aub} and is called {\em anti-integrable limit}.
The idea has been used further in \cite{Aub1,Aub2}. \\
2) If $q_0(X) \; {\rm mod} \; (2 \pi \ZZ) \geq 2$ and the metric entropy of
$(X,T,m)$ is small enough, then $\Ecal_{S_{\gamma}}(T)$ is not empty. \\
3) Given $k \in [0,1]$ and $N \in H(T)$.
If the dynamical system $(X,T,m)$ is aperiodic,
there exists an immersion with Morse index $k$. We only have to
choose the function $q_0$ such that
$m\{x \in X \; | \; V''(q_0(x))<0 \}= k$. If $h_m(T)>0$ and $q_0$
is not constant, the immersion is not trivial. \\
4) Critical points $q,q'$ with the same averaged force and Morse index
need still not belong to the same embedding:
for a fixed cohomology class $N$, there are $2^{Q}$ periodic orbits for
each of these periodic orbits.
There are $Q\!/(k\! (Q-k)\!$ orbits with the same Morse index $k$.
\begin{propo}
\label{bound}
We can choose $\gamma(N,q_0)$ satisfying
$$\gamma(N,q_0))
= 3 \cdot \max\{ |V'''|,4\} \cdot
(\max \{ |\Delta q_0 - N|_{\infty}, ||L_0^{-1}|| \})^3 \; . $$
\end{propo}
\begin{proof}
Differentiation of the Euler equation
$\epsilon \cdot \Delta(q) + V'(q) = \epsilon \cdot N$
with respect to $\epsilon$ and denoting the derivative with respect to
$\epsilon$ with a dot gives
$$ \dot{q}=-L^{-1} (\Delta(q)-N) $$
with $\dot{L}=V'''(q) \cdot \dot{q}$ so that
\begin{eqnarray*}
\frac{d}{d\epsilon} ||L^{-1}|| &\leq& ||\frac{d}{dt} L^{-1}||
\leq ||L^{-1}||^2 \cdot ||\frac{d}{dt} L|| \\
&\leq& ||L^{-1}||^3 \cdot |V'''|_{\infty} \cdot
|\Delta q-N|_{\infty} \;.
\end{eqnarray*}
We have also
\begin{eqnarray*}
\frac{d}{d\epsilon} |\Delta q-N|_{\infty}
\leq 4 \cdot | \dot{q} |_{\infty} \leq 4 \cdot ||L^{-1}||
\cdot |\Delta q-N|_{\infty} \; .
\end{eqnarray*}
These two differential inequalities
can be written with $a=||L^{-1}||$ and $b=|\Delta q-N|_{\infty}$
as
$$ \dot{a} \leq |V'''|_{\infty} b a^3, \; \; \dot{b} \leq 4 b a \; . $$
The function $c=\max\{a,b,1\}$ satisfies
$$ \dot{c} \leq \max \{ |V'''|_{\infty},4 \} \cdot c^4 $$
which has a solution for
$$ \epsilon \leq
(3 \cdot \max\{ |V'''|,4\} \cdot
(\max \{ |\Delta q_0 - N|_{\infty}, ||L_0^{-1}|| \})^3)^{-1} \; . $$
\end{proof}
\begin{coro}
For any fixed Morse function $V \in C^3(\TT^1)$, there exists a
constant $C$ such that $h_{top}(S_{\gamma}) \geq \log(C \gamma)/3$
for all $\gamma$.
\end{coro}
\begin{proof}
Proposition~\ref{bound} implies
that $\gamma(N,q_0^{(n)}) \leq M \cdot n^3$ with some constant $M$.
Take a sequence of functions $q_0^{(n)} \in L^{\infty}(X,2\pi \ZZ)$
with $|q_0^{(n)}(X)|=n$.
For $\gamma > M \cdot n^3$, we have
$$ h_{top}(S_{\gamma}) \geq \log(n) \;. $$
\end{proof}
\begin{coro}
Given a Morse function $V \in C^3(\TT^1)$.
Let $(X,T,m)$ be a dynamical system of finite entropy. For $\gamma$ large
enough, there exist simultaneous embeddings such that the averaged force
takes any value in $[0,1]$.
\end{coro}
\begin{proof}
The averaged force $\int_X N \; dm$ takes any value in $[0,1]$ as
$N$ runs through $L^{\infty}(X,\{0,2\pi\})$.
Take $\gamma$ larger than a lower bound given in Proposition~\ref{bound}
holding for all $N \in L^{\infty}(X,\{0,2\pi\})$.
\end{proof}
%Remark. Not every embedded shift is
%obtained by a homoclinic point or a horse shoe
%because there are only countably many homoclinic points and countably many
%Smale horse shoes.
%According to a theorem of Katok there exists for any $\epsilon>0$ a compact
%$T$-invariant zero-dimensional uniformly hyperbolic set $\Lambda$ such
%that the topology of $T$ restricted to $\Lambda$ is $\geq h(T)-\epsilon$. \\
\section{Immersed systems}
We defined $\Ecal_S(T)$ as the set of all $S$-invariant probability measures
$\mu$ such that $(X,T,m)$ and $(Y,S,\mu)$ are isomorphic.
This set is in general not
closed in the compact set $\Mcal_S$ of all $S$-invariant
probability measures as the following example shows:
take $X=\TT^1, T(x)=x+\alpha$ with
$\alpha$ irrational and $S(z)=e^{i\alpha} z$ on $Y=\{|z| \leq 1\}$.
The sequence $\mu_n \in \Ecal_S(T)$
of Lebesgue measures on $\{|z|=1/n\}$ converges to
the invariant measure $\mu$ on the fixed point $0$ which is no more
in $\Ecal_S(T)$. \\
We obtain however compactness if we consider
the set $\Ical_S(T)$ of all $S$-invariant probability measures $\mu$
on $Y$ such that $(Y,S,\mu)$ is a factor of $(X,T,m)$:
\begin{lemma}
\label{compact}
Given an abstract dynamical system $(X,T,m)$ and a topological
dynamical system $(Y,T)$, where $Y$ is a compact manifold.
The set $\Ical_S(T)$ is compact in $\Mcal_S$.
Moreover, if $S_n \rightarrow S$ in the uniform topology $C(Y,Y)$ and
$\mu_n \in \Ical_{S_n}(T)$. Then $\mu_n$ has an accumulation point
in $\Ical_S(T)$.
\end{lemma}
\begin{proof}
Assume
$\mu_n \in \Ical_{S_n}(T)$ converges to $\mu \in \Mcal_S$.
There exists a $T$-invariant set $X'$ of full measure and
a sequence of measurable maps $\phi_n:X \rightarrow Y \subset \RR^2$
such that $\phi_n T(x)=S \phi_n(x)$ for all $x \in X' \subset X$ and
all $n>0$.
The sequence $\phi_n$ has by Tychonov an accumulation point
$\phi$ in the space $Y^{X'}$ of all numerical functions $X' \rightarrow Y$.
Pass to a converging subsequence.
The function $\phi$ is measurable as a pointwise limit of
measurable functions. We have
$$ \phi_n(Tx) \rightarrow \phi(Tx) , x \in X' \eqno{(1)}$$
since $\phi_n$ converges pointwise and $X'$ is $T$-invariant. Since $S$
is continuous and $S_n \rightarrow S$ in $C(Y,Y)$, we obtain
$$ S_n \phi_n(x) \rightarrow S \phi(x), \; x \in X' \eqno{(2)} \; . $$
$(1)$ and $(2)$ together imply that $\phi T(x)=S \phi(x)$ pointwise
for all $x \in X'$. Because $\mu_n \rightarrow \mu$ and by
Lebesgue's dominated convergence theorem, we have for all $f \in C(Y)$
$$ 0=\int f \; d\mu_n-\int f(\phi_n) \; dm
\rightarrow \int f \; d\mu - \int f(\phi) \; dm \; . $$
Therefore
$$ \int_X f(\phi(x)) \; dm(x) = \int_Y f(y) \; d\mu(y) , \;
\forall f \in C(Y) \; .$$
This shows that $\phi$ is measure-preserving as a map from $(X,m)$ to $(Y,\mu)$.
\end{proof}
{\bf Remarks}. \\
In the twist map case, if
$\mu_n \in \Ical_S(T) \rightarrow \mu \in \Ical_S(T)$,
there exists a sequence of critical points $q_n$
which converges pointwise to a critical point $q$ with the measure
$\mu$. The set of critical points of $\Lcal_N$
in $L^{\infty}(X)$ is therefore compact in the topology of pointwise convergence.
A $S_{\gamma}$-invariant measure $\mu_{\gamma}$ is called {\em nonuniform
hyperbolic}, if it is not hyperbolic but the Lyapunov exponent of the measure
is positive.
\subsection{Nonuniform hyperbolicity}
\begin{propo}
For all $\gamma>0$, there exists $\gamma'>\gamma$ such that
$S_{\gamma'}$ has a nonuniform hyperbolic
invariant measure.
\end{propo}
\begin{proof}
Let $(X,T,m)$ be a Bernoulli shift with finite entropy and
choose $N$ with $\int_X N \; dm \notin 2 \pi \ZZ$.
The function $N$ is
not $T$-cohomologuous to a constant function and every measure
$\nu \in \Ical_S(T)$ is aperiodic since $T^n$ is ergodic. For large
$\gamma$, the hyperbolic measures $\mu_{\gamma} \in \Ecal_S(T)$ exist. Take
$\gamma'=\inf \{\gamma \; | \; \mu_{\gamma} \; {\rm hyperbolic} \; \}$.
This gives by the compactness-lemma~\ref{compact}
a measure $\mu_{\gamma'}$ which is aperiodic.
By Ruelle's formula (see for example \cite{Man}),
the Lyapunov exponent satisfies
$\lambda(\mu_{\gamma}) \geq h_{\mu_{\gamma}}(T)>0$.
On the other hand, $\mu_{\gamma}$ is not hyperbolic.
\end{proof}
{\rm Remarks}.
1) For all $\gamma>0$, there exists $\gamma'>\gamma$ such that
$S_{\gamma'}$ has a parabolic fixed point.
Proof. Given $\gamma>0$.
Choose $N \in 2\pi \ZZ$ so large that
no solution $q$ of the equation
$ \gamma \cdot V'(q)=N $
exists. For $\gamma''>\gamma$ large enough, there exists a solution.
Choose $\gamma'$ as the infimum over all
values for which there exists a hyperbolic solution.
The fixed point for $S_{\gamma'}$ is parabolic. \\
2) For all $\gamma>0$, there exists $\gamma'>\gamma$ such that
$S_{\gamma'}$ has parabolic periodic orbits of period $2$.
Proof. Fix $\gamma=1/\epsilon$.
Take $|X|=\{1,2\}$
and write $q(i)=q_i$. Choose $N_1=N(1)=0$ and $N_2=N(2)$ with
$N_2/\gamma> 2 ||V'||_{\infty}$.
Adding the two Euler equations
$$ 2 \epsilon (q_1-q_2) - V'(q_2) = 0, \;
2 \epsilon (q_2-q_1) - V'(q_1) = \epsilon N_2 $$
we obtain $V'(q_1)+V'(q_2)= \epsilon N_2$ which has no solution because
we assumed $N_2 \epsilon> 2 ||V'||_{\infty}$.
If we take $\epsilon''=1/\gamma''$
small enough, we get a hyperbolic critical point. Therefore, there exists
a value $\gamma'$ between $\gamma$ and $\gamma''$, where the hyperbolic periodic
orbit becomes parabolic. Since there are no critical points with
$q_1=q_2$, the parabolic orbit has really period two.
\section{Asymptotic behaviour of the critical points for large $\gamma$}
\begin{propo}
Fix $N$ and $(X,T,m)$. For $\epsilon=1/\gamma=0$,
instantaneously many hyperbolic critical points of $\Lcal_N$ appear.
Each of these critical points $q$ satisfies
the differential equation $\dot{q}=-L^{-1}(q) (\Delta q -N)$.
The critical points are labeled by the initial velocity
$\dot{q}_0=-L^{-1}(q)(\Delta q_0 -N)$.
\end{propo}
\begin{proof}
The motion of a hyperbolic critical point $q=q_{\epsilon}$ is given for
$\epsilon=0$ by
$ \dot{q}=-L^{-1} V'(q)/\epsilon = -L^{-1}(q) (\Delta q -N) $.
In the limiting case $\gamma=\infty$, we have still
$\dot{q}=-L^{-1}(q_0) (\Delta q_0 -N)$
which depends on $q_0$ and $N$. For
$\gamma=\infty$, many hyperbolic critical points are born if we consider
$\epsilon=1/\gamma$ as time. These critical points
can be labeled by the values of $(\Delta q_0 -N)$ by the uniqueness of
solutions of differential equations in Banach spaces.
\end{proof}
Remark. Solving the differential equation of the critical points allows to
compute numerically hyperbolic periodic orbits of $S_{\gamma}$ for
large $\gamma$.
\begin{propo}
\label{monotone}
Fix $N \in L^{\infty}(X,2\pi \ZZ)$ and
$q_0 \in L^{\infty}(X,\Sigma + 2 \pi \ZZ)$. Denote by
$q_{\epsilon}$ a critical point of $\Lcal_N$
(satisfying $q_{\epsilon} \rightarrow q_0$ for $\epsilon \rightarrow 0$),
and by $\mu_{\epsilon}$
the corresponding hyperbolic invariant measure for $\gamma=1/\epsilon$.
If $|\epsilon|$ is small enough, then
$ {\rm sign}(\dot{q}_{\epsilon})
= - {\rm sign}(V'(q_{\epsilon}) V''(q_{\epsilon}))$.
This means that for every $x \in X$, the value
$q_{\epsilon}(x)$ moves monotonically
away from $\Sigma+2 \pi \ZZ$ as $\epsilon$ is increasing.
\end{propo}
\begin{proof}
Differentiation of the Euler equations
$\epsilon(\Delta(q)-N)=V'(q)$ with respect to $\epsilon$ gives
$$ (\epsilon \Delta -V''(q)) \dot{q}=-(\Delta(q)-N)= - V'(q)/\epsilon \;. $$
Dividing both sides by $V''(q)$ gives
$$ \dot{q}=- (1-\epsilon \Delta/V''(q))^{-1} V'(q)/V''(q)
=- (1+O(\epsilon)) \frac{V'(q)}{V''(q) \epsilon} \; .$$
For small $\epsilon$,
$$ {\rm sign} (\dot{q}(x))= -{\rm sign}(V'(q(x))/V''(q(x)))
=-{\rm sign} (V'(q(x)) \cdot V''(q(x))) \; . $$
\end{proof}
\begin{lemma}
\label{Lyapunov}
Let $(X,T,m)$ be any dynamical system.
Given a one parameter family $a_t$ of functions in $L^{\infty}(X)$ satisfying
and $|a_t(x)| \geq |a_s(x)|$ for $t \geq s$. If $|b_t|_{\infty}=|1/a_t|_{\infty}$
is small enough, the Lyapunov exponent
$$ \lambda(A_t)
=\lim_{n \rightarrow \infty}
\frac{1}{n} \int_X \log ||A_t^{n-1}(x) \cdots A_t(x)|| \; dm(x) $$
of $x \mapsto A_t(x)= \left( \begin{array}{cc}
a_t(x) &-1 \\
1 & 0 \\
\end{array} \right) $
is monotonically increasing in $t$.
\end{lemma}
\begin{proof}
If $|1/a|_{\infty}$ is small enough, the discrete Schr\"odinger operator
$(L_a(x) u)_n = u_{n+1} + u_{n-1} + a(T^nx) u_n$ is invertible.
There exist two
solutions $u^{\pm} \in R^{\ZZ}$ of $L_a(x)u=0$ which are in $l^2(\pm \NN)$.
The {\em Titchmarsh-Weyl functions}
$m_a^{\pm}(x)=u^{\pm}_{n+1}/u^{\pm}_{n}$ are real (possibly infinite) and
the Lyapunov exponent satisfies
$ \lambda(A_a) = \int_X \log(m_a^-(x)) \; dm(x) $.
The function $m_a^-$ satisfies the discrete Ricatti
equation $m_a^-(T) + a +1/m_a^- =0 $.
For $b=1/a$ and $l=1/m^-$, this is equivalent to
$$ F(b,l)=b + l (T) + b \cdot l \cdot l (T) = 0 \; . $$
Because $F(0,0)=0$ and $D_2 F(0,0) u = u(T)$,
the implicit function theorem implies for small $|b|_{\infty}$
a solution $G \in L^{\infty}(X)$ of
$ F(b,G(b)) = 0$ with $m^-(a)=1/G(1/a))$.
>From $D F(b,G(b)) = 0$, we get, writing $l=G(b)$
$$ 1+Dl +l \cdot l (T) + b (Dl \cdot l (T) +l \cdot Dl (T))=0 $$
which shows that $Dl=-1+ O(||b||)$.
It follows that $b \mapsto 1/l (b)$ is monotone for small $|b|_{\infty}$.
Therefore, also the map
$$ b \mapsto \lambda(b)= \int_X \log(1/l (b)) \; dm(x) $$
is monotone as well as $t \mapsto \lambda(A_t)$.
\end{proof}
The {\em Lyapunov exponent of a critical point} $q$ is defined as the Lyapunov
exponent
$$ \lambda(\gamma)=\lim_{n \rightarrow \infty}
\frac{1}{n} \int_{\TT \times \RR} \log||dS_{\gamma}^n|| \; d\mu_{\gamma} $$
of $S_{\gamma}$ with respect to
the invariant measure $\mu_{\gamma}$ belonging to $q_{\gamma}$.
\begin{coro}
For small enough $\epsilon$, the Lyapunov exponent of $\mu_{\epsilon}$
increases monotonically as $\epsilon \rightarrow 0$.
\end{coro}
\begin{proof}
The transfer matrix
$x \mapsto A_{\epsilon}(x)= \left( \begin{array}{cc}
{1 \over \epsilon} \cdot V''(q(x))+2 &-1 \\
1 & 0 \\
\end{array} \right) \in SL(2,\CC) $
of $L_{\epsilon}$ is the Jacobean $dS_{\gamma}$ of the twist map with
$\gamma=1/\epsilon$.
Proposition~\ref{monotone} implies that for large enough
$\gamma$ and for all $x \in X$, the map
$\gamma \mapsto |\gamma \cdot V''(q_{\gamma}(x))|$ is monotone.
Apply Lemma~\ref{Lyapunov}.
\end{proof}
{\bf Remarks}.
1) The Hausdorff dimension
of the hyperbolic set ${\rm supp}(\mu_{\gamma})$ goes to zero
for $|\gamma| \rightarrow \infty$
since the Hausdorff dimension of the hyperbolic set is
by a theorem of Fathi \cite{Fat},
bounded above by $2 h_{top}(q_{\gamma})/\lambda(\gamma)$, where
$\lambda(\gamma)$ is the contraction rate of the uniform hyperbolic cocycle
$DT(\gamma)$.
It follows that the measure of the hyperbolic sets is zero for large $|\gamma|$.
For hyperbolic sets which are locally maximal, it has been observed in \cite{Aub1}
using a result of Ruelle and Bowen \cite{Bow} that the measure is zero. \\
2) If $V$ is real analytic,
the supports of two different measures $\mu_N,\mu_{N'}$ have in
general a nonempty intersection because there are only countably many
different periodic orbits of a fixed period and the support of each $\mu_N$
contains many periodic orbits. \\
\section{Generalisations and relations with other systems}
{\bf Symplectic twist maps}.\\
A part of the above discussion can be generalized in a straightforward way to
symplectic twist maps with generating function
$l (q,q')=-(q-q',q-q')/2 - \gamma \cdot V(q) $,
where $V \in C^2(\TT^d,\RR)$. It is a map on $Y=\TT^{2d}$ given by
$ S: (q,p) \mapsto (2q-p+ \nabla V(q),q)$.
An example of such a symplectic map is the three parameter
{\em Fr\"oschle family} with
$V(q)=\gamma_1 \cos(x_1) + \gamma_2 \cos(x_2)+\mu \cos(x_1+x_2)$.
%(see for example \cite{KM}.) \\
To each $S$-invariant measure $m$, there is a
function $q \in L^{\infty}(X,\RR^d)$ which is a critical point of the
functional
$ \Lcal_N(q)=\int_X l (q,q(T)) + (N, q) \; dm $,
where $N \in L^{\infty}(X,2 \pi \ZZ^d)$. The vector $ \int_X N(x) \; dm(x) $
is the averaged force vector.
There is a map from the set of invariant probability measures
to the cohomology space
$ L^{\infty}(X,2 \pi \ZZ^d)/ \sim_T $.
The second variation of $\Lcal_N$ is a
random {\em discrete Schr\"odinger operator on the strip}
$L= \tau -2 + \tau^* - \gamma \cdot V''(q)$,
where $V''(q(x))$ is the Hessian of $V$ at a point $q(x) \in \TT^d$.
The generalized Morse index is defined as before.
The topological entropy of $S_{\gamma}$ diverges like
in the one-dimensional case. \\
{\bf Twist maps on the real or complex plane}. \\
Consider a H\'enon type conservative
twist maps on $\RR^2$ or $\CC^2$ given by
$ S_{\gamma}: (x,y) \mapsto (2x-y+ \gamma \cdot V'(x),x) $,
where $V \in C^2(\RR^2)$ (rsp. $C^2(\CC^2)$)
is a Morse function having at least $2$ critical points.
In the variational functional, $N$ must vanish.
Given a dynamical system $(X,T,m)$.
A critical point of the functional
$\Lcal(q)=\int_X -(q(T)-q)^2/2 + V(q) \; dm $
corresponds to an embedding
of the dynamical system in the twist map. There is no cohomology because
we are the plane and not on the cylinder. For large enough $\gamma$,
every system embedded in the twist map with potential $\gamma \cdot V$
is embedded as a hyperbolic set.
Actually, most (with respect to Lebesgue measure)
of the orbits of these H\'enon maps escape to infinity
and there is a hyperbolic set left invariant. The generalized
Morse index of an embedded
system can be defined as before. The topological entropy of $S_{\gamma}$
is constant $\log(|\Sigma|)$ for large enough $\gamma$ (compaire \cite{Fri}).\\
{\bf Circle maps}. \\
The embedding question makes also sense for
a noninvertible dynamical system $(X,T,m)$.
%It is immersed in a topological dynamical system $(Y,S)$
%if there exists a $S$ invariant measure $\mu$
%such that $(Y,S,\mu)$ is a factor of $(X,T,m)$.
Circle maps of the form
$ x \mapsto x+\alpha + \gamma \cdot V(x) $
on $\TT^1=\RR /(2 \pi \ZZ)$, with parameters $\gamma$ and $\alpha$ and
$V \in C^2(\TT^1)$ are called {\em generalised Arnold maps}.
The variational functional does not exist unlike the case of twist maps,
but we can look at solutions
of $q \in L^{\infty}(X)$ of the equation
$ F_N(q)=q(T)-q-\alpha-\gamma \cdot V(q)-N = 0$,
with $N \in L^{\infty}(X,2 \pi \ZZ)$.
Given a zero $q$ of $F$, we have
an immersion of the dynamical system defined by $\phi(x)=q(x)$.
The implicit function theorem implies that for sufficiently large $\gamma$,
the system $(X,T,m)$, (for example a one-sided Bernoulli shift), is embedded.
The cohomology set is defined as
$ H(T)= L^{\infty}(X,2 \pi \ZZ)/ \sim_T $,
where $N \sim_T N'$ if and only if
there exists an isomorphism $U$ commuting with $S$ such that
$N(S)-N'=K(T)-K$ for
some $K \in L^{\infty}(X,2\pi \ZZ)$. The average force of an embedded
system is $\rho(q)=\int N \; dm$.
The topological entropy is diverging to $\infty$ for
$\gamma \rightarrow \infty$. \\
{\bf Polynomial maps}. \\
Consider a family of polynomial maps $\gamma \cdot P:\CC \mapsto \CC$.
There is no variational setup
any more but we can look for functions $q \in \L^{\infty}(X)$ which satisfy
$F(q)=q(T)- \gamma \cdot P(q)=0$.
The derivative of $F$ is an operator $\tau- \gamma \cdot P'$.
%As a generalised Morse
%we can take $\int_{\{Re(z)<0\}} dk(z)$, where $dk$ is the density of states
%of the operator $\tau - \gamma \cdot P'(q)$.
The limit $\gamma \rightarrow \infty$
corresponds to $F(q)=P(q)$. If $\Sigma$ is the set of nondegenerate zeros
of $P$, every $q \in L^{\infty}(X,\Sigma)$
is a solution.
For large $\gamma=1/\epsilon$, there is by the implicit function theorem
also a zero $q$ of
$\epsilon \cdot q(T) - P(q)$.
The embedded systems have their support in the {\em Julia set} of
the map $z \mapsto \gamma \cdot P(z)$. The topological entropy is
bounded from above by $\log({\rm deg}(P))$. \\
{\bf Meromorphic maps} . \\
Consider two entire maps $f,g$ and a family of meromorphic maps
$z \mapsto \gamma f(z)/g(z)$. Examples are
rational maps $z \mapsto \gamma \cdot P(z)/Q(z)$, where $P$ and
$Q$ are polynomials or $z \mapsto \gamma \cdot \tan(z)$.
To embed $(X,T,m)$, we seek solutions $q \in L^{\infty}(X)$
of
$ g(q) \cdot q(T) - \gamma \cdot f(q) = 0$.
Denote with $\Sigma_f, \Sigma_g$ the set of simple zeros of
$f$ and $g$. By the implicit function theorem, we have for
small $|\gamma|$, solutions $q$ near $L^{\infty}(X,\Sigma_g)$ and
for large $|\gamma|$ solutions $q$ near $L^{\infty}(X,\Sigma_f)$.
For meromorphic maps with infinitely many zeros and poles,
we can embed a one-sided
Bernoulli shift over any finite
alphabet $I \subset \Sigma_f$, if $|\gamma|$ is large and a one-sided
Bernoulli shift
over any finite alphabet $J \subset \Sigma_g$, if $|\gamma|$ is
small enough. The topological entropy is then growing to $\infty$ for
$\gamma \rightarrow \infty$ and $\gamma \rightarrow 0$. \\
{\bf Coupled map lattices}. \\
A {\em coupled map lattice} is obtained
by taking any not necessarily invertible
dynamical system $f=\gamma V': M \mapsto M$, where
$M = \TT^n$, extending $f$ to $M^{\ZZ}$ by
$f(x)_n=f(x_n)$ and considering the map $S$ on $M^{\ZZ}$ defined by
$Sx= \epsilon \cdot \Delta f(x) + f(x) \; {\rm mod} \; (2 \pi \ZZ)^d$.
This is an infinite chain of coupled dynamical systems which is
for $\epsilon=0$ a lattice of uncoupled noninteracting maps.
Because the time-evolution $S$ commutes with space
translation $(Ux)_n=x_{n+1}$, one can consider the system as a dynamical
system with time $\ZZ \times \NN$.
Assume, the set of hyperbolic fixed points $\Sigma$ of $V$ contains at
least two points. Take the measure-theoretical dynamical system
$(X=\Sigma^{\ZZ \times \NN},T \times R,m)$,
where $m$ is the product measure
on $X$ and $T$ rsp. $R$ is the two-sided rsp. one-sided shift.
We say that such a dynamical system
is {\em embedded} in the coupled map lattice, if there exists a measure $\mu$
on $M^{\ZZ}$ which is invariant under $S$ and $U$ such that the
systems $(X,T \times R,m)$ and
$(M^{\ZZ},U \times S,\mu)$ are isomorphic
as measure theoretical $\ZZ \times \NN$-dynamical systems.
In order to embed systems, we look for solutions $q \in L^{\infty}(X)$ satisfying
$ \tilde{F}(q)=q(R)-\epsilon \cdot \Delta_T f(q) - f(q) =0 $.
With $\epsilon'=1/\gamma$, this can be written as
$ F(q)=\epsilon' \cdot q(R) - \epsilon \epsilon' \cdot \Delta_T V'(q) -V'(q) = 0$.
For $\epsilon=\epsilon'=0$, every $q \in L^{\infty}(X,\Sigma)$ is a solution.
The implicit function theorem allows to extend this solution for small
enough $|\epsilon|,|\epsilon'|$. The map $\phi:X \mapsto M^{\ZZ}$,
$x \mapsto \{ V'(q(T^nx)) \}_{n \in \ZZ}$
embeds the system $(X, V \times T,m)$ into the coupled map lattice.
A concrete example of such a system is $M=\TT^1$, $V(x)=\cos(x)$. For
$\gamma$ large enough and $\epsilon$ small enough, there is a two-dimensional
shift embedded. \\
{\bf Discrete partial difference equations}. \\
One can also consider the problem to embed a $\ZZ^d$
dynamical systems $(X,T,m)$ with $T=(T_1, \dots , T_d)$ into a partial
difference equation $\Delta(u) = \gamma \cdot V'(u)$,
where $\Delta$ is the discrete Laplacian.
We mean with embedding that there exists a
solution $q \in L^{\infty}(X)$ of
$\Delta(q) = \gamma \cdot V'(q)$ such that the configuration map
$\phi: x \mapsto q(T^nx) \in \RR^{\ZZ^d}$ is injective on a set $X' \subset X$
of full measure. Such solutions exist for large $\gamma$ and are critical
points of the functional
$\Lcal(q) = \int \sum_{i=1}^n (q(T_i)-q)/2 - V(q) \; dm$.
The push-forward measure $\mu$ of $m$ on $\RR^{\ZZ^d}$ has a
compact support $Y$ and
$\mu$-almost every $\phi(x)$ satisfies the partial difference equation.
The map $\phi$ conjugates the $\ZZ^d$ systems $(X,T,m)$ and $(Y,S,\mu)$, where
$S_i$ are the shifts on $\RR^{\ZZ^d}$. The Hessian
at a critical point $q$ is a discrete ergodic Laplacian $L=\Delta-V''(q)$
(see \cite{Kni3}).
%\end{document}
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\end{document}