Content-Type: multipart/mixed; boundary="-------------0011121256737" This is a multi-part message in MIME format. ---------------0011121256737 Content-Type: text/plain; name="00-451.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-451.keywords" rotation vector, Aubry-Mather theory, symplectic maps ---------------0011121256737 Content-Type: application/x-tex; name="lagmap9.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lagmap9.TEX" \documentclass[10pt]{article} \pagestyle{plain} \usepackage{amstex} %\usepackage{epsfig} \setlength{\oddsidemargin} {0.2in} \setlength{\textwidth} {5.85in} \setlength{\topmargin} {0.25in} \setlength{\textheight} {8.5in} \renewcommand{\baselinestretch} {1} \setlength{\headheight} {0pt} \setlength{\headsep} {0pt} \addtolength{\footskip} {6pt} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{corr}[thm]{Corollary} \newtheorem{exam}[thm]{Example} \newtheorem{defn}{Definition}[section] \newtheorem{conj}[thm]{Conjecture} \newcommand{\beq}{\begin{eqnarray}} \newcommand{\beqn}{\begin{eqnarray*}} \newcommand{\eeq}{\end{eqnarray}} \newcommand{\eeqn}{\end{eqnarray*}} \newcommand{\dd}{\textrm{d}} \newcommand\qed{{\unskip\nobreak\hfil\penalty50 \mbox{~\fbox{\rule{0pt}{0pt}}} \parfillskip=0pt \finalhyphendemerits=0\par\medskip}} \newcommand{\proof}{\par\noindent{\bf Proof}: } \newcommand{\supp}{\textnormal{supp }} \newcounter{smallsec}[section] \renewcommand{\thesmallsec}{ \arabic{section}.\arabic{smallsec}} \newcommand{\smallsec}[1]{ \vspace{3ex} \refstepcounter{smallsec} \noindent {\bf \arabic{section}.\arabic{smallsec} #1.} \nopagebreak \medskip \nopagebreak \noindent } \newcounter{remark} \newenvironment{remark} {\medskip\par \refstepcounter{remark} \noindent{\bf Remark \arabic{remark}}:} {\par\bigskip} \newenvironment{example} {\medskip\par \refstepcounter{remark} \noindent{\bf Example \arabic{remark}}:} {\par\bigskip} \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}. 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