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rotation vector, Aubry-Mather theory, symplectic maps
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\begin{document}
\begin{titlepage}
\begin{center}
{\bf \Large Construction of Invariant Measures of Lagrangian Maps:}
{\bf \Large Minimisation and Relaxation}
\vspace{2ex}
{\large Sini\v{s}a Slijep\v{c}evi\'{c}}
\end{center}
\vspace{4ex}
\begin{abstract}
If $F$ is an exact symplectic map on the $d$-dimensional cylinder $\Bbb{T}^d \times \Bbb{R}^d$,
with a generating function $h$
having superlinear growth and uniform bounds
on the second derivative, we construct a strictly gradient semiflow $\phi^*$
on the space of shift-invariant probability measures on the space of configurations
$({\Bbb{R}}^d)^{\Bbb{Z}}$. Stationary points of $\phi^*$ are invariant measures
of $F$, and the rotation vector and all spectral invariants are invariants of $\phi^*$.
Using $\phi^*$ and the minimisation technique, we construct minimising
measures with an arbitrary rotation vector $\rho \in \Bbb{R}^d$, and with an
additional assumption that $F$ is strongly monotone, we show that the support of every
minimising measure is a graph of a Lipschitz function.
Using $\phi^*$ and the relaxation technique, assuming
a weak condition on $\phi^*$ (satisfied e.g. in the Hedlund's counter-example, and in
the anti-integrable limit)
we show existence of double-recurrent orbits of $F$ (and $F$-ergodic measures)
with an arbitrary rotation
vector $\rho \in \Bbb{R}^d$, and the action arbitrarily close to the minimal action
$A(\rho)$.
\end{abstract}
\vspace{10ex}
{\it 1991 MSC:} 58F05, 58F11
\vspace{1ex}
{\it MSC 2000:} 37J**, 37J50, 37L45
\vspace{30ex}
{\large Sini\v{s}a Slijep\v{c}evi\'{c}
Department of Mathematics
Bijeni\v{c}ka 30
10000 Zagreb, Croatia
\vspace{1ex}
fax: +385 1 6601221
e-mail: {\tt slijepce@@cromath.math.hr}
}
\end{titlepage}
\newpage
\section{Introduction}
The duality between gradient and Hamiltonian dynamics has been
a part of mathematical folklore for some time now. A beautiful
example of an application of this idea is Floer's approach to Arnold's
conjecture on the number of fixed points of a Hamiltonian map:
he studied a ``gradient'' dynamics on the space of contractible
paths, whose equilibria are the 1-periodic orbits of the Hamiltonian
flow.
Here we construct a strictly gradient semiflow $\phi^*$ on a
separable, metrisable, complete space ${\cal A}^*$ which
in a way contains all the information about a given Hamiltonian
(Lagrangian in particular) system. This means that,
in principle, {\it a question about a Lagrangian system can
be formulated as a question about its ``dual'' gradient
semiflow} $\phi^*$.
For the simplicity of arguments, here we restrict our attention
to Lagrangian maps on the cotangent bundle $T^*\Bbb{T}^d$ of
the $d$-dimensional torus $\Bbb{T}^d$. Lagrangian maps are
the discrete-time analogues of Lagrangian flows, i.e.
maps whose orbits are
extremals of the action functional $\sum h(x_i,x_{i+1})$, where
$h: \Bbb{R}^d \times \Bbb{R}^d \rightarrow \Bbb{R}$ is the generating function.
The semiflow $\phi^*$ is the semiflow on the space of shift-invariant
measures on a subset of $(\Bbb{R}^d)^{\Bbb{Z}}$, induced by
the formally gradient dynamics of the action functional.
The set of equilibria of $\phi^*$ is the set of invariant
measures of the Lagrangian map.
From the point of view of variational theory, the semiflow $\phi^*$
is a powerful new tool for construction of closed sets of measures,
such that the minimum of the action functional restricted to
the closed set is a local minimum of the action functional.
We develop two methods of use of the
semiflow $\phi^*$ for construction of invariant measures of
Lagrangian maps: the method
of {\it minimisation} (a modification of the standard variational technique), and
{\it relaxation}. We then apply these methods and address the following questions:
\vspace{1ex}
\noindent
{\bf Minimising measures.} The most successful generalisation of Aubry-Mather
theory to more degrees of freedom is by Mather \cite{Mather:91b}, who constructed
minimising measures of a Lagrangian flow with an arbitrary rotation vector, and
proved that their supports are graphs of a Lipschitz function. Using the minimisation
technique, we prove the analogues of Mather's results for Lagrangian maps. The use of
the semiflow $\phi^*$ slightly simplifies and
generalises the proof of the existence of action-minimising measures with a given rotation vector: positive definiteness
of the generating function is not a necessary condition; periodicity, superlinear
growth and e.g. uniform bounds on the second derivatives of $h$ suffice.
Using the additional assumption that $x \mapsto h_2(x,y)$, $x \mapsto h_1(y,x)$
are diffeomorphisms (where indices denote partial derivatives), we prove
the Birkhoff-Mather theorem: the support of a minimising measure is a graph of
a Lipschitz function. It is a generalisation of the well-known Birkhoff theorem
for twist maps; partial generalisation of Herman's result \cite{Herman:89}
who proved the same for measures supported on Lagrangian tori; and the analogue
of Mather's theorem \cite{Mather:91b} for positive-definite Lagrangian flows.
\vspace{1ex}
\noindent
{\bf Quasiperiodic orbits.} The Mather construction does not generalise the main
result of Aubry-Mather theory, and does not succeed to represent every rotation
vector with an {\it ergodic} minimising measure (and with a minimising orbit of
the Lagrangian flow with that rotation vector). Such an attempt, as was shown
by Hedlund \cite{Hedlund:32}, must fail. Levi \cite{Levi:97} succeeded to
construct orbits (only locally minimising)
with every rotation vector in Hedlund's counter-example. Here we
generalise Levi's construction, and demonstrate that the method of relaxation is
successful when constructing invariant measures which minimise
the action functional only locally. We introduce the ``distance'' function $d^*$
on the space of measures, with values in $[0,\infty]$,
and prove the following theorem for Lagrangian maps:
\vspace{1ex}
\noindent {\bf Theorem} \hspace{2ex}{\it
We denote the set of all ergodic measures of a given Lagrangian map $F$ by ${\cal S}^*_E$,
the set of all action-minimising invariant measures of $F$ with the rotation vector
$\rho$ by ${\cal M}^*_{\rho}$, and by $A(\rho)$ the value of the action functional
on ${\cal M}^*_{\rho}$. If there exists a neighbourhood ${\cal U}$ of ${\cal M}^*_{\rho}$
such that $d^*|_{\cal U} < \infty$, then for every $\epsilon>0$, there exists
$\mu \in {\cal S}^*_E$ such that the action $A(\mu) \leq A(\rho)+\epsilon$. }
\vspace{1ex}
We then argue why we believe that the condition of the finiteness of the distance
function in a small neighbourhood of ${\cal M}^*_{\rho}$ is typically (possibly always) satisfied.
In particular, we show that it is satisfied in the anti-integrable limit in the sense
of Aubry \cite{Aubry:92a}.
\vspace{2ex}
The paper is structured as follows: In section 2 we discuss three standing
assumptions on the generating function $h$ of a given Lagrangian map: (A1) periodicity,
(A2) the superlinear growth, and (A3)
the existence of the semiflow $\phi^*$. We show that (A3) is satisfied if
e.g. the second derivatives of $h$ are bounded, and in the case of twist maps.
In section 3 we show that $\phi^*$ is strictly gradient, and that the ``interesting''
part of the phase space of $\phi^*$ is an increasing union of $\phi^*$-invariant
compact sets. The methods of minimisation and relaxation are developed in section
4. In section 5 we show that the invariants of the semiflow $\phi^*$ are: ergodicity,
rotation vector, and all spectral invariants, including the Kolmogorov-Sinai entropy.
Minimising measures are discussed in section 6, and quasi-periodic orbits in section
7. In section 8 we apply the results to the case of twist maps ($d=1$), and discuss
open problems.
\section{Lagrangian maps}
In the following, $h : \Bbb{R}^d \times \Bbb{R}^d \rightarrow \Bbb{R}$
will denote a $C^2$-function, the discrete ``Lagrangian''. We will
impose three conditions on $h$ which will be assumed throughout the
paper. We denote by $|.|$ the Euclid metric on $\Bbb{R}^d$.
\begin{description}
\item[(A1)] {\bf Periodicity.}
For each integer vector $a \in \Bbb{Z}^d$, and for each
$x,y \in \Bbb{R}^d$, $h(x+a,y+a)=h(x,y)$.
\item[(A2)] {\bf Superlinear growth.} $\lim_{|\eta| \rightarrow \infty}
h(x,x+\eta)/|\eta| = +\infty$, uniformly in $|\eta|$.
\end{description}
The condition (A1) implies that, if $h$ is a generating function
of a map $F$, then $F$ can be restricted to a map on a torus
$\Bbb{T}^d$. The conditions (A1) and (A2) are
the discrete-time analogues of the conditions on
Lagrangian flows used by Mather in \cite{Mather:91b}.
\smallsec{Gradient dynamics of the action functional}
Before we state the third condition, we need to introduce the main
technique of the paper. In the following, the elements of
$(\Bbb{R}^d)^{\Bbb Z}$ are called configurations.
We study the system of equations
\beq
{\dd u_i \over \dd t} &=& -h_2(u_{i-1},u_i) - h_1(u_i,u_{i+1}), \ \ i\in
\Bbb{Z} \label{2e:main} \\
{\bf u}(0)& =& {\bf u}^0, \label{2e:init} \eeq
where ${\bf u}^0 \in (\Bbb{R}^d)^{\Bbb{Z}}$ is the initial condition,
indices denote partial derivatives,
and the solution is a function ${\bf u} : \Bbb{R}^+ \rightarrow
(\Bbb{R}^d)^{\Bbb{Z}}$ satisfying (\ref{2e:main}), (\ref{2e:init}).
We can formally write the equation (\ref{2e:main}) as
$${\bf u}_t = -\nabla h({\bf u}),$$
where $h({\bf u})=\sum_{i \in \Bbb{Z}} h(u_{i-1},u_i)$; therefore
we say that (\ref{2e:main}) is a formally gradient (or an extended
gradient) system of equations.
Let $\tilde{\cal B}$ denote the set of all configurations
${\bf u} \in ({\Bbb{R}^d})^{\Bbb{Z}}$, such that
$$ \sup_{i \in \Bbb{Z}} |u_{i+1} - u_i| < \infty.$$
\begin{prop} \label{2p:exist}
Assume that $h$ is $C^2$, and that it satisfies (A1), (A2).
Then the equation (\ref{2e:main}) generates a local flow on $\tilde{\cal B}$
(i.e. for each ${\bf u}^0 \in \tilde{\cal B}$, there exists
$\delta >0$, and a unique solution
$ {\bf u} : (-\delta,\delta) \rightarrow \tilde{\cal B}$ of (\ref{2e:main}),
(\ref{2e:init}) for some $\delta > 0$).
\end{prop}
\proof
For each ${\bf u}^0 \in \tilde{\cal B}$, let the space $l_{{\bf u}^0,\infty}$ be the space of all ${\bf v} \in \Bbb{R}^{\Bbb{Z}}$ such that
$\sup_{i\in \Bbb{Z}} |u_i-v_i| < \infty$, with the norm
$||v||_{{\bf u}^0,\infty}=\sup_{i\in \Bbb{Z}} |u_i-v_i|$.
Given ${\bf u}^0 \in \tilde{\cal B}$,
(A1) implies that the right-hand of (\ref{2e:main}) is locally
Lipschitz on $l_{{\bf u}^0,\infty}$. Well-known results
on existence and uniqueness of solutions of ordinary differential
equations on Banach spaces (see e.g. \cite{Dalecki:74}) now imply the claim.
\qed
In other words, Proposition \ref{2p:exist} implies existence of
a local flow on $\tilde{\cal B}$.
The third condition will be that the local flow can be extended to
a semiflow on a suitable space which contains $\tilde{\cal B}$.
Given a local flow or semiflow $\phi$ on a space ${\cal X}$,
we use the notation
$\phi^t(x)=\phi(t,x)$.
We say that a semiflow $\phi$ is backward-unique, if for each
$t$, $\phi^t : {\cal X} \rightarrow {\cal X}$ is injective. We say
that a semiflow $\phi$ is continuous, if for each $t$, the function
$\phi^t$ is continuous, and if for each $x \in {\cal X}$, the function
$t \mapsto \phi(t,x)$, $t \in [0,\infty)$, is continuous.
We denote by $T_a$, $a \in \Bbb{Z}^d$, and $S$ the translations
defined on $(\Bbb{R}^d)$:
\beqn
T_a({\bf u})_i &=& u_i+a , \\
S({\bf u})_i &=& u_{i-1}. \eeqn
The condition (A1), and the homogeneity in $i$ of the equation (\ref{2e:main})
imply
respectively that the solution of (\ref{2e:main}) commutes with the translations $T_a$, $S$. We denote by $T$ the group of all translations
$T_a$, $a \in \Bbb{Z}^d$.
For a given $C >0 $, $\tilde{\cal B}_C$ is the set of all
${\bf u} \in \Bbb{R}^{\Bbb{Z}}$ such that
$\sup_{i \in \Bbb{Z}} |u_i - u_{i+1}| \leq C$.
In the following we
always assume the topology on $\tilde{\cal B}_C$ induced by the
product topology on $({\cal R}^d)^{\Bbb{Z}}$, and
${\cal B}_C=\tilde{\cal B}_C/T$ is the quotient space with the quotient topology.
We now assume the following:
\begin{description}
\item[(A3)] {\bf Existence of a semiflow.} For each $C>0$,
there exists a complete, separable metric space $\tilde{\cal A}_C$,
such that $\tilde{\cal B}_C$ is a subspace of
$\tilde{\cal A}_C$, $\tilde{\cal
A}_C$ a subset of $(\Bbb{R}^d)^{\Bbb{Z}}$, and such that:
(i) For each $i \in \Bbb{Z}$, the projection $\pi_i :
\tilde{\cal A}_C \rightarrow
{\Bbb{R}}$, $\phi_i({\bf u})=u_i$ is continuous;
(ii) For each $D>0$, the set ${\cal Y}_D:=\{
{\bf u} \in \tilde{\cal A}_C \ : \
\forall k \in \Bbb{Z}, \ |u_{k+1}-u_k| \leq D(k^2+1)\}/T \}$ is compact;
(iii) The translations
$S$, $T_a$, $a\in \Bbb{Z}$, are homeomorphisms of $\tilde{\cal A}_C$;
(iv) the equation $(\ref{2e:main})$ generates a backward unique
continuous semiflow $\tilde{\phi}$ on $\tilde{\cal A}_C$.
\end{description}
\vspace{2ex}
\noindent {\bf Standing assumption:} In the rest of the paper (except in the section \ref{except} below),
we assume that $h$ is $C^2$, and that it satisfies (A1-3). We further assume that $\tilde{\cal A}_C$ is the
same for all $C>0$, and write $\tilde{\cal A}=\tilde{\cal A}_C$.
\begin{remark} All the results that follow could with minor modifications be proved without the assumption
that $\tilde{\cal A}_C$ is independent of $C$; we assume it for clarity of the presentation.
\end{remark}
\begin{remark}
The conditions (A3),(i),(ii) and partially (iii) are conditions
on the topology of $\tilde{\cal A}$, and say that this topology
is ``close'' to the topology induced by the product topology
on $(\Bbb{R}^d)^{\Bbb{Z}}$ in the following sense. Denote by
$\tau$ the topology on $\tilde{\cal A}$. (A3),(i)
is equivalent to the condition that the topology $\tau$
contains the induced product topology. If for each $C>0$, the induced topology $\tau$ on
${\cal Y}_C$ is the same as the induced product topology on ${\cal Y}_C$, then the Tychonoff theorem and completeness of $\tilde{\cal A}$ imply (A3),(ii).
\end{remark}
We set ${\cal A}=\tilde{\cal A}/T$, and denote by $\phi$ the
semiflow on ${\cal A}$, whose lift on $\tilde{\cal A}$ is
$\tilde{\phi}$. The condition (A1) implies that $\phi$ is
well defined. In the following, we do not distinguish between
elements of $\tilde{\cal A}$ and their equivalence classes
in ${\cal A}$, and perform the calculations in ${\cal A}$
when possible.
The role of the condition (A3) is in a way parallel to the Mather's
condition in \cite{Mather:91b}
of the completeness of the Euler-Lagrange flow, though the
relationship of these two conditions remains unclear. We immediately give two
simpler conditions which will imply (A3).
\smallsec{Examples} \label{except}
\noindent {\bf Example 1.} We can assume the following:
\begin{description}
\item[(E1)] Assume that the second derivatives $h_{11}$,
$h_{12}$, $h_{22}$ are uniformly bounded on $\Bbb{R}^d \times
\Bbb{R}^d$.
\end{description}
We choose any $\lambda > 1$, and set $\tilde{\cal A}=\tilde{\cal A}_C$ to be the Hilbert space
of all configurations ${\bf u} \in (\Bbb{R}^d)^{\Bbb{Z}}$
such that $||{\bf u}||_{\lambda} < \infty$, where
$$({\bf u},{\bf v})=\sum_{i \in \Bbb{Z}} \lambda^{-|i|} u_i v_i,$$
$||{\bf u}||_{\lambda}=\sqrt{({\bf u},{\bf u})}$.
Clearly $\tilde{\cal A}$ is
metric, complete, separable, and $\tilde{\cal B}_C$ with the product
topology is a subspace of $\tilde{\cal A}$.
\begin{prop} If $h$ satisfies (A1),(E1), then it satisfies
(A3). \end{prop}
\proof
It is easy to check that (E1) implies that the right-hand
side of (\ref{2e:main}) is uniformly Lipschitz on $\tilde{\cal A}$,
which (see e.g. \cite{Dalecki:74}) implies that
(\ref{2e:main}) generates a continuous flow on $\tilde{\cal
A}$. Since $\phi^t$ is bijection, the semiflow is
backward unique. The
remaining properties (A3),(i),(ii) and (iii) can be checked easily. \qed
The conditions (A1),(A2),(E1) and hence (A3)
are e.g. satisfied by all standard-like generating
functions
\beq h(x,y) = (x - y)^2/2 + P(x), \label{1e:stan} \eeq
where $P : \Bbb{R}^d \rightarrow \Bbb{R}$ is any $C^2$
function such that $P(x+a)=P(x)$ for all $a \in \Bbb{Z}^d$.
\vspace{2ex}
\noindent {\bf Example 2. The twist maps.} We now show that
the standard setting of the Aubry-Mather theory is sufficient.
\begin{description}
\item[(E2)] Assume that $d=1$, and that $h$ satisfies
the twist condition, i.e. that
\beq h_{12}(x,y) \leq 0, \label{twist} \eeq
where indices denote partial derivatives.
\end{description}
Since $d=1$, we can use the partial ordering on $\Bbb{R}^{\Bbb{Z}}$:
${\bf u} \leq {\bf v}$ if and only if for each $i\in \Bbb{Z}$,
$u_i \leq v_i$.
\begin{prop} \label{p2:twist}
If $h$ satisfies (A1), (E2), then it satisfies (A3).
\end{prop}
\proof
We will show that for each $N\in \Bbb{N}$, (\ref{2e:main}) generates a continuous,
backward-unique semiflow on $\tilde{B}_N$.
Proposition (\ref{2p:exist}) implies existence of a local
flow $\phi$ on $\tilde{\cal B}$.
We now show that if ${\bf u}^0 \in \tilde{\cal B}_N$ and the solution
${\bf u}(\tau)$ of (\ref{2e:main}), (\ref{2e:init})
exists on $[0,t]$, then ${\bf u}(t) \in \tilde{\cal B}_N$.
The twist condition $h_{12} \leq 0$ implies that the solution of
(\ref{2e:main}) is monotone, i.e. that ${\bf u}(0) \leq {\bf v}(0)$
implies ${\bf u}(t) \leq {\bf v}(t)$ where defined
(see e.g. \cite{Gole:92a}, Proposition 2). It is easy to check that
${\bf u} \in \tilde{\cal B}_N$ if and only if
\beq T_{-N} S {\bf u} \leq {\bf u} \leq T_N S {\bf
u}. \label{2e:relin} \eeq
Assume that ${\bf u}^0 \in \tilde{\cal B}_N$. Now monotonicity
of the solution, (\ref{2e:relin}) and the fact that the solution of
(\ref{2e:main}) commutes with $T_N$, $S$, imply
$ T_{-N} S {\bf u}(t) \leq {\bf u}(t) \leq T_N S {\bf u}(t)$, hence
${\bf u}(t) \in \tilde{\cal B}_N$.
Since the right-hand side of (\ref{2e:main}) is uniformly Lipschitz
on $l_{{\bf u}^0,\infty} \cap \tilde{\cal B}_N$ in
$||.||_{{\bf u}^0,\infty}$ norm; we conclude that (\ref{2e:main})
generates a semiflow $\phi$ on $\tilde{\cal B}_N$.
Let $||.||_{\gamma}$ be the same norm as in Example 1; this norm
on $\tilde{\cal B}_N$ is compatible with the induced product topology.
The right-hand side of (\ref{2e:main}) is uniformly Lipschitz on
$\tilde{\cal B}_N$ in $||.||_{\gamma}$,
which implies that $\phi$ is a continuous semiflow (see \cite{Dalecki:74}).
We now show backward uniqueness. Assume that for some
${\bf x},{\bf y} \in \tilde{\cal B}_N$, $t>0$,
$\phi^t({\bf x})= \phi^t({\bf y})$. Let $t_0$ be the
smallest such $t \geq 0$. Now $t_0 > 0$
is in contradiction to the local uniqueness of the solution
(Proposition \ref{2p:exist}). We deduce that
$t_0=0$, hence ${\bf x}={\bf y}$.
Now we set $\tilde{\cal A}_C=\tilde{\cal B}_N$, where $N$ is any integer greater than $C$. \qed
\smallsec{Symplectic and Lagrangian maps}
We denote by $\Bbb{T}^d=\Bbb{R}^d/\Bbb{Z}^d$ the $d$-dimensional torus,
and by $\Bbb{A}^d \cong T^*(\Bbb{T}^d) \cong \Bbb{T}^d \times \Bbb{R}^d$
the $d$-dimensional cylinder. Let $(\varphi,p)$ be the canonical
coordinates on $\Bbb{A}^d$, $v=\sum_{j=1}^d p_j d\varphi_j$ the Liouville
1-form and $w=dv=\sum_{j=1}^d d\varphi_j \wedge dp_j $ the canonical
symplectic form on $\Bbb{A}^d$. We denote by $\tilde{\Bbb{A}}^d \cong
\Bbb{R}^d \times \Bbb{R}^d$ the cover of $\Bbb{A}^d$ with
canonical coordinates $(x,p)$, $\varphi = x \ ( \textnormal{mod} \ 1)$. Given
a diffeomorphism $F$ of $\Bbb{A}^d$, We denote by $\tilde{F}$ its
lift, $\tilde{F} : \tilde{\Bbb{A}}^d \rightarrow \tilde{\Bbb{A}}^d$.
\begin{defn}
We say that a diffeomorphism $F$ on $\Bbb{A}^d$ is Lagrangian, if
it satisfies the following:
\begin{description}
\item[(L1)] F is {\bf exact symplectic}; i.e. the form $F^*v-v$
is exact;
%\item[(L2)] F is {\bf monotone}; i.e. if we write
%$$ DF(\varphi,p)= \left( \begin{array}{cc} a(\varphi,p) & b(\varphi,p) \\
%c(\varphi,p) & d(\varphi,p) \end{array} \right),$$
%then $\det b(\varphi,p) \neq 0$ for all $(\varphi,p) \in \Bbb{A}^d$;
\item[(L2)] F is {\bf strongly monotone}; i.e. if
we write $\tilde{F}$ in the form $\tilde{F}(x,p)
=(\tilde{F}_1(x,p),\tilde{F}_2(x,p))$, then
$r \rightarrow \tilde{F}_1(x,r)$ is a diffeomorphism of $\Bbb{R}^d$ for all
$x \in \Bbb{R}^d$.
\end{description}
\end{defn}
We say that a diffeomorphism $F$ on $\Bbb{A}^d$ is generated by
a $C^2$ function $h : \Bbb{R}^d \times \Bbb{R}^d \rightarrow \Bbb{R}$
(or, equivalently, that $h$ is a generating function of $F$),
if its lift $(x,p) \mapsto (x',p')$ satisfies the relations:
\beq \begin{array}{rcl}
p&=& -h_1 (x,x') \\ p' & =& h_2(x,x'). \end{array} \label{2r:lagdef}\eeq
We formulate the following additional condition on $h$:
\begin{description}
\item[(P)] For all $x \in \Bbb{R}^d$, the functions
$y \mapsto h_2(y,x)$ and $y \mapsto h_1(x,y)$ are diffeomorphisms of
$\Bbb{R}^d$.
\end{description}
\begin{prop}
Each Lagrangian diffeomorphism $F$ on $\Bbb{A}^d$ is generated by
a $C^2$ function $h$ satisfying (A1), (P). Each $C^2$ function
$h$ satisfying (A1), (P), is a generating function of a Lagrangian diffeomorphism $F$ on $\Bbb{A}^d$. \end{prop}
\proof \cite{Herman:89}, section 8. \qed
A Lagrangian map $F$ whose generating function satisfies (A2) is
called monotone, globally positive (terminology by Herman). A
sufficient condition on a Lagrangian map to be monotone, globally
positive is given in \cite{Herman:89}, 8.12. We will need only
(A1-2) and (P) when proving an important property of constructed invariant
measures (see remark \ref{r:her} for details); these are the same assumptions as those in \cite{Herman:89}.
\smallsec{Stationary configurations}
Given a semiflow $\psi$ on ${\cal X}$, we say that a point
$x \in {\cal X}$ is stationary, if for each $t>0$, $\psi^t(x)=x$.
The stationary points of the semiflow $\phi$ generated by
(\ref{2e:main}) are ${\bf u} \in {\cal A}$ satisfying
$\nabla h({\bf u})=0$, i.e. for each $i \in \Bbb{Z}$,
\beq h_2(u_{i-1},u_i)+h_1(u_i,u_{i+1})=0. \label{2r:statdef} \eeq
We denote by ${\cal S}$ the set of all stationary points in ${\cal A}$.
If the function $h$ satisfies (P), then (\ref{2r:lagdef}) implies
that for any $C>0$, the shift $S$ on the set ${\cal S} \cap {\cal B}_C$
is conjugate to the Lagrangian map $F$ generated by $h$, restricted to the set of points $(\varphi,p) \in \Bbb{A}^d$ such that
\beq \sup_{n \in \Bbb{Z}} (\tilde{F}^n_1(x,p)-
\tilde{F}^{n-1}_1(x,p)) \leq C, \label{2r:condition} \eeq
where $\tilde{F}=(\tilde{F}_1,\tilde{F}_2$ is a lift of $F$, and
$x = \varphi \ (\textrm{mod} \ 1)$. Therefore in the following we restrict
our attention to the dynamical system shift $S$ on ${\cal S} \cap {\cal B}_C$ for $C$ large enough.
\begin{remark} The orbits of $F$ not satisfying (\ref{2r:condition})
for any $C>0$ can fill a significant part of the phase portrait
(if $F$ is ``far'' from integrable), for results on such orbits
when $d=1$ see \cite{Slijepce:99b}. The techniques of this paper could
be extended to these orbits, adding two additional assumptions:
(i) that ${\cal A}$ contains all the solutions of (\ref{2r:statdef});
and (ii) that the topology on ${\cal S}$ induced by ${\cal A}$ is the same
as the induced product topology on ${\cal S}$.
\end{remark}
\section{The induced semiflow on the space of measures}
The main tool in the following will be the induced semiflow
on the space of Borel probability measures on ${\cal A}$.
In this section we define the induced semiflow, show that it is
gradient, and that each measure of interest is contained in
a compact set invariant for the semiflow.
Let ${\cal A}^*$ denote the set of all Borel probability measures
on ${\cal A}$. We always assume the weak$^*$-topology on ${\cal A}$.
It is important to note that we study simultaneously two dynamical
systems on ${\cal A}$: the shift $S$, which corresponds to
the dynamics of the Lagrangian map; and the semiflow $\phi$.
We say that a subset ${\cal X}$ is $S$-invariant, (respectively
$\phi$-invariant), if $S({\cal X})=S^{-1}({\cal X})={\cal X}$
(respectively if for each $t \geq 0$, $\phi^{-t}({\cal X})={\cal X}$).
Analogously, we say that a measure $\mu \in {\cal A}^*$ is
$S$-invariant, if $\mu \circ S^{-1} = \mu = \mu \circ S$.
A measure $\mu \in {\cal A}^*$ is $S$-ergodic, if for each
$S$-invariant Borel-measurable set ${\cal X} \subset {\cal A}$,
$\mu({\cal X}) = 0$ or $1$.
The theory of $\phi$-invariant or ergodic measures will not be
needed in the following; we only note that all $\phi$-invariant
measures are supported on ${\cal S}$, which was in a more
general framework of ``extended gradient systems''
proved in \cite{Slijepce:99c}.
Now we list further notation:
\vspace{1ex}
\hspace{5ex} ${\cal A}^*_S$, ${\cal A}^*_E$ - all $S$-invariant (resp. $S$-ergodic)
measures in ${\cal A}^*$,
\hspace{5ex} ${\cal S}^*_S$, ${\cal S}^*_E$ - all $S$-invariant (resp. $S$-ergodic)
measures supported on ${\cal S}$
\vspace{1ex}
\noindent (we say that a measure $\mu$ is supported on ${\cal S}$, if $\mu({\cal S})=1$).
The induced semiflow $\phi^*$ on ${\cal A}^*$ is defined as:
$$\phi^{*t}(\mu)= \mu \circ \phi^{-t}.$$
\smallsec{Properties of the weak$^*$ topology}
(A3) implies that ${\cal A}^*$ is complete, metrisable and
separable (\cite{Parathasarathy:67}), and all the measures
in ${\cal A}^*$ are regular. Given a continuous
function $f : {\cal A} \rightarrow {\cal A}$, the definition of weak$^*$-topology
implies immediately that the induced function $f^* : {\cal A}^*
\rightarrow {\cal A}^*$, $f^*(\mu)=\mu \circ f^{-1}$ is
weak$^*$-continuous. We conclude that $\phi^{*t}$ is weak$^*$-continuous
for each $t \geq 0$.
Given a sequence of Borel measurable functions $f_n : {\cal X}
\rightarrow {\cal X}$, where ${\cal X}$ is a metrisable space,
and a probability measure $\mu$ on ${\cal X}$,
we say that $f_n$ converges in probability, if for each $\epsilon > 0$,
$\mu\{ x \ : \ d(f_n(x),f(x) \} > \epsilon) \rightarrow 0$, where
$d$ is a metric on ${\cal X}$. A sequence $f_n$ converges in
distribution, if $\mu \circ f_n$ weak$^*$-converges to $\mu \circ f$.
The following is a well-known fact (see e.g. \cite{Billingsley:68}).
\begin{prop} \label{p3:weak}
Given a sequence $f_n : {\cal X} \rightarrow {\cal X}$
of Borel-measurable functions on a complete
separable metrisable space ${\cal X}$, and a Borel probability measure
$\mu$ on ${\cal X}$,
if the sequence $f_n$ converges $\mu$-a.e. to $f$, it converges to $f$
in probability.
If the sequence $f_n$ converges in probability to $f$, it converges to $f$
in distribution.
\end{prop}
Given a sequence of positive times $t_n \rightarrow t$, we
conclude that $\phi^{*t_n}\mu \rightarrow \phi^{*t}\mu$ in
weak$^*$-topology; hence $t \mapsto \phi^{*t}\mu$ is continuous.
We conclude the following:
\begin{prop} The semiflow $\phi^{*}$ is a continuous semiflow on ${\cal A}^*$.
\end{prop}
Since $S$ is continuous and ${\cal S}$ closed, ${\cal A}^*_S$,
${\cal S}^*_S$ are weak$^*$-closed and $\phi^*$-invariant sets.
In the following, we restrict our attention to the semiflow
$\phi^*$ on ${\cal A}^*_S$.
\smallsec{The semiflow $\phi^*$ is gradient}
\begin{defn} A semiflow $\psi$ on a set ${\cal A}$ is strictly gradient, if there
exists a function $L : {\cal A} \rightarrow \Bbb{R}$
(called Lyapunov function), such that for each non-stationary
$x \in {\cal A}$, and any $t > 0$, $A(\psi(t,x)) < A(x)$.
\end{defn}
One of main difficulties when applying the semiflow $\phi$
to construction of stationary configurations with given properties,
is that it is only formally gradient (the natural ``Lyapunov''
function $A({\bf u})=\sum h(u_i,u_{i+1})$ is divergent;
for discussion of various properties of such systems we again refer
the reader to \cite{Slijepce:99c}). We can, however, overcome
this difficulty by working in the space ${\cal A}^*_S$.
\begin{thm} \label{3t:gradient}
The semiflow $\phi^*$ is strictly gradient on ${\cal A}^*_S$,
with Lyapunov function
$$ A(\mu)=\int_{\cal A} h(u_0,u_1) d\mu.$$
The set of stationary points of $\phi^*$ is ${\cal S}^*_S$.
\end{thm}
\proof
Clearly ${\cal S}^*_S$ is a subset of the set of stationary points
of $\phi^*$.
Assume that $\mu \in {\cal A}^*_S$ is a measure not
supported on ${\cal S}$. It is sufficient to show that $A$ is
strictly decreasing along the $\phi^*$-orbit of $\mu$.
In the following, we use the notation $x_i^t=\phi^t({\bf x})_i$.
Given $T > 0$, we calculate, using the $S$-invariance of $\mu$ in
the fourth line below:
\beq
A(\phi^{*T}(\mu)) - A(\mu) &=&
\int_{\cal A}h(x_0,x_1)d\phi^{*T}\mu - \int_{\cal A}h(x_0,x_1)d\mu
\nonumber \\
&=& \int_{\cal A}h(x^T_0,x^T_1)d\mu -
\int_{\cal A}h(x_0,x_1)d\mu \nonumber \\
&=& \int_{\cal A} \int_0^T (h_1(x^t_0,x^t_1)\dot{x}_0 +
h_2(x^t_0,x^t_1)\dot{x}_1) dt d\mu \nonumber \\
&=& \int_{\cal A} \int_0^T h_1(x^t_0,x^t_1)\dot{x}_0 dt d\mu
+ \int_{\cal A} \int_0^T h_2(x^t_0,x^t_1)\dot{x}_1 dt d(\mu \circ S)
\nonumber \\
&=& \int_{\cal A} \int_0^T (h_1(x^t_0,x^t_1)\dot{x}_0 + h_2(x^t_{-1},x^t_0)
\dot{x}_0)dt d\mu \nonumber \\
&=& \int_{\cal A} \int_0^T -(\dot{x}^t_0)^2 dt d\mu. \label{rel:4relgr}\eeq
Assume that the right-hand side above is equal to $0$. Then
$\int_0^T (\dot{x}^t_0)^2 dt = 0$ $\mu$-a.e., and then
$\phi^t({\bf x})_0=\phi({\bf x})_0$ $\mu$-almost
everywhere, for all $t \in [0,T]$. Since
$\mu$ is $S$-invariant, $\phi^t({\bf x})=\phi({\bf x})$
for each $t \in [0,T]$, $\mu$ a.e., which is a contradiction to the
assumption that $\mu \not\in {\cal S}^*_S$. \qed
\smallsec{The action of a measure}
The function $A(\mu)=\int_{\cal A} h(u_0,u_1) d\mu$ defined in Theorem
\ref{3t:gradient} is called the action of $\mu$. Restriction
of $A$ to ${\cal S}^*_S$ is the same action defined by
Mather (\cite{Mather:89}) in the case of twist maps, and the analogue
of the action for Lagrangian flows (\cite{Mather:91b}).
Note that for each $\mu \in {\cal A}^*_S$, and each $i\in \Bbb{Z}$,
the $S$-invariance of $\mu$ implies that
$A(\mu)=\int_{\cal A} h(u_i,u_{i+1}) d\mu$.
We define the action $A({\bf x})$ of a configuration ${\bf x}$
as $A({\bf x})=1/N \cdot \lim_{i=-N}^{N-1} h(x_i,x_{i+1})$,
if the expression is convergent. If a measure $\mu\in {\cal A}^*_S$
is $S$-ergodic, and $A(\mu)<\infty$, then for $\mu$-a.e. ${\bf x}$,
$A({\bf x})=A(\mu)$.
Since by (A3),(i), ${\bf u} \mapsto (u_0,u_1)$ is continuous,
so is ${\bf u} \mapsto h(u_0,u_1)$. We conclude:
\begin{lemma} \label{3l:semi}
$A : {\cal A}^*_S \rightarrow \Bbb{R}$ is lower-semi continuous.
\end{lemma}
\proof Let $A_C(\mu)= \int_{\cal A} \min (h(x_0,x_1),C) d\mu$, for some $C \in \Bbb{R}$.
Since $\min (h(x_0,x_1),C) : {\cal A} \rightarrow \Bbb{R}$ is bounded and continuous,
$A_C$ is continuous, and $A_C \uparrow A$ as $C \rightarrow \infty$, which implies
the claim. \qed
Given a real number $M$, we denote by
${\cal X}^*_M$ the set of all $\mu \in {\cal A}^*_S$ such
that $A(\mu) \leq M$. Theorem \ref{3t:gradient} implies
that ${\cal X}^*_M$ is $\phi^*$-invariant, and the lower semi-continuity
of $L$ implies that it is closed.
Conditions (A1) and the continuity of $h$ imply that there exists a constant
$K \geq 0$ such that for each $(x,y) \in \Bbb{R}^2$,
$h(x,y) \geq -K$. We conclude that for each $\mu \in {\cal A}^*_S$,
\beq A(\mu) \geq -K. \label{3r:lowbound} \eeq
\smallsec{Compactness of ${\cal X}^*_M$}
If we define ${\cal B}_C$, $C \in \Bbb{R}$, to be the set of all
${\bf u} \in {\cal A}$ such that $\sup_{i \in \Bbb{Z}} |u_{i+1}-u_i|
\leq C$, then ${\cal B}_C$ is homeomorphic to
$$ ([-C,C]^d)^{\Bbb{N}} \times \Bbb{T}^d \times ([-C,C]^d)^{\Bbb{N}}.$$
The Tychonoff theorem implies that ${\cal B}_C$ is compact.
We will show that each $\mu \in {\cal X}^*_M$ is ``almost''
supported on ${\cal B}_C$ for large $C$.
We say that a family $\Pi$ of Borel probability measures
on a metric space ${\cal X}$ is tight, if for each $\epsilon$, there
exists a compact set ${\cal C} \subset {\cal X}$ such that for each
$m \in \Pi$, $m(C) \geq 1 - \epsilon.$ The following is a
well known result that we need
(proof is in \cite{Billingsley:68}, Theorems 6.1 and 6.2).
\begin{thm} \label{3t:Prokhorov}
A family of Borel probability measures on a metrisable, complete,
separable space ${\cal X}$ is relatively compact if and only if it is tight.
\end{thm}
\begin{thm} \label{3t:tight}
For each $M \in \Bbb{R}$,
the set ${\cal X}^*_M$ is tight, hence compact.
\end{thm}
\proof
{\it Step 1.} The condition (A2) implies existence of constants
$C,d > 0$ such that for each $x,y \in \Bbb{R}^d$, $|x-y|>d$ implies
\beq h(x,y) \geq C|x-y|. \label{3r:replace} \eeq
Choose $\epsilon$, $0 < \epsilon < {M+K \over Cd}$, where
$K$ is the constant from (\ref{3r:lowbound}). Given
$m \in {\cal X}^*_M$, we get
for any $i \in \Bbb{Z}$, applying (\ref{3r:replace})
and $S$-invariance of $m$:
\beqn
\lefteqn{ M \geq \int_{{\cal A}} h(x_i,x_{i+1})dm
= \int_{|x_{i+1}-x_i|\geq {M+K \over C\epsilon}} h(x_i,x_{i+1})dm} \\
&& + \int_{|x_{i+1}-x_i| \leq {M+K \over C\epsilon}} h(x_i,x_{i+1})dm
\geq C {M+K \over C\epsilon} m\left( |x_{i+1}-x_i|\geq {M+K \over
C\epsilon}\right) - K, \eeqn
and therefore
\beq m\left( |x_{i+1}-x_i|\geq {M+K \over
C\epsilon}\right) \leq \epsilon. \label{rel:4tigA} \eeq
\vspace{1ex}
\noindent {\it Step 2.} Define the set
$${\cal Y}_{\epsilon} = \{ {\bf x}\in {\cal A}, \
|x_{k+1}-x_k| \leq {M + K \over C \epsilon }(|k|^2+1),
\ k \in\Bbb{Z} \}/T.$$
(A3),(ii) implies that ${\cal Y}_{\epsilon}$ is compact, hence measurable. Then, applying (\ref{rel:4tigA}), we get
\beq
m({\cal Y}_{\epsilon})&=& 1-m({\cal Y}_{\epsilon}^c )
\geq 1 - \sum_{k \in \Bbb{Z}} m\left( |x_{i+1}-x_i|
\geq {M+K \over C\epsilon }(|k|^2+1) \right) \nonumber \\
&\geq & 1 - \epsilon \sum_{k\in \Bbb{Z}}
{1 \over |k|^2+1} \geq 1 - \epsilon a, \label{rel:4tigB} \eeq
where $a= \sum_{k\in \Bbb{Z}} 1/(|k|^2+1)$. (A3),(ii) implies
that ${\cal Y}_{\epsilon}$ is compact.
Since $\epsilon$ was arbitrary,
Theorem \ref{3t:Prokhorov} implies now that ${\cal X}^*_M$ is relatively compact.
Lower semi-continuity of $A$ implies that ${\cal X}^*_M$ is closed, hence compact.\qed
\begin{remark} The statement of Theorem \ref{3t:tight} is
still true if we replace (A2) with a weaker assumption
that for some $C,d > 0$, $|x-y|>d$ implies (\ref{3r:replace}).
\end{remark}
We can summarise the results of this section in the following:
\begin{corr} \label{3c:maincor}
For each $M \in \Bbb{R}$, the set ${\cal X}^*_M$
is compact, invariant for the semiflow $\phi^*$.
\end{corr}
\section{Construction of invariant measures of Lagrangian maps}
We denote by ${\cal X}^* = \cup_{M \in \Bbb{R}} {\cal X}^*_M$
the set of all measures in ${\cal A}^*_S$ with finite action.
Then $A : {\cal X}^* \rightarrow \Bbb{R}$ is well-defined and lower semi-continuous.
Recall that $\phi^*$ is a strictly gradient flow on ${\cal X}^*$ with
the Lyapunov function $A$, so we can use the following well-known elementary
properties of strict gradient semiflows:
\begin{thm} \label{4t:tool}
Given a semiflow $\psi$ on a metrisable space ${\cal X}$ with a strict
Lyapunov function $L$, every local minimum of $L$ on ${\cal X}$ is
a stationary point of $\psi$.
If for each $M \in \Bbb{R}$, the set $\{ x \in {\cal X}, \ : \
L(x) \leq M \} $ is compact, then for each $x \in {\cal X}$, the
$\omega$-limit set $\omega(x)$ is non-empty and consists of
stationary points.
\end{thm}
\smallsec{Construction by minimisation}
\begin{thm} \label{4t:min}
Let ${\cal Y}^*$ be a closed, non-empty, $\phi^*$-invariant subset of
of ${\cal X}^*$. Then the set of local minima of $A|_{{\cal Y}^*}$ is a non-empty
subset of ${\cal S}_S^*$. In particular, $A$ attains its minimum on ${\cal Y}^*$.
\end{thm}
\proof
Since ${\cal Y}^*$ is $\phi^*$-invariant, and $A$ is a strict Lyapunov
function for the semiflow $\phi^*$, every local minimum of $A$
is a stationary point of $\phi^*$. Theorem \ref{3t:gradient}
implies that the set of local minima of $A|_{{\cal Y}^*}$ is a subset
of ${\cal S}^*_S$. Since for all $M > 0$, ${\cal Y}^* \cap {\cal X}^*_M$
is compact and $L$ lower semi-continuous,
it attains its minimum on ${\cal Y}^*$. \qed
We construct measures in ${\cal S}^*_S$ using the minimisation
method in section \ref{s:min}.
\smallsec{Construction by relaxation}
In the following, $\omega(\mu)$ denotes the $\omega$-limit set of a
measure $\mu \in {\cal X}^*$ with respect to the semiflow $\phi^*$.
\begin{thm} \label{t4:relax}
Given any $\mu \in {\cal X}^*$, $\omega(\mu)$ is
non-empty and $\omega(\mu) \subset {\cal S}^*_S$.
\end{thm}
\proof
Theorem \ref{3t:tight} implies that the set $\{ \mu \in {\cal X}^*
\ : \ A(\mu) \leq M \} = {\cal X}^*_M$ is compact; now Theorems
\ref{3t:gradient} and \ref{4t:tool} imply the claim. \qed
We construct measures in ${\cal S}^*_S$ using the relaxation
method in section \ref{s:quasi}.
\begin{remark} {\it Construction by the fixed point argument.}
Given a closed, $\phi^*$-invariant subset ${\cal Y}^*$
of ${\cal X}^*$, the
third possible method of construction of a measure ${\cal S}^*_S
\subset {\cal Y}^*$ is to apply the Tychonoff fixed-point theorem
to the function $\phi^{*1}$
on the convex hull of ${\cal Y}^*$, and an application of
the Choquet representation theorem and the Krein-Milman theorem
then implies that ${\cal S}^*_S \cap {\cal Y}^* \ne \emptyset $.
Since we do not know an example of a new result obtained with
this method, details are omitted, and can be found in \cite{Slijepce:99}.
\end{remark}
\section{Invariants of the induced semiflow}
In this section we show that all ergodic-theoretical properties
are $\phi^*$-invariant. We discuss as well which properties
are preserved in the limit; i.e. whether they extend to the closure of the
$\phi^*$-orbit of a measure.
\smallsec{Ergodicity}
We first prove that ergodicity is an invariant of the semiflow
$\phi^*$. This will follow from a more general Theorem \ref{3t:iso}
below, but we first give an elementary proof.
\begin{lemma} \label{prop:4elem}
(i) for each ${\cal U} \subset {\cal A}$, $\phi^{-t}\phi^t({\cal U})
=\phi^t\phi^{-t}({\cal U}) = {\cal U}$;
(ii) for each ${\cal U} \subset {\cal A}$,
$\phi^{-t}S ({\cal U})=S \phi^{-t} ({\cal U})$;
(iii) if ${\cal U}$ is $S$-invariant, so is $\phi^{-t}({\cal U})$.
\end{lemma}
\proof
(i) follows from injectivity of $\phi^t$, assumed in (A3),(iv).
Applying (i), we get:
$$S\phi^{-t}({\cal U})= \phi^{-t}\phi^tS\phi^{-t}({\cal U})
=\phi^{-t}S\phi^t\phi^{-t}({\cal U})=\phi^{-t}S({\cal U});$$
(iii) follows from (ii). \qed
\begin{prop} \label{prop:4erg}
The set ${\cal A}^*_E$ is invariant for semiflow $\phi^*$.
\end{prop}
\proof Choose an $S$-ergodic measure $\mu$, and
an $S$-invariant set ${\cal U}$. Proposition \ref{prop:4elem}, (iii) implies
that $\phi^{-t}({\cal U})$ is $S$-invariant, hence
$\phi^{*t}\mu({\cal U})=\mu(\phi^{-t}({\cal U})) \in \{ 0,1 \}$.
Therefore $\phi^{*t}\mu$ is $S$-ergodic. \qed
The set of $S$-ergodic measures is not in general
weak$^*$-closed, which will be the major difficulty when constructing
measures in ${\cal A}^*_E$ with prescribed properties.
\begin{remark}
The following conjecture is still open: if $\mu \in {\cal X}^*$ is
$S$-ergodic, then each $m \in \omega(\mu)$ is $S$-ergodic.
If it is true, it would in particular imply immediately existence
of quasiperiodic orbits with an arbitrary rotation vector (see
section \ref{s:quasi} for details). A counterexample to an analogous
claim in the case of Lagrangian flows is the measure constructed
in \cite{Slijepce:99c}, Example 8. \end{remark}
\smallsec{Ergodic invariants}
Recall that, given probability spaces
$(X_1,{\cal B}_1,m_1)$ and $(X_2,{\cal B}_2,m_2)$, and
measure-preserving transformations
$S_1 : X_1 \rightarrow X_1$, $S_2 : X_2 \rightarrow X_2$, we
say that the dynamical system $(X_2,{\cal B}_2,m_2,S_2)$ is a factor
of the dynamical system $(X_1, {\cal B}_1,m_1,S_1)$, if there
exist measurable sets $M_1 \subset X_1$, $M_2 \subset X_2$
of full measure, $S_1$- (resp. $S_2$-) invariant, and
a measurable, measure-preserving map $\theta : M_1 \rightarrow
M_2$ (called factor map), such that
$$ \theta \circ S_1 = S_2 \circ \theta.$$
We say that $(X_1, {\cal B}_1,m_1,S_1)$ and
$(X_2,{\cal B}_2,m_2,S_2)$ are isomorphic, if there exists a
factor map which is a bijection, with measurable inverse.
Invariant measures of isomorphic dynamical systems have a number
of properties (spectral invariants) in common, to be specified
in the following. We first prove that the flow $\phi^t$ is the
isomorphism. (Below ${\cal B}$ denotes the Borel $\sigma$-algebra on
${\cal A}$.)
\begin{thm} \label{3t:iso}
For each measure $\mu \in {\cal A}^*_S$ and $t \geq 0$,
the dynamical systems $({\cal A},{\cal B},\mu,S)$ and
$({\cal A},{\cal B},\phi^{*t}\mu,S)$ are isomorphic.
\end{thm}
\proof
Let $M_1 = {\cal A}$, and $M_2 = \phi^t({\cal A})$.
The map $\theta:=\phi^t|_{M_1}$, $\theta: M_1 \rightarrow M_2$
is measure-preserving
(because of the definition of $\phi^{*t}\mu$), measurable
(since it is continuous), surjective, injective (because of
(A3),(iv)) and therefore bijective. Since ${\cal A}$ is separable,
metrisable and complete, the Kuratovski theorem (see \cite{Parathasarathy:67}, Theorem 3.9) implies that the inverse of $\theta$ is measurable.
Since $\phi^t$ commutes with $S$, so does $\theta$. We conclude that
$\theta$ is isomorphism. \qed
Standard results in ergodic theory (see \cite{Walters:82} for
definitions; and Theorems 2.13 and 4.11 therein for proofs)
now imply:
\begin{corr}
The flow $\phi^*$ preserves the following properties and invariants
of a measure $\mu \in {\cal A}_S^*$ with respect to $S$:
(i) the property of weak-mixing; (ii) the property of strong-mixing;
(iii) the Kolmogorov-Sinai entropy; (iv) spectrum.
\end{corr}
\begin{remark} None of the properties above are necessarily
preserved in the limit. An example follows: assume $d=1$,
and $h(x,y)=(x-y)^2$. In this (integrable) case,
all measures in ${\cal S}^*_S$ have $0$
entropy and are not mixing. We can easily find a
strongly-mixing measure with positive entropy in ${\cal X}^*$,
say the Bernoulli measure $\mu_B$ supported
on the set of configurations $\{ {\bf x} \ : \ x_i=0 \textnormal{ or }1\}$;
and then $\omega(\mu_B) \subset {\cal S}^*_S$.
\end{remark}
\smallsec{Linearity of the flow}
The definition of the semiflow $\phi^*$ implies that the map $\phi^{*t}$
is affine, i.e. that, given $p,q\geq 0$, $p+q=1$,
$\phi^{*t}(p\mu_1+q\mu_2)=p\phi^{*t}\mu_1 + q \phi^{*t}\mu_2$.
Given an $S$-invariant measure $\mu \in {\cal A}^*_S$,
the Choquet representation theorem
(see \cite{Phelps:66}) implies that there exists a measure
$\tau$ on the Borel subsets of the space
${\cal A}^*_S$, such that $\tau({\cal A}^*_E)=1$, and
such that for each $f \in L_1({\cal A},\mu)$,
\beq
\int_{{\cal A}} f(x) \dd \mu(x)= \int_{{\cal A}^*_S}
\left( \int_{{\cal A}} f(x) \dd m(x) \right) d\tau(m).
\label{rel:4ergdec} \eeq
The measure $\tau$ is called an ergodic decomposition of $\mu$.
The definition of Lebesgue integral and Proposition \ref{prop:4erg}
imply the following:
\begin{corr} \label{3c:ergdec}
If $\tau$ is an ergodic decomposition of $\mu \in {\cal A}^*_S$,
then $\tau \circ (\phi^*)^{-t}$ is an ergodic decomposition of
$\phi^{*t}\mu$, i.e. for each $f \in L_1({\cal A},\phi^{*t}\mu)$,
\beq
\int_{{\cal A}} f(x) d\phi^{*t}\mu(x)= \int_{{\cal A}^*_S}
\left( \int_{{\cal A}} f(x) d \phi^{*t}m(x) \right) d\tau(m).
\label{rel:4decB} \eeq
\end{corr}
\smallsec{Rotation vectors}
The rotation vector as defined below is a natural generalisation of
the rotation number in the Aubry-Mather theory. Rotation number
of a measure was introduced by Mather (\cite{Mather:89}).
\begin{defn}
Given a measure $\mu \in {\cal A}^*_S$, we define the rotation
vector \index{rotation vector}
$\rho(\mu)$ of the measure $\mu$ as
$$ \rho(\mu)=\int_{{\cal A}} (x_1-x_0) d\mu.$$
Given ${\bf x}\in {\cal A}$, we define its rotation vector
as
$$\rho({\bf x})=\lim_{|n| \rightarrow \infty} {x_n - x_0 \over n},$$
if the expression is convergent.
\end{defn}
Birkhoff ergodic theorem implies that, given $\mu \in {\cal
A}^*_S$ such that $|\rho(\mu)| < \infty$, then $\mu$-almost every
${\bf x}$ has well defined rotation vector. If $\mu \in {\cal A}^*_E$
and $\rho(\mu)=\rho$, then $\mu$-a.e. configuration ${\bf x}$ has rotation
vector $\rho$.
We will need the following result on continuity of $\rho$
(see also \cite{Mather:91b}, Lemma on p177).
\begin{lemma} \label{5l:rocont} The function $\mu \rightarrow \rho(\mu)$ is
continuous on ${\cal X}^*_M$, for all $M > 0$.
\end{lemma}
\proof
Applying (A2), for each $\epsilon > 0$, we can find $C(\epsilon) > 0$ such that
$|x_1 - x_0| \geq C(\epsilon)$ implies $|h(x_0,x_1)|/|x_1-x_0| \geq 1/\epsilon$.
We define
$$ \rho_{\epsilon}({\bf x}) = \min \left\{ 1, { C(\epsilon) \over |x_1 - x_0|}
\right\} (x_1-x_0) .$$
Then $\rho_{\epsilon} : {\cal A} \rightarrow \Bbb{R}^d$ is bounded and continuous,
hence $\rho^*_{\epsilon} :
\mu \mapsto \int \rho_{\epsilon}d \mu $ is continuous. Given $\mu \in {\cal X}^*_M$,
applying the bound $\int_{\cal A} |h(x_0,x_1)| d\mu \leq M+K$ (where $K$ is the bound from
(\ref{3r:lowbound})), we get:
\beq |\rho(\mu)-\rho^*_{\epsilon} | & \leq &
\int_{\cal A} |x_1-x_0 - \rho_{\epsilon}| d\mu \leq
\int_{|x_1-x_0| \geq C(\epsilon)}
|x_1-x_0| d \mu \leq \int_{|x_1-x_0| \geq C(\epsilon)} |x_1-x_0| d \mu \nonumber \\
&\leq & \int_{|x_1-x_0| \geq C(\epsilon)} \epsilon |h(x_0,x_1)| d \mu \leq
\epsilon (M + K). \label{5r:robound} \eeq
We conclude that $\rho : {\cal X}^*_M \rightarrow \Bbb{R}^d$
can be uniformly approximated by continuous functions $\rho^*_{\epsilon}$, hence
it is continuous. \qed
Note that relation (\ref{5r:robound}) implies that for each $\mu \in
{\cal X}^*$, $|\rho(\mu)| < \infty$.
We now prove that the rotation vector is invariant for the semiflow
$\phi^*$.
\begin{thm} \label{5t:rotvec}
If $\mu \in {\cal X}^*$, then for
each $t \geq 0$, $\rho(\phi^{*t}\mu)=\rho(\mu)$.
\end{thm}
\proof
First, assume that $\mu \in {\cal A}^*_E$. Note that
(\ref{3r:lowbound}) implies that $A(\mu)$ is bounded from below,
$A(\mu) \geq -K$. We use relation (\ref{rel:4relgr}), and for
each $t>0$, $i\in \Bbb{Z}$, we get
\beqn \int_{\cal A} \int_0^t(\dot{x}_i(\tau))^2 \dd \tau \dd \mu \leq
A(\mu)+K. \eeqn
The inequality of arithmetic and geometric mean for integrals implies
that for each $t>0$, $i \in \Bbb{Z}$,
\beqn
\int_{\cal A} {1 \over t}(x_i(t)-x_i(0))^2 \dd \mu \leq
A(\mu)+K. \eeqn
Given $t>0$, we conclude that for each $\epsilon >0$
we can find $C>0$ large enough such that for each $n \in \Bbb{N}$,
$\mu (|x_n(t)-x_n(0)|\leq C) \geq 1 - {\epsilon \over 2}$, hence
\beq
&& \mu(|x_n(t)-x_n(0)|\leq C,|x_0(t)-x_0(0)|\leq C) \geq 1 - \epsilon,
\nonumber \\
\textnormal{and } &&
\mu\left( \left| {x_n(t)-x_0(t) \over n} -{x_n(0)-x_0(0) \over n} \right|
\leq {2C \over n} \right) \geq 1-\epsilon. \label{rel:4andand} \eeq
Since $\phi^{*t}\mu$ is ergodic, $\phi^{*t}\mu$-a.e. configuration
${\bf x}$ has the rotation number $\rho'=\rho(\phi^{*t}\mu)$.
Since $\epsilon$ can be arbitrarily small, and $n$ arbitrarily large,
(\ref{rel:4andand}) implies $\rho'=\rho(\mu)$.
If $\mu\in {\cal A}^*_S$, relation (\ref{rel:4decB}) implies
the claim. \qed
Now Lemma \ref{5l:rocont} and Theorem \ref{5t:rotvec} imply the following:
\begin{corr}
For each $\mu \in {\cal X}^*$, and any $m \in \omega(\mu)$, $\rho(\mu)=\rho(m)$.
\end{corr}
\section{Minimising measures} \label{s:min}
In this section we prove analogues of Mather's results on minimising
measures of Lagrangian flows from \cite{Mather:91b}. The novelty is the
method of construction of minimising measures: their existence is a
direct corollary of the fact that the flow $\phi^*$ is gradient, and in
particular does not depend on (P) ((P) is the analogue of the condition
of the positive definiteness of Lagrangian flows, used by Mather). Most
of the remaining proofs (except partially the proof of the generalisation of the Birkhoff theorem) follow closely \cite{Mather:91b}; we present only
the proofs which are not completely analogous.
\smallsec{Existence of minimising measures} \label{ss7:ex}
We denote by ${\cal X}^*_{\rho}$ the set of all measures $\mu \in {\cal X}^*$
(i.e. with finite action $A(\mu)$) such that $\rho(\mu)=\rho$. We
first show that ${\cal X}^*_{\rho}$ is non-empty. The function
$\tilde{\pi}_{\rho} : \Bbb{R}^d \rightarrow \tilde{\cal A}$ defined as
\beq \tilde{\pi}_{\rho}(x)_k=x+k\rho, \ \ k \in \Bbb{Z}, \label{6r:defpi}
\eeq
commutes with translations $x \mapsto x+a$ and $T_a$,
on $\Bbb{R}^d$ and $\tilde{\cal A}$ (i.e. $\tilde{\pi}_{\rho}(x+a)=
T_a(\tilde{\pi}_{\rho}(x)$) for all $a\in \Bbb{Z}^d$, hence we
can define $\pi_{\rho} : \Bbb{T}^d \rightarrow {\cal A}$ such that
its lift is $\tilde{\pi}_{\rho}$. Since $\pi_{\rho}(\Bbb{T}^d)
\subset {\cal B}_{|\rho|}$, (A3) implies that $\pi_{\rho}$ is continuous,
hence Borel-measurable. If $R_{\rho}$ is the $x \mapsto x+\rho$
translation on the torus $\Bbb{T}^d$, (\ref{6r:defpi}) implies that $\pi_{\rho}
\circ R_{\rho} = S \circ \pi_{\rho}$. We now define the measure
$\mu_{\rho}= \lambda \circ \pi^{-1}_{\rho}$, where $\lambda$ is
the Lebesgue-Haar measure on $\Bbb{T}^d$. Since $\lambda$ is
$R_{\rho}$-invariant, we deduce that $\mu_{\rho} \in {\cal A}^*_S$.
Since $\supp \mu_{\rho} \subset {\cal B}_{|\rho|}$, and
since ${\bf x} \mapsto h(x_0,x_1)$ is continuous, hence bounded on
${\cal B}_{|\rho|}$, $A(\mu_{\rho}) < +\infty$ and $\mu_{\rho} \in
{\cal X}^*_{\rho}$.
\begin{thm} The function $A$ attains its minimum on ${\cal X}_{\rho}^*$
for any $\rho \in \Bbb{R}^d$, and every such minimum is a measure
in ${\cal S}_S^*$.
\end{thm}
\proof
Let $M = A(\mu_{\rho})$. Corollary \ref{3c:maincor},
Theorem \ref{5t:rotvec} and Lemma \ref{5l:rocont}
imply that the set ${\cal Y}^*={\cal X}_M^* \cap {\cal X}^*_{\rho}$ is $\phi^*$-invariant and compact, and the choice of $M$ implies that it
is non-empty. Theorem \ref{4t:min} now implies the claim. \qed
We denote the set of all measures which minimise $A$ on ${\cal X}^*_{\rho}$
by ${\cal M}^*_{\rho}$, and let ${\cal M}^* = \cup_{\rho \in \Bbb{R}^d}
M_{\rho}^*$. The elements of ${\cal M}^*$ are called minimising measures.
In the rest of this section, we discuss properties of measures
in ${\cal M}^*$ and their supports.
\smallsec{Minimising configurations}
Analogously as in Aubry-Mather theory, we say that a segment $(x_i,...x_j)$, $i0$ there exists $M(K) >K$ such that if
$(x_i,...,x_j)$ is a minimising segment, and $|x_j-x_i|/(j-i) \leq K$, then
for $i \leq i' \leq j' \leq j$ we have $|x_{j'}-x_{i'}|/|j'-i'| \leq M(K)$.
\end{lemma}
The proof follows from (A2) (i.e. the superlinear growth of $h$); it is
completely analogous to \cite{Mather:91b}, Lemma, p182, and is omitted.
We denote by ${\cal M}$ the set of all minimising configurations. (A3),(i) implies
that for each $i0$ for
some segment $(y_i,...y_j)$, $y_i=x_i$, $y_j=x_j$. Let ${\cal U}$ be a small
neighbourhood of $(x_i,...x_j)$ such that ${\bf z} \in {\cal U}$ implies
$|h(z_i,...,z_j)-h(x_i,...,x_j)| \leq \epsilon$. Choose a $\mu$-generic
configuration ${\bf z}$, i.e. such that
\beqn \lim_{N \rightarrow \infty } { 1 \over 2N} \sum_{k=-N}^N {\bf 1}_{\cal U}
(S^{k(j-i+1)}{\bf z}) &=& \mu ({\cal U}) \eeqn
(where ${\bf 1}_{\cal U}$ is the characteristic function of ${\cal U}$),
$A({\bf z})=A(\mu)$, and $\rho({\bf z})=\rho(\mu)$.
Let ${\bf z}^N$ be any configuration such that the segment
$(z^N_{-N},...,z^N_N)$ is minimising, and such that $z^N_{-N}=z_{-N}$, $z^N_N=z_N$.
Furthermore, applying Lemma \ref{l6:bound}, for $N$ large enough we can choose
${\bf z}^N \in {\cal B}_{M|\rho|+1}$. Let $m^N$ be the measure supported uniformly
on $\{S^{-N}{\bf z}^N,...,S^N{\bf z}^N \}$, and $m$ a weaką$^*$-limit point
of the sequence $m^N$ (it exists, because ${\cal B}_{M|\rho|+1}$ is compact), and
then $m \in {\cal A}^*_S$. It
is easy to see that $\rho(m)=\rho({\bf z})=\rho(\mu)$, and $A(m) \leq A(\mu)-\epsilon \mu({\cal U})$,
which is in contradiction to $\mu \in {\cal M}^*$. \qed
\smallsec{Mather's $\alpha$ and $\beta$ functions}
Following Mather, we define $\beta(\rho)=A(\mu)$, where $\rho \in \Bbb{R}^d$, and $\mu \in {\cal M}^*_{\rho}$.
\begin{lemma} $\beta$ is convex, and has superlinear growth, i.e.
$\beta(\rho)/|\rho| \rightarrow \infty$ as $|\rho| \rightarrow
\infty$.
\end{lemma}
\proof Choose $p,q \geq 0$, $p+q=1$, $\rho_1,\rho_2 \in \Bbb{R}^d$
and $m_1 \in {\cal M}^*_{\rho_1}$, $m_2 \in {\cal M}^*_{\rho_2}$.
Then $\beta(p\rho_1+q\rho_2) \leq A(pm_1+qm_2)=pA(m_1)+qA(m_2)
= p \beta(\rho_1)+q\beta(\rho_2)$, hence $\beta$ is convex.
Superlinear growth follows directly from the definition of $\beta$ and
(A2). \qed
Let $\alpha : \Bbb{R}^d \rightarrow \Bbb{R}$ be the conjugate function
of $\beta$ in the sense of convex analysis, i.e.
\beq \alpha(c)=-\min_{\rho \in \Bbb{R}^d} \{ \beta(\rho) - (c,\rho) \},
\label{r6:alpha} \eeq
where $c \in \Bbb{R}^d$, and $(c,h)$ is the canonical scalar product in
$\Bbb{R}^d$. From a basic result of convex analysis (see \cite{Rockafellar:70})
it follows that, since $\beta$ is everywhere
finite and has superlinear growth, the same is true for $\alpha$,
and $\beta(\rho)=-\min_{c \in \Bbb{R}^d}\{ \alpha(c)-(c,\rho)\}$.
If we define
$$A_c(\mu)=A(\mu)-(c,\rho(\mu)),$$
where $\mu \in {\cal X}^*$, the definitions of $\alpha$ and $\beta$
imply that $\alpha(c) = - \min\{A_c(\mu) \ : \ \mu\in {\cal X}^* \}$.
We denote by ${\cal M}^{c*}$ the set of measures $\mu \in {\cal X}^*$
which minimise $A_c(\mu)$, the discussion above implies
that for all $c \in \Bbb{R}^d$, ${\cal M}^{c*}$ is non-empty, and that
${\cal M}^*=\cup_{\rho \in \Bbb{R}^d} {\cal M}_{\rho}^*
= \cup_{c \in \Bbb{R}^d} {\cal M}^{c*}$. (Geometrically, if
$\mu \in {\cal M}_{\rho}^*$, then $\mu \in {\cal M}^{c*}$, where $c$
is the slope of the supporting hyperplane of the epigraph of $\beta$ at $\rho(\mu)$.)
An important question (discussed further in section \ref{s:quasi})
is for which $\rho \in {\Bbb R}^d$, there exists a stationary ergodic measure
$\mu$ such that $\rho(\mu)=\rho$. The following result follows
from the fact that ergodic measures are extremal points of the set
of invariant measures (see \cite{Mather:91b}, p179 for details).
\begin{prop}
Given $\rho \in \Bbb{R}^d$, either of the following conditions are sufficient for existence
of an ergodic measure in ${\cal M}^*_{\rho}$:
(i) $(\rho,\beta(\rho))$ is an extremal point of the epigraph
of $\beta$;
(ii) there exists $c\in \Bbb{R}^d$ such that
${\cal M}^{c*} \subset {\cal M}_{\rho}^*$ (and then
${\cal M}^{c*} = {\cal M}_{\rho}^*$).
\end{prop}
Finally, it is easy to check that ${\cal M}^{c*}$ is a convex
set, and that its extremal points are ergodic measures. We write
$\supp {\cal M}^{c*} = Cl \cup_{\mu \in {\cal M}^{c*}}
\supp \mu$ (where $Cl$ is the closure).
\begin{corr} \label{c7:corr}
There exists $M > 0$ such that $\supp {\cal M}^{c*} \subset
{\cal B}_M$. In particular, $\supp {\cal M}^{c*}$ is compact.
\end{corr}
\proof
Since $\beta : \Bbb{R}^d \rightarrow \Bbb{R}$ has superlinear
growth, there exists $K>0$ such that for each $\mu \in {\cal M}^{c*}$,
$|\rho(\mu)| \leq K$. If $m$ is an ergodic measure in
${\cal M}^{c*}$, then for $m$-a.e. ${\bf x}$, $|\rho({\bf x})| \leq K$,
and then Lemma \ref{l6:bound} implies that for $m$-a.e. ${\bf x}$,
and any $i \in \Bbb{Z}$, $|x_{i+1}-x_i| \leq M(K)$. But the
set of such orbits is dense in ${\cal M}^{c*}$, hence
$\supp {\cal M}^{c*} \subset {\cal B}_{M(K)}$. \qed
\smallsec{The Birkhoff-Mather Theorem}
The following Lemma is a version of
Aubry's Crossing Lemma. The usual form of it, a result about twist
maps, is not valid in our more general setting. We propose the
following statement as a suitable substitute. It is analogous to
Mather's Crossing Lemma, \cite{Mather:91b}, Lemma, p186 (Mather's condition of
positive definiteness of the flow is replaced with (P)).
\begin{lemma} \label{l7:amcl}
{\bf Aubry-Mather Crossing Lemma.} Assume that $h$
satisfies (P). For every $K>0$, there exist $\eta, \delta, C>0$
such that, if $(x_{-1},x_0,x_1)$ and $(y_{-1},y_0,y_1)$ are stationary
segments such that $|x_0-y_0| < \delta$,
$|x_1-y_1| > C|x_0-y_0|$ (or $|x_{-1}-y_{-1}| > C|x_0-y_0|$),
and $|x_i-x_{i-1}|\leq K$, $|y_i-y_{i-1} | \leq K$, $i \in \{ 0,1 \}$,
then there exist $a,b \in \Bbb{R}^d$ such that $|x_0 - b|\leq 1$,
$|y_0-a| \leq 1$, and
\beq h(y_{-1},y_0,y_1)+h(x_{-1},x_0,x_1) - h(y_{-1},a,x_1)-h(x_{-1},b,y_1)
> \eta |x_0-y_0|^2. \label{r6:No} \eeq
\end{lemma}
\proof Assume that for some $C>0$, $|x_1-y_1| > C|x_0 - y_0|$
(the case $|x_{-1}-y_{-1}| > C|x_0-y_0|$ is analogous), and
that $|x_0-y_0| < \delta$, for some $\delta,C>0$ to be specified later.
Let $C_1^{-1}$ be the uniform bound
on the norm of the inverse of functions $x \mapsto h_1(y,x)$,
$x \mapsto h_2(x,y)$, restricted to the
set $\{ x \ : \ |x-y|\leq K+1\}$ (it exists because of (P)).
Then
\beq |h_1(y_0,x_1)-h_1(y_0,y_1)| \geq C_1 |x_1-y_1| \geq
C_1C|x_0-y_0|. \label{r6:A} \eeq
If we denote by $g$ the function $a \mapsto g(a) = h(y_{-1},a,x_1)$,
(\ref{r6:A}) and $h_2(y_{-1},y_0)+h_1(y_0,y_1) = 0$ imply
$|g'(y_0)| \geq C_1 C |x_0-y_0|$. Let $C_3$ be the bound on
$h_{11}(x,y)$, $h_{22}$ on the set $\{(x,y), \ : \ |x-y| \leq K+2 \}$,
and then $|(g''(a)x,x)| \leq C_3x^2$ on the set
$\{ a \ : \ |a-y_{-1}|,|x_1-a| \leq K + 2 \}$. Then if $\delta$ small
enough (i.e. such that $C_1C\delta / 2C_3 \leq 1$), we choose
$$a=y_0 - {g'(y_0) \over |g'(y_0)|}{C_1 C \over 2C_3} |x_0 - y_0|,$$
and then $|a-y_0| \leq 1$. Then Taylor's formula, together with all the
bounds from above, implies:
\beq
h(y_{-1},y_0,x_1)-h(y_{-1},a,x_1) & = & g'(y_0)(y_0-a)
+ (g''(\xi)(y_0-a),(y_0-a)) \nonumber \\
& \geq & {(C_1C)^2 |x_0-y_0|^2 \over 4C_3}. \label{r6:B} \eeq
We set $b=y_0$, and then
\beq h(x_{-1},y_0,y_1)-h(x_{-1},b,y_1)=0. \label{r6:C} \eeq
Applying again Taylor's formula, we get:
\beq
\lefteqn{h(x_0,x_1) - h(y_0,x_1)+ h(x_{-1},x_0)- h(x_{-1},y_0) =
h_1(x_0,x_1)(x_0-y_0)} \hspace{80ex} \nonumber \\ +(h_{11}(\xi,x_1)(y_0-x_0),(y_0-x_0))
+ h_2(x_{-1},x_0)(x_0-y_0) \nonumber \\ +(h_{22}(x_{-1},\xi')(y_0-x_0),(y_0-x_0))
\geq -2C_3(x_0-y_0)^2. \label{r6:D} \eeq
Summing (\ref{r6:B}), (\ref{r6:C}), and (\ref{r6:D}) we get
$$ h(y_{-1},y_0,y_1)+h(x_{-1},x_0,x_1) - h(y_{-1},a,x_1)-h(x_{-1},b,y_1)
\geq ({C^2C_1^2 \over 4C_3 } - 2C_3)|x_0-y_0|^2.$$
We choose now $\delta \leq \min \{ 1,2C_3/C_1C \}$,
$C > \sqrt{8}C_3/C_1$ and $\eta <
C^2C_1^2/4C_3 - 2C_3$. We proved the claim in the case $|x_0-y_0|>0$.
When $|x_0-y_0|=0$, with the same choice of $\delta, C, \eta$, we set (in the
case $|x_1-y_1|> 0$, the case $|x_{-1}-y_{-1}|> 0$ is analogous)
$a=y_0-\delta_0 g'(y_0)/|g'(y_0)|$,
where $\delta_0>0$ is small enough, and $b=y_0$. Then the left-hand side of
(\ref{r6:B})$>0$, (\ref{r6:C})=0, (\ref{r6:D})=0, and their sum (the left-hand
side of (\ref{r6:No})) is strictly positive. \qed
\begin{thm} \label{thm:BMT} {\bf The Birkhoff-Mather Theorem.} Assume that $h$ satisfies
(P). Then projection $\pi_0 : \supp {\cal M}^{c*} \rightarrow \Bbb{T}^d$ is injective, and
there exists a constant $C>0$ such that, if $x_0,y_0 \in \pi_0(
\supp {\cal M}^{c*})$, and ${\bf x}=\pi_0^{-1}(x_0)$, ${\bf y}=
\pi_0^{-1}(y_0)$, then
\beqn |x_1-y_1| &\leq & C|x_0-y_0| , \\
|x_{-1}-y_{-1}| &\leq & C |x_0-y_0|. \eeqn
\end{thm}
\proof Let $K$ be the constant from Corollary \ref{c7:corr} such that $\supp {\cal M}^{c*}
\subset {\cal B}_K$, and $\eta, \delta, C >0$ the constants from Lemma \ref{l7:amcl}.
Assume that there exist ${\bf x}, {\bf y} \in \supp {\cal M}^{c*}$ such that
$|x_0-y_0| < \delta$ and $|x_1 - y_1| > C|x_0 - y_0|$ (or
$|x_{-1} - y_{-1}| > C|x_0 - y_0|$, the proof is analogous). We apply Lemma
\ref{l7:amcl}, find $a$, $b$ such that (\ref{r6:No}) is satisfied, and denote the
left-hand side of (\ref{r6:No}) by $5\epsilon > 0$.
{\it Step I.} We can find $\delta_0 > 0$ such that for $\xi,\xi',z_{-1},z_1,z'_{-1},z'_1 \in \Bbb{R}^d$,
if $|x_0 - \xi| \leq 1$, $|y_0 - \xi'| \leq 1$, and $|x_{-1}-z_{-1}|<\delta_0$,
$|y_{-1}-z'_{-1}|<\delta_0$, $|x_1-z_1|<\delta_0$, $|y_1-z'_1|<\delta_0$, then
\beq \begin{array}{rcl} |h(x_{-1},\xi,y_1)-h(z_{-1},\xi,z'_1)| & \leq & \epsilon \\
|h(y_{-1},\xi',x_1)-h(z'_{-1},\xi',z_1)| & \leq & \epsilon.\end{array} \label{r7:AA} \eeq
We for the moment work in the space $\tilde{\cal A}$, let $\tilde{\bf x}$, $\tilde{\bf y}$ be
configurations from the equivalence classes ${\bf x}$, resp. ${\bf y}$.
Let $\tilde{\cal U}_1$, $\tilde{\cal U}_2$ be neighbourhoods of $\tilde{\bf x}$,
$\tilde{\bf y}$
such that $\tilde{\bf z} \in \tilde{\cal U}_1$, $\tilde{\bf z}' \in \tilde{\cal U}_2$,
$|\tilde{x}_0-\xi| \leq 1$, $|\tilde{y}_0- \xi'| \leq 1$ imply
$|\tilde{x}_{-1}-\tilde{z}_{-1}|<\delta_0$, $|\tilde{y}_{-1}-\tilde{z}'_{-1}|<\delta_0$,
$|\tilde{x}_1-\tilde{z}_1|<\delta_0$, $|\tilde{y}_1-\tilde{z}'_1|<\delta_0$,
and
\beq \begin{array}{rcl} |h(\tilde{x}_{-1},\xi,\tilde{x}_1)-
h(\tilde{z}_{-1},\xi,\tilde{z}_1)| & \leq & \epsilon \\
|h(\tilde{y}_{-1},\xi',\tilde{y}_1)-h(\tilde{z}'_{-1},\xi',\tilde{z}'_1)| & \leq & \epsilon.\end{array} \label{r7:BB} \eeq
Now Lemma \ref{l7:amcl}, and relations (\ref{r7:AA}) and (\ref{r7:BB}) imply that,
if $\tilde{\bf z} \in \tilde{\cal U}_1$, $\tilde{\bf z}' \in \tilde{\cal U}_2$,
then
\beq h(\tilde{z}_{-1},\tilde{z}_0,\tilde{z}_1)+h(\tilde{z}'_{-1},\tilde{z}'_0,\tilde{z}'_1) - h(\tilde{z}_{-1},b,\tilde{z}'_1)-h(\tilde{z}'_{-1},a,\tilde{z}_1)
\geq \epsilon. \label{r7:CC} \eeq
{\it Step II.} We set ${\cal U}_1=\tilde{\cal U}_1/T$, ${\cal U}_2=\tilde{\cal U}_2/T$.
Let $m_1, m_2\in {\cal M}^{c*}$ be $S$-ergodic measures such that
$m_1(U_1) > 0$, $m_2(U_2)>0$, and ${\bf u}$, ${\bf v}$ $m_1$- (resp. $m_2$-) generic
points in the following sense: $\rho({\bf u})=\rho(m_1)$, $\rho({\bf v})=\rho(m_2)$,
$A({\bf u})=A(m_1)$, $A({\bf v})=A(m_2)$, and
\beqn \lim_{N \rightarrow \infty } { 1 \over 2N} \sum_{k=-N}^N
{\bf 1}_{{\cal U}_1}(S^k({\bf u}))
&=& \mu ({\cal U}_1), \\
\lim_{N \rightarrow \infty } { 1 \over 2N} \sum_{k=-N}^N {\bf 1}_{{\cal U}_2}(S^k({\bf v}))
&=& \mu ({\cal U}_2). \eeqn
We now apply Mather's ``surgery''. Let $\tilde{\bf u}, \tilde{\bf v} \in \tilde{\cal A}$
be configurations from the equivalence classes ${\bf u}$, resp. ${\bf v}$.
We can find increasing sequences $m_k$, $n_k$, $k \in \Bbb{Z}$, and sequences
$a_k,b_k \in \Bbb{Z}^d$,
such that $T_{a_k}S^{m_k}\tilde{\bf u} \in \tilde{\cal U}_1$,
$T_{b_k}S^{n_k}\tilde{\bf v} \in \tilde{\cal U}_2$,
and $(m_k-m_{-k})/2k \rightarrow \mu({\cal U}_1)$,
$(n_k-n_{-k})/2k \rightarrow \mu({\cal U}_2)$, as $k \rightarrow \infty$. We now find
sequences $m'_k, n'_k \in \Bbb{Z}$, $a'_k, b'_k \in \Bbb{Z}^d$, such that
\beqn m'_{2k}-m'_{2k-1}&=& m_{2k}-m_{2k-1}, \\
m'_{2k+1}-m'_{2k}&=& n_{2k+1}-n_{2k}, \\
n'_{2k}-n'_{2k-1}&=& n_{2k}-n_{2k-1}, \\
n'_{2k+1}-n'_{2k}&=& m_{2k+1}-m_{2k}, \\
a'_{2k}-a'_{2k-1} & = & b'_{2k-1}-b'_{2k} = b_{2k}-a_{2k}, \\
a'_{2k+1}-a'_{2k} & = & b'_{2k}-b'_{2k+1} = a_{2k+1}-b_{2k}.
\eeqn
We now define the
configurations $\tilde{\bf u}'$, $\tilde{\bf v}'$, as
\beqn \begin{array}{rcll}
\tilde{u}'_{m'_{2k-1}+i} &=& \tilde{u}_{m_{2k-1}+i} + a'_{2k-1},
\ \ \ & i=1,...,m_{2k}-m_{2k-1}-1, \\
\tilde{u}'_{m'_{2k}+i} &=& \tilde{v}_{n_{2k}+i} + a'_{2k}, \ \ \ & i=1,...,n_{2k+1}-n_{2k}-1, \\
\tilde{v}'_{n'_{2k-1}+i} &=& \tilde{v}_{n_{2k-1}+i} +b'_{2k-1}, \ \ \ &
i=1,...,n_{2k}-n_{2k-1}-1, \\
\tilde{v}'_{n'_{2k}+i} &=& \tilde{u}_{m_{2k}+i} + b'_{2k}, \ \ \ & i=1,...,m_{2k+1}-m_{2k}-1, \\
\end{array} \eeqn
and let $\tilde{u}'_{m'_k}$, $\tilde{v}'_{n'_k}$ be such that the segments
$(\tilde{u}'_{m'_k-1},\tilde{u}'_{m'_k},\tilde{u}'_{m'_k+1})$, resp.
$(\tilde{v}'_{n'_k-1},\tilde{v}'_{n'_k},\tilde{v}'_{n'_k+1})$ are minimising
(i.e. we ``switch'' between ${\bf u}$, ${\bf v}$, when they come into ${\cal U}_1$,
resp. ${\cal U}_2$).
Let ${\bf u}'$, ${\bf v}' \in {\cal A}$
be the equivalence classes of $\tilde{\bf u}'$, $\tilde{\bf v}'$.
Now the construction and (\ref{r7:CC}) imply that
\beq \begin{array}{rcl} \limsup_{N \rightarrow \infty}{1 \over 2N}
(h(u'_{-N},...,u'_N)+ h(v'_{-N},...,v'_N) & \leq &
{A(\mu_1) + A(\mu_2) \over 2} - \epsilon {\mu_1({\cal U}_1)+\mu_2({\cal U}_2) \over 2}, \\
\lim_{N \rightarrow \infty} {1 \over 2N} ((u'_{N}-u'_{-N})+(v'_{N}-v'_{-N})) & = &
{\rho(\mu_1)+\rho(\mu_2) \over 2}. \end{array} \label{r8:meas} \eeq
in ${\cal A}=\tilde{\cal A}/T$.
Let $m_N$, $N\in \Bbb{N}$, be the measure uniformly distributed on the set
$$\{ S^{-N}{\bf u}',S^{-N+1}{\bf u}',...,S^N{\bf u}', S^{-N}{\bf v}',S^{-N+1}{\bf v}',
...,S^N{\bf v}'\}.$$ The construction and Lemma \ref{l6:bound} imply the existence
of $K'>0$ such that ${\bf u}',{\bf v}' \in {\cal B}_{K'}$, which is a subset of ${\cal A}$
and compact, hence the sequence $m_N$ has a limit point in ${\cal A}_S$. Relation
(\ref{r8:meas}) implies that $\rho(m) = (\rho(\mu_1)+\rho(\mu_2))/2)$, and
$A(m) < (A(\mu_1)+A(\mu_2))/2$. But ${\cal M}^{c*}$ is convex, hence $\beta(
(\rho(\mu_1)+\rho(\mu_2))/2)=(\beta(\rho(\mu_1))+\beta(\rho(\mu_2)))/2=(A(\mu_1)+A(\mu_2))/2$.
The definition of $\beta$ implies that $A(m) \geq \beta(
(\rho(\mu_1)+\rho(\mu_2))/2)$, which is a contradiction.
\vspace{1ex}
We proved the claim only when $|x_0-y_0| \leq \delta$. But since the domain of $\pi_0^{-1}$
is a subset of $\Bbb{T}^d$, which is compact, the claim follows.
\qed
\begin{remark} {\it On Herman's Theorem}. \label{r:her} Herman \cite{Herman:89}, Theorem 8.14,
proved that, given a monotone, globally positive exact symplectomorphism $F$, every
$F$-invariant Lagrangian torus is a graph of a Lipschitz function.
Note that in Theorem \ref{thm:BMT}, the standing assumption (A3) could be replaced
with ``assume that for a given $c$, ${\cal M}^{c*}$ is non-empty''. In other words, in
the proof we used only (A1),
(A2) and (P), which are the same assumptions as those used by
Herman. Under the same assumptions,
Herman (\cite{Herman:89}, section 12) proved that every invariant Lagrangian torus
consists of minimising orbits. Proposition \ref{p7:close} below now implies
that every $F$-invariant measure supported on the Lagrangian torus is minimising.
If such a measure $\mu$ is unique, and $\supp \mu$ is equal to the Lagrangian
torus, Theorem \ref{thm:BMT} implies the same result as Herman, Theorem 8.14.
\end{remark}
\smallsec{Periodic approximations of minimising measures}
Finally we show that, for every $\rho \in \Bbb{R}^d$, we can find
a periodic orbit arbitrarily close (in weak$^*$-topology) to the support
of ${\cal M}_{\rho}$; Bernstein and Katok \cite{Bernstein:87} proved
a similar result for Lagrangian tori and symplectic maps close to integrable.
\begin{prop} \label{p7:close}
Assume that $\rho \in \Bbb{R}^d$, and that $(x^n_{i_n},x^n_{i_n+1},...,x^n_{j_n})$
is a sequence of minimising segments, such that $j_n-i_n \rightarrow \infty$,
and $(x^n_{j_n}-x^n_{i_n})/(j_n-i_n) \rightarrow \rho$, as $n \rightarrow \infty$.
Then $h(x^n_{i_n},...,x^n_{j_n})/(j_n-i_n) \rightarrow \beta(\rho)$ as
$n \rightarrow \infty$.
\end{prop}
The proof is completely analogous to \cite{Mather:91b}, Proposition 1, and is
omitted.
We say that a configuration $\tilde{\bf x} \in \tilde{\cal A}$
is of type $(a,q)$, $a \in \Bbb{Z}^d$,
$q \in \Bbb{N}$, if $T_aS^q \tilde{\bf x}=\tilde{\bf x}$. A configuration
${\bf x} \in {\cal A}$ is of type $(a,q)$, if any $\tilde{\bf x}$
in the equivalence class ${\bf x}$ is of type $(a,q)$.
Let ${\cal P}^*_{a,q}$ denote the set of all $S$-invariant measures
supported on $S$-images of a single configuration of type $(a,q)$.
Elements of ${\cal P}^*_{a,q}$ are called periodic measures of type ${a,q}$.
Then ${\cal P}^*_{a,q} \subset {\cal X}^* \cap {\cal A}^*_E$. It is easy
to show that ${\cal P}^*_{a,q}$ is weak$^*$-closed, and $\phi^*$-invariant.
Theorem \ref{4t:min} now implies that $A$ attains its minimum on ${\cal P}^*_{a,q}$.
We denote the set of minima of $A|_{{\cal P}^*_{a,q}}$ by ${\cal M}^*_{a,q}$. Then
every measure in ${\cal M}^*_{a,q}$ is stationary, and supported on $S$-images
of a single stationary orbit of type $(a,q)$.
\begin{corr} \label{c7:per}
For every $\rho \in \Bbb{R}^d$, in every weak$^*$-neighbourhood of ${\cal M}^*_{\rho}$
there exists a stationary periodic measure.
\end{corr}
\proof
Choose a sequence $(a_n,q_n) \in \Bbb{Z}^d \times \Bbb{Z}$ such that
$a_n/q_n \rightarrow \rho$, as $n \rightarrow \infty$. We choose a sequence
of measures $m_n \in {\cal M}^*_{a_n,q_n}$. We can uniformly bound $|a_n/q_n|$
by some constant $K$, and then Lemma \ref{l6:bound} implies that
for all $n \in \Bbb{N}$, $m_n \in {\cal B}^*_{M(K)}$, which is compact; hence $m_n$
has a limit point $m$. Lemma \ref{5l:rocont} (i.e. continuity of the rotation
vector) shows that
$\rho(m)=\rho$, and Proposition \ref{p7:close} implies that $A(m)=\beta(\rho)$.
By the definition of ${\cal M}^*_{\rho}$, we get $m \in {\cal M}^*_{\rho}$,
which implies the claim. \qed
\section{The distance functions and quasiperiodic orbits} \label{s:quasi}
\smallsec{The distance and length functions}
We define the following functions $l,d : {\cal A} \rightarrow [0,\infty]$,
with
\beqn d({\bf x}) &= & \sup_{t \rightarrow \infty} |x_0^t-x_0| , \\
l({\bf x}) & = & \sup_{t \rightarrow \infty} \int_0^t |\dot{x}_0^{\tau}|d\tau.
\eeqn
Since the semiflow $\phi$ and the projection $\pi_0 : {\bf x} \rightarrow
x_0$ are continuous, functions $d,l$ are measurable. Clearly,
$d({\bf x}) \leq l({\bf x})$.
\begin{example} \label{e8:twist}
Assume (E2) (i.e. the case of twist maps, $d=1$), and let
${\bf x}$, ${\bf y}$ be two minimising configurations. Assume
further that ${\bf x}$, ${\bf y}$ are not such that one of them is on the boundary of
the same Birkhoff region of instability where the other one lies (an exceptional case).
We set ${\bf z} = (\max (x_i,y_i))_{i\in \Bbb{Z}}$
(or ${\bf z} = (\min (x_i,y_i))_{i\in \Bbb{Z}}$). The main
result of \cite{Slijepce:99a} is that $d({\bf z})<\infty$
if and only if ${\bf x}$ and ${\bf y}$ are in the same Birkhoff
region of instability.
\end{example}
We define functions $d^*, l^* : {\cal A}^*_S \rightarrow \Bbb{R}$,
with
\beqn d^*(\mu) &=& \int_{\cal A}d({\bf x})\mu, \\
l^*(\mu) &=& \int_{\cal A}l({\bf x})\mu. \eeqn
\begin{prop}
Functions $d, l, d^*, l^*$ are lower semi-continuous.
\end{prop}
\proof For some $N \in \Bbb{N}$,
we define
\beqn d_N &=& \min \{ \sup_{t \in [0,N]} |x_0^t-x_0|,N \} \\
l_N & =& \min \{ \sup_{t \in [0,N]} \int_0^t |\dot{x}_0^{\tau} d\tau|,
N \}. \eeqn
It is easy to check that $d_N, l_N : {\cal A} \rightarrow
\Bbb{R}$ are continuous, bounded, and that $d_N \uparrow d$,
$l_N \uparrow l$, hence $d,l$ are lower semi-continuous.
We define $d^*_N(\mu) = \int_{\cal A}d_N({\bf x})d\mu$,
$l^*_N(\mu) = \int_{\cal A}l_N({\bf x}) d\mu$, and
then $d^*_N$, $l^*_N$ are continuous.
The Lebesgue monotone convergence theorem implies that
$d^*_N \uparrow d^*$, $l^*_N \uparrow l^*$ which implies
the lower semi-continuity of $d^*,l^*$. \qed
\smallsec{Integrability of $d^*$ and $l^*$}
Now we show some important consequences of integrability of
$l^*$ and $d^*$; the following results will be tools for construction
of invariant measures when using the relaxation technique.
We first need the following technical result; we denote by
$Cl{\cal O}({\bf x})$ the closure of the $\phi$-orbit
of ${\bf x}$.
\begin{lemma} \label{l8:compact}
If $\mu \in {\cal X}^*$, and $\int_{\cal A}d^*({\bf x}) d\mu < \infty$,
then for $\mu$-a.e. ${\bf x} \in {\cal A}$, $Cl{\cal O}({\bf x})$ is compact.
\end{lemma}
\proof Let ${\cal Y}_C = \{ {\bf x} \in {\cal A} \ : \ \forall k\in \Bbb{Z}, \
|x_{k+1}-x_k| \leq C( k^2+1) \}$. Choose $\epsilon > 0$, and the same reasoning as
in the proof of Theorem \ref{3t:tight} implies the existence of $C_{\epsilon} > 0$
such that $\mu ({\cal Y}_{C_{\epsilon}}) \geq 1 - \epsilon$. Since $\mu$ is
$S$-invariant, $\int_{\cal A}d(S^n{\bf x}) d\mu = \int_{\cal A}d({\bf x})d\mu$, and
similarly we deduce the existence of a constant $D_{\epsilon}$ and the set
${\cal Y}'_{D_{\epsilon}} = \{ {\bf x} \in {\cal A} \ : \ \forall k \in \Bbb{Z}, \
d(S^k{\bf x}) \leq D_{\epsilon}(k^2+1) \}$ such that $\mu({\cal Y}'_{D_{\epsilon}})
\geq 1 - \epsilon$. Then if ${\cal Z}_{\epsilon}={\cal Y}_{C_{\epsilon}} \cap
{\cal Y}'_{D_{\epsilon}}$, then $\mu({\cal Z}_{\epsilon}) \geq 1-2\epsilon$, and
for all ${\bf x} \in {\cal Z}_{\epsilon}$ and $t \geq 0$, ${\bf x}^t \in
{\cal Y}_{C_{\epsilon}+2D_{\epsilon}}$, which is by (A3),(ii) compact. Now the set
${\cal Z}=\bigcup_{n \in \Bbb{N}}Z_{1/n}$ is the set such that $\mu({\cal Z})=1$,
and for all ${\bf x} \in {\cal Z}$, $Cl{\cal O}({\bf x})$ is compact. \qed
\begin{prop}
Assume that $\mu \in {\cal X}^*$, and that $l^*(\mu) < \infty$.
Then $\omega(\mu)$ consists of a single measure $m$, and the dynamical system
$({\cal A},{\cal B},m,S)$ is a factor of the dynamical system
$({\cal A},{\cal B},\mu,S)$ (where ${\cal B}$ is the Borel
$\sigma$-algebra on ${\cal A}$).
If $\mu$ is $S$-ergodic, so is $m$.
\end{prop}
\proof
Since $\mu$ is $S$-invariant, for all $i\in \Bbb{Z}$,
$\int_{\cal A}|\dot{x}_i^t| < \infty$, and there exists a set ${\cal Z}$,
$\mu({\cal Z})=1$, such that for each ${\bf x} \in {\cal Z}$,
$n \in \Bbb{Z}$, $x^t_n$ is convergent. Lemma \ref{l8:compact} implies
the existence of a set ${\cal Z}'\subset {\cal Z}$, $\mu({\cal Z}')=1$,
such that for all ${\bf x} \in {\cal Z}'$, $Cl{\cal O}({\bf x})$
is compact. Therefore for all ${\bf x} \in {\cal Z}'$,
${\bf x}^t$ is convergent.
We now define the map $\theta : {\cal Z}' \rightarrow
{\cal A}$ as: $\theta({\bf x})=\lim_{t \rightarrow \infty}{\bf x}^t$
We claim that $\theta$ is Borel-measurable. Indeed, if ${\cal U}$
is an open subset of ${\cal A}$,
then $\theta^{-1}({\cal U}) = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty}
\phi^{-k}({\cal U})$, hence $\theta^{-1}({\cal U})$ is a measurable set.
Since $\phi^t \rightarrow \theta$, $\mu$-a.e., Proposition \ref{p3:weak} implies
that $\phi^{*t}\mu \rightarrow m$, $m = \mu \circ \theta^{-1}$, and $m$
is the unique measure in $\omega(\mu)$.
It follows immediately that $\theta$ is the factor
map. A factor of an ergodic dynamical
system is always ergodic, hence if $\mu$ is $S$-ergodic, so is $m$. \qed
\begin{prop} \label{p8:tool2}
Assume that $\mu \in {\cal X}^*$ is $S$-ergodic,
and that $d^*(\mu) < \infty$. Then for each $m \in \omega(\mu)$,
and for $m$-a.e. ${\bf x} \in {\cal A}$, $\rho({\bf x})=\rho(\mu)$.
If $\nu$ is a Borel measure on ${\cal S}^*_S$, an ergodic decomposition of
$m \in \omega(\mu)$, then for $\nu$-almost every measure $m'$, $\rho(m')=\rho(\mu)$.
\end{prop}
\proof Let $\rho=\rho(\mu)$, and
choose $\delta > 0$. We define the set
\beqn {\cal Z}_{n,\delta} = \{ {\bf x} \in {\cal A} \ : \left| {x_n - x_0 \over n} -
\rho \right| \leq \delta \}, \eeqn
and then it is a closed set, and Birkhoff ergodic theorem implies that
for all $\epsilon > 0$,
\beq \exists n'_0 \ \ \textnormal{such that} \ \ \forall n \geq n'_0, \
\mu ({\cal Z}_{n,\delta}) \geq 1- \epsilon. \label{r8:p1} \eeq
Let $d^*(\mu)= C$, and then, since $\mu$ is $S$-invariant, for all $k\in \Bbb{Z}$,
$\int_{\cal A} d(S^k({\bf x})) \dd \mu = C$, and $\mu(d(S^k({\bf x}) \geq D) \leq C/D$.
Therefore for each $\epsilon > 0$
\beq \exists n_0'' \ \ \textnormal{such that} \ \ \forall n \geq n''_0,
\ \forall k \in \Bbb{Z}, \
\mu \left( \left| d(S^k({\bf x})) - d({\bf x}) \over n \right| \leq \delta \right)
\geq 1- \epsilon . \label{r8:p2} \eeq
Now (\ref{r8:p1}) and (\ref{r8:p2}) imply that for all $n \geq n_0 = \max(n'_0,n_0'')$,
for all $t \geq 0$, $\phi^{*t} \mu({\cal Z}_{n,3\delta}) \geq 1 - 3\epsilon $. Since
$Z_{n,3\delta}$ is closed, we conclude that
\beq \forall \ \epsilon > 0, \ \exists n_0, \ \textnormal{such that} \ \forall n>n_0,
\ m({\cal Z}_{n,3\delta}) \geq 1-3\epsilon. \label{r8:p3} \eeq
By the Birkhoff Ergodic Theorem, $m$-a.e. ${\bf x}$ has well defined rotation vector,
and (\ref{r8:p3}) implies that for $m$-a.e. ${\bf x}$, $|\rho({\bf x})- \rho |\leq
3 \delta$, and the set ${\cal Z}_{\delta}= \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty}
{\cal Z}_{n,4\delta}$ has full $m$-measure. The definition of ergodic decomposition
implies that for $\nu$-a.e. measure $m'$, $m'({\cal Z}_{\delta})=1$, and
then $|\rho(m')-\rho| \leq 4\delta$.
Since $\delta$ can be arbitrarily small, the claim is proved. \qed
\smallsec{Quasiperiodic orbits}
The construction in section \ref{s:min} does not ensure existence
of $S$-ergodic stationary measures with every rotation number
$\rho \in \Bbb{R}^d$, and in particular existence of stationary
configurations with every rotation number. If $d \geq 2$,
the Hedlund counter-example \cite{Hedlund:32} suggests that
an attempt to represent every rotation vector with a minimising
$S$-ergodic measure can not in general succeed.
We say that a stationary configuration is quasiperiodic, if it has a well-defined
irrational rotation number. We construct
here, under some assumptions on the function $d$,
$S$-ergodic measures with an arbitrary (irrational) rotation vector $\rho \in \Bbb{R}^d$
(which may be only locally minimal in ${\cal X}^*$), and, as a consequence,
(quasiperiodic) stationary configurations with that rotation vector.
We say that a configuration ${\bf x} \in {\cal A}$ is
double-recurrent, if there exist two sequences $m_i \rightarrow
-\infty$, and $n_i \rightarrow \infty$, such that
$\lim_{i \rightarrow \infty} S^{m_i}{\bf x}={\bf x}$,
$\lim_{i \rightarrow \infty} S^{n_i}{\bf x}={\bf x}$.
\begin{thm} \label{t8:quasi} Given $\rho \in \Bbb{R}^d$,
the following two conditions are equivalent:
(i) There exists a stationary $S$-ergodic measure in ${\cal X}^*_{\rho}$;
(ii) there exists a $S$-ergodic
$\mu \in {\cal X}_{\rho}^*$ such that $d^*(\mu) < \infty$.
If the equivalent conditions above are true, then there exists a
double-recurrent stationary configuration ${\bf x}$ such that
$\rho({\bf x})=\rho$.
\end{thm}
\proof The implication (i)$\Rightarrow$(ii) is trivial, since for every
$\mu \in {\cal S}^*_S$, $d^*(\mu)=0$. Assume (ii); Theorem \ref{t4:relax}
implies that we can choose a measure
$m \in \omega(\mu)$ supported on ${\cal S}$.
Then by Proposition \ref{p8:tool2}, there exists
a measure $m'$, such that $\rho(m')=\rho(\mu)$, and since $m \in {\cal S}^*_S$,
we can choose $m' \in {\cal S}^*_S$.
Existence of a double recurrent stationary configuration with rotation
vector $\rho$ follows now from Birkhoff ergodic theorem and Poincar\'{e}
recurrence theorem applied to the stationary $S$-ergodic measure $\mu$ with the
rotation vector $\rho$. \qed
For example, Theorem \ref{t8:quasi} implies that if for a given
$\rho \in \Bbb{R}^d$, $d^*(\mu_{\rho}) < \infty$ (where $\mu_{\rho}$
is the measure constructed in subsection \ref{ss7:ex}), then there exists
a quasiperiodic double-recurrent orbit with rotation vector $\rho$.
\vspace{1ex}
\begin{thm} \label{t8:low}
Given $\rho \in \Bbb{R}^d$,
let $M=M(|\rho|+2)$ be the constant defined in Lemma \ref{l6:bound}, and
assume that there exists a neighbourhood ${\cal U}$
of ${\cal M}^*_{\rho}$ for each $\mu \in {\cal U}$ supported on ${\cal B}_M$,
$d^*(\mu) < \infty$. Then for every $\epsilon>0$, there exists
a measure $\mu \in {\cal S}^*_E$ such that
$\rho(\mu)=\rho$ and $A(\mu) \leq \beta(\mu)+\epsilon$.
\end{thm}
If there exists an $S$-ergodic minimising measure with rotation number $\rho$ (i.e.
that $(\rho,\beta(\rho))$ is an extremal point of the epigraph of $\beta$),
the claim is trivial. We now assume that such a measure does not exist.
We first construct a measure which satisfies all the properties demanded above except
stationarity.
\begin{lemma} \label{l8:small}
For every $\epsilon > 0$ and $\rho \in \Bbb{R}^d$, there exists an $S$-ergodic
measure ${\mu} \in {\cal X}^*$,
supported on ${\cal B}_M$, $M=M(|\rho|+2)$,
such that $\rho(\mu)=\rho$, and $A(\mu) \leq \beta(\mu)+\epsilon$.
\end{lemma}
\proof
{\it Step I. Construction of a configuration.}
We find $n_0$ large enough such that for $n \geq n_0$, $|h(x,y)| \leq n\epsilon/4$
when $|x-y| \leq M$, and $M=M(|\rho|+2)$ is the constant from Lemma \ref{l6:bound}.
Let $(y^n_0,y^n_1,...,y^n_n)$, $n\in \Bbb{N}$ be a sequence of
minimising segments, such that $y^n_0=0$ and
$x^n_n=n\rho$. Proposition \ref{p7:close} implies that we can find
$n > n_0$ and ${\bf y}=(y_0,...,y_n)=(y^n_0,...,y^n_n)$ such that
\beq h(y_0,y_1,...,y_n)/n \leq \beta(\rho) + \epsilon/2. \label{r8:hamb} \eeq
Now we define a configuration ${\bf x}$ in the following way: we set
\beqn x_{nk+r}=y_r + a_k, \ \ k\in \Bbb{Z}, \ r \in \{0,1,...,n-1 \}, \eeqn
where we choose $a_k \in \Bbb{Z}^d$ such that
\beqn |x_{nk} + a_k - nk\rho| < 1. \eeqn
The construction of ${\bf x}$ now implies that there exists a constant $C$ such that
$\sup |x_i - i \rho| \leq C/2$, which implies that, for all $m \in \Bbb{Z}$, $k \in \Bbb{N}$,
\beq \left| {x_{m+k}-x_m \over k } - \rho \right| & \leq & C/|k|. \label{r8:rotb} \eeq
Since the ``cost of switching'' between two translates of the segment ${\bf y}$
in ${\bf x}$ are at most $|h(y_{r-1}+a_{k-1},y_r+a_{k-1})-h(y_{r-1}+a_{k-1},y_0+a_k)|
\leq n \epsilon / 2$, relation (\ref{r8:hamb}) implies that
\beq
\limsup_{m \rightarrow \infty} h(x_{-m},x_{-m+1},...,x_m)/2m \leq \beta(\rho)+\epsilon.
\label{r8:hamb2} \eeq
\noindent {\it Step II. Construction of the measure.} Let $\mu_m$ be the uniform measure
on the set
$$\{ S^{-m}{\bf x},S^{-m+1}{\bf x},...,S^m{\bf x} \}.$$ Since for all $m$,
$\mu_m$ is supported on the compact set ${\cal B}_M$,
we can find a limit point $\mu$ of
$(\mu_m)_{m \in \Bbb{Z}}$, which is in ${\cal A}_S$. Relation (\ref{r8:hamb2})
implies that
\beq A(\mu) \leq \beta(\rho)+\epsilon. \label{r8:finA} \eeq
We define the set ${\cal Z}_k = \{ {\bf x} \in {\cal A} \ : \
|(x_k-x_0)/k - \rho| \leq C/k \}$, and (\ref{r8:rotb}) implies that $\mu_m({\cal Z}_k) = 1$
for all $m \in \Bbb{R}$. Since ${\cal Z}_k$ is closed, it follows that for all $k$,
$\mu({\cal Z}_k)=1$. Let ${\cal Z}=\cap_{k \in \Bbb{N}} {\cal Z}_k$, and then
$\mu({\cal Z})=1$. Now (\ref{r8:finA}) and the ergodic decomposition theorem
applied to $\mu$ imply that we can find an $S$-ergodic measure $\mu'$ such that
$A(\mu')\leq \beta + \epsilon$ and $\mu'({\cal Z})=1$. The later and
Birkhoff ergodic theorem imply that $\rho(\mu')=\rho$. \qed
\vspace{1ex}
\noindent {\bf Proof of Theorem \ref{t8:low}:} The assumptions imply that
for $\epsilon$ small enough, we can find a measure $m$ described in Lemma
\ref{l8:small} such that $d(m)< \infty$. We choose $\mu_1 \in \omega(m)$,
and then by Theorem \ref{3t:gradient}, $A(\mu_1) \leq \beta(\rho) + \epsilon$, and
$\mu_1 \in {\cal S}^*_S$.
Proposition \ref{p8:tool2} and ergodic decomposition theorem
applied to $\mu_1$
now imply that we can find $\mu \in {\cal S}^*_E$, such that
$\rho(\mu)=\rho$, and $A(\mu) \leq \beta(\rho)+\epsilon$. \qed
\smallsec{Anti-integrable limit with non-degenerate potential}
In this section we construct a family of examples which satisfy the conditions of
Theorem \ref{t8:low}. The examples are symplectic maps very far from integrable; in
the sense called by Aubry \cite{Aubry:92a} the anti-integrable limit.
\begin{defn} \label{8d:deg}
We say that a $C^1$ function $P:\Bbb{R}^d \rightarrow \Bbb{R}$ satisfying
$P(x+a)=P(x)$ for every $a \in \Bbb{Z}^d$ is a non-degenerate potential, if there exist
constants $D, \epsilon > 0$
such that for every $C^1$ curve $\gamma : [0,T] \rightarrow \Bbb{R}^d$
satisfying for all $t \in (0,T)$,
\beq \inf_{\eta > 0} | \eta d\gamma(t)/dt - \nabla P(\gamma(t))| \leq \epsilon,
\label{8r:eps} \eeq
the following is true: $|\gamma(T)-\gamma(0)| < D$.
\end{defn}
The following Proposition is a simple exercise, we leave it to the reader.
\begin{prop} \label{8p:noproof}
If a $C^1$ function $P:\Bbb{R}^d \rightarrow \Bbb{R}$ satisfying
$P(x+a)=P(x)$ for every $a \in \Bbb{Z}^d$ has finitely many critical
poins (i.e. $x$ such that $\nabla P(x)=0$) in the unit cube, then it is non-degenerate.
\end{prop}
\begin{lemma} \label{8l:stays}
Assume that $h$ is a generating function satisfying (A1-3), and that $P$ is
a non-degenerate potential. Then for each $C >0$ there exists $\lambda > 0$, such that,
if $\phi_{\lambda}$,
is the gradient semiflow
of the generating function $h_{\lambda}(x,y)=h(x,y)-\lambda P(x)$, and $d$ the corresponding
distance function, then for each ${\bf x} \in {\cal B}_C$,
$$ d({\bf x}) \leq D+1,$$
where $D$ is the constant from Definition \ref{8d:deg}.
\end{lemma}
\proof
Let $C' = \sup \{h_1(x,y), h_2(x,y), \ : \ |x-y| \leq C+2D+2$ \}.
Let $\lambda$ be large enough, such that $\lambda \epsilon > 2C'$.
We will show that every ${\bf y} \in {\cal B}_C$, if
${\bf x}(t)$ is the solution of (\ref{2e:main}), ${\bf x}(0)=y$, then for each $t \geq 0$,
$||x(t)-x(0)||_{\infty} < D+1$.
Assume the contrary, and let $T=\inf \{ t \geq 0, : ,
||x(t)-x(0)||_{\infty} \geq D+1$. Then $||x(T)-x(0)||_{\infty}= D+1$, choose $i$
such that $|x_i(T)-x_i(0)| \geq D$, and let $\gamma : [0,T] \rightarrow \Bbb{R}$
be the curve $\gamma(t)=x_i(t)$. Then, applying (\ref{2e:main}) and the
definition of $C'$, for all $t \in [0,T]$,
\beq |d\gamma(t)/dt - \lambda \nabla P(\gamma(t))| &=& |h_2(x_{i-1},x_i)+h_1(x_i,x_{i_1})|
\leq 2C'. \label{8r:lam} \eeq
But (\ref{8r:eps}) implies that there exists $t \in [0,T]$ such that $|{1 \over \lambda}
d\gamma(t)/dt - \nabla P(\gamma(t)) | \geq \epsilon$, and (\ref{8r:lam}) now
implies that $\epsilon \lambda \leq 2C'$, which is in contradiction to the choice of $\lambda$.
\qed
\begin{corr} \label{8c:forreferee}
Assume that $h$ is a generating function satisfying (A1-3), and that $P$ is
a non-degenerate potential. Then for each $C >0$
there exists $\lambda > 0$, such that, given the generating function $h_{\lambda}(x,y)=h(
x,y)+\lambda P(x)$,
for all $\rho \in \Bbb{R}^d$, $|\rho| \leq C$, there exists $\mu \in {\cal S}^*_E$
such that $\rho(\mu)=\rho$ and $A(\mu) \leq \beta(\mu)+\epsilon$.
\end{corr}
\proof
Let $M=M(|\rho|+2)$ be the constant from Lemma \ref{l6:bound} associated to the generating
function $h_{\lambda}$; note that this constant is independent of $\lambda$.
Using Lemma \ref{8l:stays}, we
choose $\lambda$ large enough such that for all ${\bf x} \in {\cal B}_M$,
$d({\bf x}) \leq D+1$. Then for each measure $\mu \in {\cal A}^*_S$ supported on
${\cal B}_M$, $d^*(\mu)\leq D+1$, hence Theorem \ref{t8:low} applies. \qed
\begin{example} Proposition \ref{8p:noproof} implies that e.g. the potential $P_c(x_1,x_2)=\sin x_1+\sin x_2+\sin (x_1-x_2)$,
$P:\Bbb{R}^2 \rightarrow \Bbb{R}$, is nondegenerate, and that one can
apply Corollary \ref{8c:forreferee} to $h_{\lambda}=|x-y|^2+P_c(x)$, $x,y\in \Bbb{R}^2$.
\end{example}
We note that in dimension $d=3$, one can
construct, adjusting the Hedlund's counter-example (see e.g. \cite{Levi:97}),
a generating function $V$ satisfying
(A1-3) and a non-degenerate potential $P$, such that the only minimising configurations
of the generating function $h(x,y)+\lambda P(x)$ for all $\lambda \geq 0$ have integer
rotation vectors. This implies that the measures constructed in Corollary
\ref{8c:forreferee} are not necessarily minimising.
In the following two remarks we explain why we believe that the conclusion of Theorem
\ref{t8:low} is true in much more general situations than the non-degenerate anti-integrable
limit described above.
\begin{remark} {\it On Hedlund's counter-example and Levi's construction.}
We now comment on Levi's construction of
orbits with an arbitrary rotation vector in Hedlund's counterexample \cite{Levi:97},
ignoring for the moment that in this case we deal with a Lagrangian
(geodesic in particular) flow, and not a map. Levi constructed a closed family
of continuous curves (an analogue to a set of configurations in ${\cal A}$ here),
called ``pseudogeodesics'', and showed (\cite{Levi:97}, Lemma D) that (using
the terminology developed here) the function $d$ is uniformly bounded
and very small on that set. Furthermore, the set of Levi's pseudogeodesics
is rich enough, so that one can construct an $S$-ergodic
measure with an arbitrary rotation vector, supported on that set, and
recover the analogue of Theorem \ref{t8:quasi}, (ii). One could now
slightly improve Levi's construction, and construct invariant measures described
in Theorems \ref{t8:quasi} and \ref{t8:low}, as well as double-recurrent
quasi-periodic orbits with an arbitrary rotation vector.
\end{remark}
\begin{remark} {\it On the assumptions of Theorem \ref{t8:low}.} We conjecture
that the assumptions of Theorem \ref{t8:low} are satisfied in, if not all,
then in all but some degenerate cases. Assume the contrary, i.e.
that for all measures $\mu$ constructed in Lemma \ref{l8:small},
$d^*(\mu)=\infty$. It would mean that, even though the measure $\mu$ has
arbitrary small ($\leq \epsilon$)
relaxation energy $A(\mu)-A(\omega(\mu))$ available, almost every configuration
in $\supp \mu$ gets arbitrarily far away during the relaxation. Furthermore, a result
from \cite{Slijepce:99c} implies that for $\mu$-a.e. ${\bf x}$, $\omega({\bf x})$
(with respect to $\phi$) is a subset of ${\cal S}$; it now implies a degeneracy
of the phase space of $F$ in a neighbourhood of $\supp {\cal M}^*_{\rho}$ in some
sense, yet to be understood.
\end{remark}
\section{Discussion}
\smallsec{The twist maps and Aubry-Mather theory}
In the following we discuss
the case of twist maps on $\Bbb{T} \times \Bbb{R}$, i.e. we assume (E2). We
first comment on Gol\'{e}'s alternative proof \cite{Gole:92a} of the following
fundamental result of Aubry-Mather theory.
\begin{thm} \label{t9:AM} {\bf The Aubry-Mather theorem.}
Assume (E2). Then for every $\rho \in \Bbb{R}$, we can
find a configuration ${\bf x} \in {\cal S}$, such that $\rho({\bf x})=\rho$.
\end{thm}
An important step in Gol\'{e}'s proof (which was based on the study of
the semiflow induced by (\ref{2e:main})) is the following claim:
\begin{description}
\item[(*)] Assume (E2), and $N>0$. The $\omega$-limit set with respect to the semiflow
$\phi$ of each ${\bf x}\in {\cal B}_N$ contains a configuration in ${\cal S}$.
\end{description}
However, Gol\'{e}'s proof of (*) is incorrect (the statement of \cite{Gole:92a},
Lemma 2, does not extend to a ``neighbourhood of $x$''). Gol\'{e} later
replaced (*) with a weaker statement, and recovered the proof (Erratum to
\cite{Gole:92a} will appear in \cite{Gole:99}). One can prove (*) using the detailed
understanding of the energy flow of the equation (\ref{2e:main}), the proof
is in \cite{Slijepce:99c} ((*) is true even without the assumption (E2), i.e. $d=1$,
if the $\phi$-orbit of ${\bf x}$ is contained in ${\cal B}_C$ for some $C>0$).
We now give a short proof of \ref{t9:AM} which uses only Proposition \ref{p2:twist}
and Theorem \ref{3t:gradient}, and complete the argument from \cite{Gole:92a}.
\vspace{1ex}
\noindent {\bf Proof of Theorem \ref{t9:AM}:} For a given $\rho \in \Bbb{R}$,
we define the set $\tilde{\cal Y}_{\rho}$, of all ${\bf x} \in \Bbb{R}^{\Bbb{Z}}$
such that, given $p \in \Bbb{Z}$ and $q \in \Bbb{N}$, $T_pS^q{\bf x} \geq {\bf x}$
if $p/q > \rho$, and $T_pS^q{\bf x} \leq {\bf x}$ if $p/q < \rho$, and
let ${\cal Y}_{\rho}=\tilde{\cal Y}_{\rho}/T$. Then for each ${\bf x} \in {\cal Y}_{\rho}$,
$\rho({\bf x})=\rho$. If $N$ is the smallest integer greater than $|\rho|$,
${\cal Y}_{\rho}$ is a closed subset of ${\cal B}_N$, hence compact, and the semiflow
$\phi$ is by Proposition \ref{p2:twist} well-defined on ${\cal Y}_{\rho}$.
Monotonicity of the dynamics (\ref{2e:main}) (see the proof of Proposition
\ref{p2:twist}) implies that ${\cal Y}_{\rho}$ is $\phi$-invariant, hence the
set ${\cal Y}^*_{\rho}$ of $S$-invariant measures supported on ${\cal Y}_{\rho}$
is non-empty, compact (because of the compactness of ${\cal Y}_{\rho}$),
and $\phi^*$-invariant.
We choose $\mu \in {\cal Y}^*_{\rho}$, and $m \in \omega(\mu)$, then because of
Theorem \ref{3t:gradient} and closedness of ${\cal Y}^*_{\rho}$,
$m \in {\cal Y}^*_{\rho} \cap {\cal S}^*_S$. Then $m$-a.e. configuration
${\bf x}$ is stationary and in ${\cal Y}_{\rho}$, hence $\rho({\bf x})=\rho$. \qed
One can, however, use the remaining results of the paper and recover stronger results
of the Aubry-Mather theory including
the detailed description of the set of minimising measures and their supports,
by proving that the function
$\beta : \Bbb{R} \rightarrow \Bbb{R}$ is strictly convex. This was
already done by Mather, \cite{Mather:91b}, section 6.
We conjecture that the Mather's minimising measure (whose support is the
Aubry-Mather's set) is the unique measure in the $\omega$-limit set
(with respect to $\phi^*$) of the measure $\mu_{\rho}$, constructed in
subsection \ref{ss7:ex}. Gol\'{e}'s ghost tori \cite{Gole:92b} would then be
the attractor of $\supp \mu_{\rho}$ (i.e. the set $\bigcap_{T \geq 0}
Cl \bigcup_{t \geq T} \phi^t(\supp \mu_{\rho})$); this idea might lead
to a generalisation of the results from \cite{Gole:92b} to more degrees of
freedom.
\smallsec{Open problems}
The results of this paper are a contribution towards the description
of the phase-space of the semiflow $\phi^*$, as a tool for understanding the
phase space of the corresponding Lagrangian system. Here we list further possible
directions of this program.
\vspace{1ex}
\noindent {\bf Hyperbolicity.} A fascinating relationship between
a Lagrangian map $F$ and its ``dual'' semiflows $\phi$, $\phi^*$ is the result
by Aubry, Baesens, MacKay \cite{Aubry:92} (assuming only (A1) and (E1)):
a closure of an orbit of $F$ is uniformly hyperbolic, if and only if the corresponding
equilibrium of the flow $\phi$ is hyperbolic (in $||.||_p$ norm, for any $p \in [1,\infty]$).
It would be interesting to find a characterisation of hyperbolic invariant
measures of $F$ in terms of the semiflow $\phi^*$.
\vspace{1ex}
\noindent {\bf Birkhoff region of instability.} We propose the following generalisation
of Birkhoff region of instability to dimesions higher than $1$. We say that a
Birkhoff equivalence class is the set of all configurations in ${\cal S}$,
which is a subset of a connected component of $\{ {\bf x} \in {\cal A} \ : \
d({\bf x})< \infty \}$ (see Example \ref{e8:twist} for the case $d=1$). We say that
a Birkhoff region of instability is interior of a Birkhoff equivalence class.
The author has already succeeded to prove the existence of a heteroclinic orbit joining
any two $\supp {\cal M}^*_{\rho}$, $\supp {\cal M}^*_{\rho'}$ in the same
Birkhoff region of instability. Therefore the problem of existence of
heteroclinic orbits (and ``Arnold's diffusion'') can possibly be reduced to the study of
the distance function $d$ on ${\cal A}$.
\vspace{1ex}
\noindent {\bf The distance function and the Peierls Barrier.} Mather (\cite{Mather:91b})
and Fathi (\cite{Fathi:99}) constructed heteroclinic orbits
in Lagrangian systems, assuming certain topological properties of the Peierls
set (i.e. the set where the Peierls barrier is $0$). Since the Peierls barrier acts
as a ``barrier'' for the flow $\phi$, we conjecture that
finiteness of the functions $d, d^*$ in a small neighbourhood of
$\supp {\cal M}^*_{\rho}$, ${\cal M}^*_{\rho}$ is closely related (and possibly equivalent)
to these properties.
\vspace{1ex}
\noindent {\bf Measures with positive entropy.} We conjecture that each
set ${\cal M}^*_{\rho}$ not supported on a Lagrangian torus
(+ possibly additional assumptions on the Peierls set, as above) can be
weak$^*$-approximated by a positive-entropy invariant measure of the
Lagrangian map (generalisation of \cite{Forni:96}); we already have
initial results in that direction.
\vspace{1ex}
\noindent {\bf The Morse-Floer homology.} One might try to construct the
analogue of the Morse-Floer homology for the flow $\phi^*$, which might in
particular lead to better topological understanding of break-ups of invariant
tori, and KAM theory.
\section*{Acknowledgments} A part of this work was completed while I
was a PhD student in DAMTP, University of Cambridge; and is in
\cite{Slijepce:99}. I would like to thank my supervisor, Robert MacKay
for his continuing support. My thanks are due to Dimitris Gatzouras and Anthony
Quas for their help with ergodic-theoretical arguments, and to the referee for suggesting
numerous improvements of the paper. I also thank the organisers of the Winter School on Smooth Ergodic Theory (participation supported by ESF/PRODYN) and CIME Summer School on Hamiltonian Dynamics (participation supported by UNESCO-ROSTE) for inviting me to these most useful workshops.
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\end{document}
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