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Aubry-Mather theory, KAM tori, Lagrangian graphs, Schwartzman unique ergodicity
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\begin{document}

\title{Uniqueness of Invariant Lagrangian Graphs in a Homology or 
a Cohomology Class.}
\author{\sc{Albert Fathi, Alessandro Giuliani and Alfonso Sorrentino}}

\date{22 January 2008}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%
\begin{center}
\parbox[c]{4.5 in}
{\small Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$
on the cotangent space ${\rm T}^*M$, strictly convex and superlinear in the 
momentum variables, we prove uniqueness of certain ``ergodic'' invariant 
Lagrangian graphs within a given homology or cohomology class. In particular, 
in the context of quasi-integrable Hamiltonian systems, our result implies 
global uniqueness of Lagrangian KAM tori with rotation vector $\r$. This result 
extends generically to the $C^0$-closure of KAM tori.}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% SECTION 1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}\label{intro}

%%%%%%%%%%%%%%%%%%%%s+\t}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A particularly interesting and fruitful approach to the study of local and 
global properties of dynamical systems is concerned with the study of invariant
submanifolds, rather than single orbits, paying particular 
attention to their existence, their ``fate'' and their geometric properties.
In the context of quasi-integrable Hamiltonian systems, one of the most 
celebrated breakthroughs in this kind of approach was KAM theory, which 
provided a method to construct invariant submanifolds diffeomorphic 
to tori, on which the dynamics is conjugated to a quasi-periodic motion
with rotation vector $\r$, sometimes referred to as {\it KAM tori}. 
KAM theory finally settled the old question about {\it existence} of such 
invariant submanifolds 
in ``generic'' quasi-integrable Hamiltonian systems, dating back at least
to Poincar\'e, and opened the way to a new understanding of the nature of 
Hamiltonian systems, of their stability and of the onset of chaos in classical 
mechanics. However, the natural question about the {\it uniqueness} of 
these invariant submanifolds for a fixed rotation vector $\r$
remained open for many more years and, quite 
surprisingly, even nowadays, for many respects it is still unanswered. 
A possible reason for this is that the analytic methods, which the 
KAM algorithm is based on, are not well suited for studying global
questions, while, on the other hand, the natural variational methods to 
approach this problem have been developed only much more recently and 
they are still not so widely well-known.

In this paper we address the above problem and prove global uniqueness of 
certain ``ergodic'' invariant Lagrangian graphs 
within a given homology or cohomology class, for a large class 
of convex Hamiltonians, known as Tonelli Hamiltonians.
Our work will be based on the variational 
approach provided by the so-called Aubry-Mather theory, as well as its 
functional side called weak KAM theory.

The paper is organized as follows. In Section \ref{sec1} we  
define the geometric objects we shall look at (the invariant Lagrangian 
graphs), we introduce some concepts (homology class of an 
invariant measure and Schwartzman ergodicity), which will turn out to be useful
for illustrating their dynamical properties, and we state our main 
uniqueness results.
In Section \ref{sec2} we review the main definitions and
results of Aubry-Mather and weak KAM theories, to the extent of 
what we need for our proofs. In Section \ref{sec3} 
we prove our main results. In Section \ref{sec4} we discuss in detail
the implications of our results for KAM theory and compare
them with some previous local uniqueness theorems for KAM 
tori. In Appendix \ref{app1} we discuss some details
concerning the definition of Schwartzman ergodicity, give 
some examples and describe some properties of Schwartzman ergodic flows.
In Appendices \ref{app2} and \ref{app3} we prove some properties of
Lagrangian graphs and the Ma\~n\'e set.
%In Appendix \ref{app2} we prove some properties of 
%Lagrangian graphs and their relation with KAM tori. 
%In Appendix \ref{app3} we prove a connection between invariant measures supported on minimizing orbits and Aubry sets.

\vspace{.3truecm}

{\bf Acknowledgements.} We would like to thank Giovanni Gallavotti for 
having drawn our attention to this problem and for useful discussions. 
AF and AS are grateful to John Mather for many fruitful conversations.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% SECTION 1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Setting and Main results}\label{sec1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $M$ be a compact and connected 
smooth manifold without boundary of dimension $n$.
Denote by ${\rm T}M$ its tangent bundle and ${\rm T}^*M$ the cotangent one. A
point of ${\rm T}M$ will be denoted by $(x,v)$, where $x\in M$ and $v\in
{\rm T}_xM$, and a point of ${\rm T}^*M$ by $(x,p)$, where 
$p\in {\rm T}_x^*M$ is a linear form on the vector space ${\rm T}_xM$. 
Let us fix a Riemannian metric $g$ on $M$ and let
$\|\cdot\|_x$ be the norm induced by $g$ on ${\rm T}_x M$; we shall use the 
same notation for the norm induced on the cotangent space
${\rm T}_x^*M$. This cotangent space ${\rm T}^*M$ can be canonically endowed 
with a symplectic structure, given by the exact $2$-form 
$\o=dx\wedge dp = -d(pdx)$, where $({\mathcal U},\,x)$ is a local coordinate 
chart for $M$ and $({\rm T}^*{\mathcal U},\,x, p)$ the associated 
cotangent coordinates. The $1$-form $\l = pdx$ is also called 
{\it tautological form} (or {\it Liouville form}) and is intrisically defined, 
\ie independently of the choice of local coordinates (see Appendix \ref{app2}).
A distinguished role in the study of the geometry of a symplectic space is 
played by the so-called {\it Lagrangian submanifolds}.

\begin{Def}\rm
Let $\Lambda$ be an $n$-dimensional $C^1$ submanifold of 
$({\rm T}^*M,\omega)$. We 
say that $\Lambda$ is {\it Lagrangian} if for any $(x,p)\in \Lambda$, 
${\rm T}_{(x,p)}\Lambda$ is a Lagrangian subspace, \ie
$\omega \big|_{{\rm T}_{(x,p)}\Lambda} = 0$. 
\end{Def}

We shall mainly be concerned with Lagrangian graphs, that is Lagrangian 
manifolds $\Lambda\subset T^*M$ such that $\Lambda=\{(x,\eta(x))\,,\ x\in M\}$.
It is straightforward to check that the graph $\Lambda$ is Lagrangian if 
and only if $\eta$ is a closed $1$-form (see Appendix \ref{app2}). 
The element $c=[\eta]\in \rH^1(M;\R)$ is called the {\it 
cohomology class}, or {\it Liouville class}, of $\Lambda$.
This motivates the following extension of the notion of 
Lagrangian graph to the continuous case.

\begin{Def}\rm
A continuous section $\L$ of ${\rm T}^*M$ is a $C^0$-Lagrangian graph
if it locally coincides with the graph of an exact differential.
\end{Def}

We will consider the dynamics on ${\rm T}^*M$ generated by a Tonelli 
Hamiltonian.

\begin{Def}\rm A function $H:\,{\rm T}^*M\longrightarrow \R$ is called a 
{\it Tonelli (or optical) Hamiltonian} if:
\begin{itemize}
\item[i)]  the Hamiltonian $H$ is of class $C^k$, with $k\geq 2$;
\item[ii)]  the Hamiltonian $H$ is strictly convex in the fiber in the $C^2$ 
sense, \ie the second partial 
vertical derivative ${\dpr^2 H}/{\dpr p^2}(x,p)$ is positive definite, 
as a quadratic form, for any $(x,p)\in {\rm T}^*M$;
\item[iii)] the Hamiltonian $H$ is superlinear in each fiber, \ie
$$\lim_{\|p\|_x\rightarrow +\infty} \frac{H(x,p)}{\|p\|_x} = + \infty$$
(by the compactness of $M$, this condition is independent of the choice of the 
Riemannian metric).
\end{itemize}
\end{Def}

Given $H$, we can define the associated {\it Lagrangian},
as a function on the tangent bundle:
%
\bea L:\; {\rm T}M &\longrightarrow & \R \nn\\
(x,v) &\longmapsto & 
\sup_{p\in {\rm T}^*_xM} \{\langle p,\,v \rangle_x -H(x,p)\}\,\nn \eea 
%
where $\langle \,\cdot,\,\cdot\, \rangle_x$
represents the canonical pairing between the tangent and cotangent
space.

If $H$ is a Tonelli Hamiltonian, one can easily prove that $L$ is
finite everywhere, of class $C^k$, superlinear and strictly
convex in the fiber in the  $C^2$ sense (\ie $L$ is a 
{\it Tonelli Lagrangian}) and the 
associated Euler-Lagrange flow $\Phi^L_t$ of $L$ is conjugated to the 
Hamiltonian flow $\Phi^H_t$ of $H$ via
the {\it Legendre transform}:
%
\bea \cL:\; {\rm T}M &\longrightarrow & {\rm T}^*M \\
(x,v) &\longmapsto & \left(x,\,\frac{\dpr L }{\dpr v}(x,v)
\right).\label{2.1} \eea
%
From now on we shall fix $H$ and denote by $L$ its conjugated Lagrangian and,
when referring to an ``invariant'' measure or set, 
we will understand ``invariant with respect to 
the Hamilton flow generated by $H$'' or ``with respect to the 
Euler-Lagrange flow generated by $L$''. 

Given an invariant probability measure $\m$ on ${\rm T}M$ one can 
associate to it an element $\r(\m)$ of the homology group $\rH_1(M;\RRR)$, 
known as {\it rotation vector} or {\it Schwartzman asymptotic cycle},
which generalizes the notion of rotation vector given by 
Poincar\'e and describes how, asymptotically, a 
$\m$-average orbit winds around ${\rm T}M$. See Section \ref{sec2}
and Appendix \ref{app1} for a precise definition. This allows us to 
define the homology class of certain invariant Lagrangian graphs. 

\begin{Def}\rm
A Lagrangian graph $\L$ is called {\it Schwartzman uniquely ergodic} if all 
invariant 
measures supported on $\L$ have the same rotation vector $\r$, which 
will be called {\it homology class of} $\L$. Moreover, if there exists
an invariant measure with full support, $\L$ will be called {\it Schwarztman 
strictly ergodic}.
\end{Def}

We are now ready to state our main results.\\

{\bf{Main Result.}}
{\it For any given $\r\in \rH_1(M;\RRR)$, there exists at most one 
Schwarzman strictly ergodic invariant Lagrangian graph with homology class 
$\r$} [Theorem \ref{uniqschw}, Section \ref{sec3}]\\

For sake of completeness, we also recall the following well-known 
result, which is a corollary of the results in \cite{Mather91}
(see also Section \ref{sec3} for a proof).\\
\\
{\bf{Well-known Result.}} {\it For any given $c\in \rH^1(M;\RRR)$, there 
exists at most one invariant Lagrangian
graph $\Lambda$ with cohomology class $c$, carrying an invariant measure 
whose support is the whole of $\Lambda$.}  [Theorem \ref{uniqcom}, 
Section \ref{sec3}]\\

If $M=\TTT^n$, it is natural to ask for the implications of our result 
for KAM theory. In this case, the homology group $\rH_1(\TTT^n;\RRR)$  
is canonically identified with $\R^n$, and  the invariant manifolds of 
interest are the so-called KAM tori, defined as follows.  

\begin{Def}\label{kamtorus}\rm
$\TT\subset\TTT^n\times\RRR^n$ is a (maximal) KAM torus with 
rotation vector $\r$ if:\\
\0i) $\TT\subset\TTT^n\times\RRR^n$ is a continuous graph over $\TTT^n$;\\
\0ii) $\TT$ is invariant under the Hamiltonian flow $\Phi_t^H$ 
generated by $H$;\\
\0iii) the Hamiltonian flow on $\TT$ is conjugated to a uniform 
rotation on $\TTT^n$; \ie there exists a diffeomorphism 
$\f: \TTT^n\to \TT$
such that $\f^{-1}\circ\Phi_t^H\circ\f= R_\r^t$, $\forall t\in\RRR$,
where $R_\r^t: x \to x+\r t$.
\end{Def}

The celebrated KAM Theorem, whose statement will be recalled in Section 
\ref{sec4}, gives sufficient conditions on $H$ and on the rotation
vector $\r$, allowing one to construct a KAM torus with rotation vector $\r$
and prescribed regularity (depending on the regularity class of $H$). Its 
proof is constructive and the invariant torus one manages to construct is 
locally unique (in a sense that will be clarified in Section \ref{sec4}). 
In spite of the long history and the huge literature dedicated to the KAM 
theorem, the issue of global uniqueness of such tori is still object of some
debate and study, see for instance \cite{BroerTakens}. Our main result 
settles such question in the case of Tonelli Hamiltonians.

\begin{Cor}[Global uniqueness of KAM tori]\label{corollary1}
Every Tonelli Hamiltonian $H$ on ${\rm T}^*\TTT^n$ possesses at most one 
Lagrangian KAM torus for any given rotation vector $\r$. In particular, 
if $H$ and $\r$ satisfy the assumptions of the KAM Theorem, then there exists 
one and only one KAM torus with rotation vector $\r$.
\end{Cor}

The property of being Lagrangian plays a crucial role. As was observed by 
Herman 
\cite{Herman}  (see also Appendix \ref{app2}), when $\r$ is
 {\it rationally independent}, \ie 
$\media{\r,\n}\neq 0$, $\forall \n\in\ZZZ^n\setminus\{0\}$, as assumed in the 
KAM theorem, every KAM torus with frequency $\r$ 
is automatically Lagrangian. On the other hand, the existence of
Lagrangian KAM tori with rationally dependent frequency is not typical. 
In some cases, a variant of the classical KAM algorithm allows one 
to construct {\it resonant} invariant tori with a given rational 
rotation vector $\r$, also known as low dimensional tori 
\cite{GG1,GG2}. However, typically they do not foliate any Lagrangian 
submanifold. Therefore, 
the question of uniqueness of resonant tori is more subtle 
and, to our knowledge, apart from a few partial results \cite{CGGG}, 
it is still open.

Note that the fact that the orbits on a 
KAM torus are action-minimizing is a key remark used in the proof of 
the Corollary. We deduce this property  from the fact that such a Lagrangian 
torus gives a solution to the Hamilton-Jacobi equation. It can be also 
deduced from a classical result by 
Weierstrass, as already pointed out by J\"urgen Moser. 
We thank John Mather for having drawn our attention to the remark in 
\cite{Mather91}. [In fact, Weierstrass method or the use of the 
Hamilton-Jacobi Equation 
are essentially two sides of the same coin].

In Section \ref{sec4} we will extend Corollary \ref{corollary1} 
to generic invariant 
tori contained in the $C^0$-closure of the set of KAM tori. 
As remarked by Herman \cite{HermanTori}, 
generically this set is much larger than the
set of KAM tori, and typically the flow on such invariant manifolds is not 
conjugated to a rotation. See Section \ref{sec4} for a more detailed 
discussion of these issues. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% SECTION 2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{An introduction to Aubry-Mather theory}\label{sec2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section is meant to provide a brief (but comprehensive)
introduction to Mather's theory for Lagrangian systems and 
weak KAM theory. We shall recall most of the results that we 
are going to 
use, trying to give general ideas rather 
than rigourous proofs 
(for which we refer the reader to \cite{Fathibook, Mather91}).

The setting is the same introduced in Section \ref{sec1}. Let $M$ be a compact 
and connected 
smooth manifold without boundary of dimension $n$. We denote by $H$ a 
fixed Tonelli Hamiltonian 
and $L$ the associated Tonelli Lagrangian.
Before entering into the details of Mather's theory, let us make a crucial 
remark.
Observe that if $\eta$ is a $1$-form on $M$, we can easily define a function 
on the tangent space (linear on each fiber)
\beqano 
\hat{\eta}: {\rm T}M &\longrightarrow&  \R \\
(x,v) &\longmapsto& \langle \eta(x),\, v\rangle_x
\eeqano
and consider a new Tonelli Lagrangian $L_{\eta}:= L - \hat{\eta}$. The 
associated Hamiltonian will be 
$H_{\eta}(x,p) = H(x,\eta(x) + p)$.
Moreover, if $\eta$ is closed, then $\int L\, dt$ and $\int L_{\eta} dt$ will 
have the same extremals and therefore the Euler-Lagrange flows on ${\rm T} M$ 
associated to $L$ and $L_{\eta}$ will be the same. This last point may be seen 
by observing that the variational equations $\d \int L\, dt =0$ and $\d \int 
(L-\hat{\eta})\, dt =0$ for the fixed end-point problem clearly have the same 
solutions, since $\eta$ is closed.
Although the extremals are the same, this is not generally true for the orbits

``minimizing the action'' (we shall give a precise definition of  
``minimizers'' later in this section).
What one can say is that they stay the same when we change the 
Lagrangian by an exact $1$-form. Thus, for a fixed $L$, the minimizers will 
depend only on the de Rham cohomology class $c=[\eta] \in \rH^1(M;\R)$. 
Considering modified Lagrangians corresponding to 
different cohomology classes represents a keystone of Mather's
theory of Lagrangians systems.\\


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In order to generalize to more degrees of freedom Aubry and Mather's 
variational approach to twist maps, a first important notion is that of 
{\it minimal measures}, which replaces that of action minimizing orbits: 
Aubry-Mather theory in higher dimension cannot deal with such orbits, due to a 
lack of them; there is, in fact, a classical example due to Hedlund (see 
\cite{Hedlund}) of a 
Riemannian metric on a three-dimensional torus, for which minimal geodesics 
exist only in three directions. Instead, Mather proposed to look at the 
closely related notion of action minimizing invariant probability measures.
Let $\calM(L)$ be the space of probability measures on 
${\rm T}M$  invariant  under the Euler-Lagrange flow of $L$. To every 
$\m \in \calM(L)$, we may 
associate its {\it average action}
$$ A_L(\m) = \int_{{\rm T}M} L\,d\m\,. $$
Since $L$ is bounded below (because of the superlinear growth condition), this 
integral exists although it might be $+\infty$. In \cite{Mather91}, 
Mather showed the 
existence of $\m \in \calM(L)$ such that $A_L(\m)<+\infty$. The argument is 
mainly the same as Krylov-Bogoliubov's theorem concerning existence of 
invariant measures for flows on compact spaces. This argument is applied to 
a one-point 
compactification of ${\rm T}M$, and the main step consists in showing that the 
measure provided by this construction has no atomic part supported at $\infty$ 
(which is a fixed point for the extended system). Note that Mather's approach 
works for periodic time-dependent Lagrangians. For time independent 
Lagrangians, finding such a $\mu$ is much easier. By conservation of energy, 
the levels of the energy function are compact  and invariant under the 
Euler-Lagrange flow $\Phi^L_t$, and therefore carry such measures.
In case $A_L(\m)<\infty$, thanks to the superlinearity of $L$, for any 
closed 1-form $\eta$ on $M$, the integral  $ \int_{{\rm T}M} \hat{\eta} d\m$ 
is well defined and finite, see \cite{Mather91}. Moreover, it is quite easy 
to show (again see \cite{Mather91}) that since $\m$ is invariant by 
the Euler-Lagrangian flow $\Phi^L_t$, if $\eta=df$ is an exact form, then 
$\int{\widehat{df}} d\m =0$. Therefore, we can define a linear functional
\beqano
{\rm H}^1(M;\R) &\longrightarrow& \R \\
c &\longmapsto& \int_{{\rm T}M} \hat{\eta} d\m\,,
\eeqano
where $\eta$ is any closed $1$-form on $M$, whose cohomology class is $c$. By 
duality, there 
exists $\rho (\m) \in {\rm H}_1(M;\R)$ such that
$$
\int_{{\rm T}M} \hat{\eta} d\m = \langle c,\rho(\m) \rangle
\qquad \forall\,c\in {\rm H}^1(M;\R)$$ 
(the bracket on the right--hand side denotes the canonical pairing between 
cohomology and 
homology). We call $\rho(\m)$ the {\it rotation vector} of $\m$. It is 
the same as 
the
Schwartzman's asymptotic cycle of $\mu$ (see Appendix \ref{app1} and 
\cite{Schwartzman}).
One can show that the action functional $A_L: \calM(L) 
\longrightarrow \R\cup \{+\infty\}$ is lower semicontinuous and that for every 
$h\in {\rm H}_1(M;\R)$ 
there exists $\m\in \calM(L)$ with $\rho(\m)=h$. In other words
the map
$\rho: \calM(L) \longrightarrow {\rm H}_1(M;\R)$ is surjective.\\

Following Mather, let us consider
the minimal value of the average action $A_L$ over the 
probability measures with rotation vector $h$:
\bea
\b: {\rm H}_1(M;\R) &\longrightarrow& \R \nn\\
h &\longmapsto& \min_{\m\in\calM(L):\,\rho(\m)=h} A_L(\m)\,.\label{2.2}
\eea
In fact, the minimum above is actually achieved (see \cite{Mather91}).
This function $\beta$ is what is generally known as {\it Mather's 
$\beta$-function}.
A measure $\m \in \calM(L)$ realizing such a minimum, \ie $A_L(\m) = 
\b(\rho(\m))$, 
is called a {\it minimal} (or a {\it Mather}) {\it measure}. The 
$\beta$-function is convex, and therefore one can consider its {\it conjugate} 
function (given by Fenchel's duality)
$ \a: {\rm H}^1(M;\R) \longrightarrow \R $ defined by
\beqano
\a(c) &:=& \max_{h\in {\rm H}_1(M;\R)} \left(\langle c,h \rangle - \b(h) 
\right)
= \\
&=& - \min_{h\in {\rm H}_1(M;\R)} \left(\b(h) - \langle c,h \rangle\right) =\\
&=& - \min_{\m \in \calM(L)} \left( A_L(\m) - \langle c, \rho(\m)\rangle\right)
=\\
&=& - \min_{\m \in \calM(L)} A_{L-c}(\m).\\
\eeqano
Note that for a given $\mu$ invariant under the Euler-Lagrange flow $\Phi^L_t$,
we have $\langle c, \rho(\m)\rangle= \int_{{\rm T}M} \hat{\eta}\, d\m$, for any 
closed 1-form $\eta$ on $M$ in the cohomology class  $c$. Therefore, we have  
$A_{L-c}(\m)= \int_{{\rm T}M} (L-\hat{\eta})\, d\m$, for $\eta$ a closed 
$1$-form on $M$ such that $[\eta]=c$.

It is important to notice that the value of this $\a$-function coincides 
with what is called {\it Ma\~n\'e's critical value}, which will be introduced 
later in this section.

An important fact is the next Lemma. To state it, recall that, like any convex 
function on a
finite-dimensional space, the Mather function  $\b$
admits a subderivative at any point $h\in \rH_1(M;\R)$, \ie we can find $c\in 
\rH^1(M;\R)$ such that
$$\forall h'\in \rH_1(M;\R), \quad \b(h')-\b(h)\geq \langle c,h'-h\rangle.$$
As it is usually done, we will denote by $\partial \b(h)$ the set of $c\in 
\rH^1(M;\R)$ which 
are subderivatives of $\b$ at $h$, \ie the set of $c$ which satisfy the 
inequality above. By Fenchel's duality,  we have
$$c\in \partial \b(h) \Longleftrightarrow \langle c,h\rangle=\a(c)+\b(h).$$
\begin{Lem}\label{CaracMinim}
If $\m \in \calM(L)$, then $A_L(\m)=\b(\rho(\m))$ if and only if there 
exists $c\in {\rm H}^1(M;\R)$ such that $\m$ minimizes $A_{L-c}$  
(\ie $A_{L-c}(\mu)=-\a(c)$).
Moreover, if $\m \in \calM(L)$ satisfies $A_L(\m)=\b(\rho(\m))$, and $c \in 
\rH^1(M;\R)$, then $\m$ minimizes $A_{L-c}$ if and only if $c\in \partial 
\b(\rho(\mu))$ {\rm(}or  $\langle c,h\rangle=\a(c)+\b(\rho(\m)\rm{)}$.
\end{Lem}
\begin{Proof} We will prove both statement at the same time.
Assume $A_L(\m_0)=\b(\rho(\m_0))$. Let $c\in \partial \b(\rho(\mu_0))$, by 
Fenchel's duality this is equivalent to
\begin{align*}
\a(c)&=\langle c,\rho(\m_0)\rangle-\b(\rho(\m_0))\\
&=\langle c,\rho(\m_0)\rangle-A_L(\m_0)\\
&=-A_{L-c}(\m_0).
\end{align*}
Therefore by what was obtained above $A_{L-c}(\m_0)=\min_{\m \in \calM} 
A_{L-c}(\m)$.

Assume conversely that $A_{L-c}(\m_0)=\min_{\m \in \calM} A_{L-c}(\m)$, for some 
given cohomology class $c$. Therefore by what was obtained above
$$\a(c)=-A_{L-c}(\m_0),$$
which can be written as 
$$ \langle c,\rho(\m_0)\rangle=\a(c)+A_L(\m_0).$$
It now suffices to use the Fenchel inequality $ \langle c,\rho(\m_0)\rangle\leq 
\a(c)+\b(\rho(\m_0)$, and the inequality $\b(\rho(\m_0)\leq A_L(\m_0)$, given 
by the definition of $\b$, to obtain the  equalities 
$$ \langle c,\rho(\m_0)\rangle=\a(c)+A_L(\m_0).$$
In particular, we have $A_L(\m_0)=\b(\rho(\m_0))$.
 \end{Proof}
If $\m\in\calM(L)$ and $\m$ minimizes 
$A_{L-c}$, we shall say that $\m$ is a {\it $c$-action minimizing measure}.
For such $\m$, $c$ is a subderivative of $\b$ 
at $\rho(\m)$, \ie the slope of a supporting hyperplane of the epigraph of 
$\b$ at $\rho(\m)$.\\
The above discussion leads to two equivalent formulations for the minimality 
of a measure $\m$:
\begin{itemize}
\item there exists a homology class $h \in {\rm H}_1(M;\R)$, namely its 
rotation vector $\rho(\m)$, such that $\m$ minimizes $A_L$ amongst all 
measures 
in $\calM(L)$, with rotation vector $h$; \ie $A_L(\m)=\b (\rho(\m))$;
\item there exists a cohomology class $c \in {\rm H}^1(M;\R)$, namely any 
subderivative of $\b$ at $\rho(\m)$, such that $\m$ minimizes $A_{L-c}$ 
amongst all measures in $\calM(L)$; \ie $A_{L-c}(\m)=-\a (c)$.
\end{itemize}

For $h \in {\rm H}_1(M;\R)$ and $c \in {\rm H}^1(M;\R)$, let us define
\beqano
\calM^h &:=& \{\m \in \calM(L): \; A_L(\m)<+\infty, \;\rho(\m)= h \; 
{\rm and}\; 
A_L(\m)=\beta(h)\}\\
\calM_c &:=& \{\m \in \calM(L): \; A_{L}(\m)<+\infty \;{\rm and}\; A_{L-c}(\m)
=-\a (c)\}.
\eeqano 
Observe that because of the superlinear growth condition in the fiber, 
$A_L(\m)< +\infty$
implies $A_{L-c}(\m)< +\infty$. 

We have
$$
\bigcup_{h \in {\rm H}_1(M;\R)} \calM^h = \bigcup_{c \in {\rm H}^1(M;\R)} 
\calM_c\,.
$$

This leads to the definition of a first important family of invariant sets: 
{\it Mather sets}. For a cohomology class $c \in {\rm H}^1(M;\R)$, we call 
{\it Mather set of 
cohomology class} $c$ the set:
%
\be \widetilde{\cM}_c := \overline{\bigcup_{\m \in \calM_c} {\rm supp}\,\m} 
\subset {\rm T}M\,;\label{2.3}\ee
%
the projection on the base manifold $\cM_c = \pi \left(\widetilde{\cM}_c\right)
\subseteq M$ is called {\it projected Mather set} (with cohomology class $c$).
In \cite{Mather91}, Mather proved the celebrated {\it graph theorem}:

\begin{Teo}\label{Theograph}Let $\widetilde{\cM}_c$ be defined as in 
(\ref{2.3}). 
The set $\widetilde{\cM}_c$ is compact, invariant 
under the Euler-Lagrange flow and $\pi|{\widetilde{\cM}_c}$ is an injective 
mapping of $\widetilde{\cM}_c$ into $M$, and its inverse $\pi^{-1}: \cM_c 
\longrightarrow \widetilde{\cM}_c$ is 
Lipschitz. Moreover this set is contained 
in the energy level corrisponding to the value $\a(c)$, \ie
%
\be H \circ \cL(x,v) = \a(c) \qquad \forall\,(x,v)\in \widetilde{\cM}_c\,.
\label{2.4}\ee
\end{Teo}
%

\begin{Rem}{\rm The last statement, corresponding to (\ref{2.4}), is due to 
Dias-Carneiro \cite{Carneiro}.}
\end{Rem}
 
Analogously, one can consider the {\it Mather set corresponding to a rotation 
vector} $h\in{\rm H}_1(M;\R)$ as
%
\be \widetilde{\cM}^h := \overline{\bigcup_{\m \in \calM^h} {\rm supp}\,\m} 
\subset {\rm T}M\,,\label{2.5}\ee
%
and the projected one $\cM^h = \pi \left(\widetilde{\cM}^h\right) 
\subseteq M$. 

Notice that by Lemma \ref{CaracMinim}, if $c\in\partial\b(h)$, we have
$$\widetilde{\cM}^h\subseteq \widetilde{\cM}_c.$$
Therefore, although this was not shown in \cite{Mather91}, the set 
$\widetilde{\cM}^h$
also has  a Lipschitz graph over the basis. 

\begin{Teo}\label{matherrotation} 
Let $\widetilde{\cM}^h$ be defined as in (\ref{2.5}). $\widetilde{\cM}^h$
is compact, invariant under the Euler-Lagrange flow and 
$\pi|{\widetilde{\cM}^h}$ is an injective 
mapping of $\widetilde{\cM}^h$ into $M$ and its inverse $\pi^{-1}: \cM^h 
\longrightarrow \widetilde{\cM}^h$ is Lipschitz.
\end{Teo}

\begin{Rem} Though the graph property for $\widetilde{\cM}^h$ is not proved in 
\cite{Mather91}, it was shown there that the support of an $h$-minimizing 
measure has the graph property. The  graph property for $\widetilde{\cM}^h$ 
can be easily deduced from
this last property. In fact, since the space of 
probability 
measures on ${\rm T}M$ is a separable metric space, one can take
a countable dense set $\{\m_n\}_{n=1}^{\infty}$ of Mather's 
measures with rotation vector $h$ and consider the new measure
$\tilde{\m} = \sum_{n=1}^\infty \frac{1}{2^n}\m_n$. This is still an invariant 
probability measure with rotation vector $h$ and ${\rm supp}\,\tilde{\m} = 
\widetilde{\cM}_h$. Therefore, as the support of a single $h$-minimizing 
measure, $\widetilde{\cM}^h $ has the graph property.

Moreover, this remark points out that there always exist Mather's measures
$\m^h$ and $\m_c$ of full support, \ie 
${\rm supp}\,{\m^h} = \widetilde{\cM}^h$ and ${\rm supp}\,{\m_c} = 
\widetilde{\cM}_c$.

We will say that an $h$-minimizing (resp.~$c$-minimizing)  measure $\mu$ has 
maximal
support if ${\rm supp}\,{\m} = \widetilde{\cM}^h$ (resp.~${\rm supp}\,{\m} = 
\widetilde{\cM}_c$).
\end{Rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In addition to the Mather sets, one can construct other compact invariant 
sets, that play an interesting role both from a dynamical system and a 
geometric 
point of view: the {\it Aubry sets} and the {\it Ma\~n\'e  sets}. Instead of 
considering action minimizing invariant probability measures, we now shift our 
attention to $c$-{\it minimizing curves}.


Let us fix $\eta$ a closed $1$-form on $M$,
with cohomology class $c$.
As done by Mather in \cite{Mather93}, it is convenient to
introduce, for $t>0$ and $x,\,y \in M$, the following quantity:
$$ h_{\eta,t}(x,y) = \inf \int_0^t L_{\eta}(\g(s),\dot{\g}(s))\,ds\,,$$
where the infimum is taken over all piecewise $C^1$ paths $\g:
[0,t]\longrightarrow M$, such that $\g(0)=x$ and $\g(t)=y$.
We define the {\it Peierls' barrier} as:
$$ h_{\eta}(x,y) = \liminf _{t \rightarrow +\infty} (h_{\eta,t}(x,y) + \a(c)t)
\,.$$
%where $\a:\,H^1(M;\R) \longrightarrow \R$ is Mather's $\a$ function 
%(see \cite{Mather91}). 
It can be shown that this function is Lipschitz and that, only in the 
autonomous case, this $\liminf$ can be 
replaced with a $\lim$ (see \cite{Fathibook}).
Observe that $h_{\eta}$ does not depend only on the cohomology class $c$, but 
also on the choice of the representant; namely, if $\eta' = \eta + df$, then 
$h_{\eta'}(x,y) = h_{\eta}(x,y) + f(y) - f(x)$.

\begin{Prop}
The values of the map $h_{\eta}$ are finite. Moreover, the following
properties hold:
\begin{itemize}
\item[{\rm i)}]   $h_{\eta}$ is Lipschitz;
\item[{\rm ii)}]  for each $x\in M$, $h_{\eta}(x,x)\geq 0$;
\item[{\rm iii)}]  for each $x,\,y,\,z \in M$, $h_{\eta}(x,y) \leq h_{\eta}
(x,z) + h_{\eta}(z,y)$;
\item[{\rm iv)}]  for each $x,\,y \in M$, $h_{\eta}(x,y) + h_{\eta}(y,x) 
\geq 0$.
\end{itemize}
\end{Prop}

For a proof of the above claims and more, see \cite{Mather93, Fathibook, 
ContrerasIturriaga}.
Inspired by these properties, one can consider its symmetrization:
\beqano \d_c:\; M\times M &\longrightarrow & \R \\
(x,y) &\longmapsto & h_{\eta}(x,y) + h_{\eta}(y,x) \eeqano
%
(observe that this function does depend only on the cohomology class).
This function is positive,
symmetric and satisfies the triangle inequality; therefore, it is a
pseudometric on $$\cA_{c} =\{x\in M : \; \d_c(x,x)=0 \}\,. $$

$\cA_{c}$ is called the {\it projected Aubry set} 
associated to $L$ and $c$, and $\d_c$ is {\it Mather's pseudometric}. In
\cite{Mather93}, Mather has showed that this is a non-empty compact
subset of $M$, that can be Lipschitzly lifted to a compact invariant subset of 
${\rm T}M$.

\begin{Rem}
One can easily construct a metric space out of it. We call {\it quotient 
Aubry set}, or
{\it Mather quotient},  the metric space $(\cbA_{c},\, \bd_c)$
obtained by identifying two points in $\cA_{c}$, if their $\d_c$-pseudodistance
is zero.
We shall denote an element of this quotient by $\bar{x} = \{y\in
\cA_{c}:\; \d_c(x,y)=0 \}$. These elements (that are also called {\it
$c$-static classes}, see \cite{ContrerasIturriaga}) provide a partition of 
$\cA_{c}$ into compact subsets,
that can be lifted to invariant subsets of ${\rm T}M$.
\end{Rem}

We say that an absolutely continuous curve $\g:\R \longrightarrow M$ is a 
$c$-{\it minimizer}, if for any interval $[a,b]$ and any other absolutely 
continuous 
curve $\g_1: [a,b] \longrightarrow M$ such that $\g(a)=\g_1(a)$ and $\g(b)=
\g_1(b)$, we have
$$ \int_a^b L_{\eta}(\g(t),\dot{\g}(t))dt \leq \int_a^b L_{\eta}(\g_1(t),
\dot{\g}_1(t))dt\,.$$
In other words, $\g:\R \longrightarrow M$ is a $c$-minimizer, if for any $a<b$ 
$$ \int_a^b L_{\eta}(\g(t),\dot{\g}(t))dt = h_{\eta,b-a}(\g(a),\g(b))\,.$$
This curves are also called {\it semi-static curves}.
A peculiar property of these curves is that their $\a$-limit set is contained 
in a static class and their 
$\omega$-limit set too   
(possibly in a different one). See \cite{Mather93, Fathibook, 
ContrerasIturriaga} for more details.\\

We define the {\it Ma\~n\'e set} (with cohomology class $c$) as:
%
\be 
\widetilde{\cN}_{c}= \bigcup \left\{(\g(t),\dot{\g}(t)):\; \text{$\g$ is a
$c$-minimizer and}\;t\in\R\right\}.\label{maneset}\ee

In particular, amongst these minimizers, one can consider particular ones that 
satisfy some extra conditions on the $\a$ and
$\omega$-limit sets. Namely, let $\g:\R \longrightarrow M$ be a $c$-minimizer 
and $\a,\a'$ be in the $\a$-limit set of $\g$ and let $\omega,\omega'$ be in 
the $\omega$-limit set of $\g$. From the discussion in \cite{Mather93}, it 
follows that $\d_c(\a,\a')= \d_c(\omega,\omega')=0$ and $\d_c(\a,\omega) = 
\d_c(\a',\omega')$. We say that $\g$ is a $c$-{\it regular minimizer} if 
a further condition is satisfied: $\d_c(\a,\omega)=0$.
An equivalent condition is: $\g$ is semi-static and for any interval $a<b$, 
we have that 
$ h_{\eta,b-a}(\g(a),\g(b)) = - h_{\eta,b-a}(\g(b),\g(a))\,.$
These curves are also called {\it static curves}.
Observe that the above condition can be interpreted saying that the $\a$ and 
$\omega$-limit sets of $\g$ are both contained in the same $c$-static class.\\

We define the {\it Aubry set} (with cohomology class $c$) as:
%
\be\widetilde{\cA}_{c}= \bigcup \left\{(\g(t),\dot{\g}(t)):\; \text{$\g$ is a
$c$-regular minimizer and}\;t\in\R\right\}.\label{2.6}\ee
%
Similarly to the Mather set, the Aubry set is also 
a graph over the basis, see \cite{Mather93,Fathibook}. 

\begin{Teo}\label{Mane} Let $\widetilde{\cA}_{c}$ be defined as in (\ref{2.6}).
By definition $\widetilde{\cA}_{c} \subseteq \widetilde{\cN}_{c}$. Both 
$\widetilde{\cA}_{c}$ and  $\widetilde{\cN}_{c}$ are compact subsets of $TM$ 
invariant under the Euler-Lagrange flow $\Phi^L_t$. Moreover,
$\pi \big(\widetilde{\cA}_{c}\big) 
= \cA_c$ and $\pi|\widetilde{\cA}_{c}$ is a bi-Lipschitz homeomorphism.
\end{Teo}
%

Note that the graph property does not hold for the Ma\~n\'e set. Moreover, 
there is a clear relation between the support of Mather's measures and these 
sets.

\begin{Teo}\label{emmeinenne}
For any $c\in \rH^1(M;\RRR)$, $\widetilde{\cM}_{c} \subseteq 
\widetilde{\cA}_{c}$. Moreover:
\beqa{numero}
\m \in \calM_c \; \Longleftrightarrow \; \m\in\calM(L) \;
{\rm and}\;{\rm supp}\,\m \subseteq \widetilde{\cA}_c
\; \Longleftrightarrow \; \m\in\calM(L) \;
{\rm and}\;{\rm supp}\,\m \subseteq \widetilde{\cN}_c\,.
\eeqa
\end{Teo}

The fact that an invariant measure whose support is contained in 
$\widetilde{\cN}_c$ (or $\widetilde{\cA}_c$) is $c$-minimizing might be seen,
for example, as a consequence of the finiteness of the Peierls' barrier. 
The proof of the first equivalence in 
(\ref{numero}) goes back to Ma\~n\'e \cite{ManeI}. As far as the second 
equivalence is concerned, it follows from the fact that the 
{\it non-wandering set} of 
$\Phi^L_t\big|\widetilde{\cN}_c$ is contained in $\widetilde{\cA}_c$, see 
Appendix \ref{app3}. \\

We can summarize what discussed so far in the following diagram:

$$\xymatrix{
\widetilde{\cM}_c \ar@{->}[d] \ar@{}[r]|\subseteq & \widetilde{\cA}_c
\ar@{->}[d] \ar@{}[r]|\subseteq  & 
\widetilde{\cN}_c \ar@{}[r]|\subseteq & \widetilde{\cE}_c \ar@{}[r]|
\subseteq & {\rm T}M \ar@{->}[d]^{\pi}
\\  
\cM_c \ar@{}[r]|\subseteq & \cA_c \ar@{}[r]|\subseteq  &&&  \ar@{}[l]|
\subseteq M\\}
$$

where $\widetilde{\cE}_c$ is the energy level corresponding to $\a(c)$.\\


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Another interesting approach to the study of these invariant sets
is provided by {\it weak KAM theory}.
This is mainly based on the concept of ``critical'' subsolutions and ``weak'' 
solutions of Hamilton-Jacobi equation and can be interpreted, from a symplectic
geometric point of view, as the study of particular Lagrangian graphs 
and their non-removable intersections (see \cite{Paternain-Siburg}). This 
latter approach 
is particularly interesting, since it relates the dynamics of the system to the 
geometry of the space and might potentially open the way to a ``symplectic'' 
definition of Aubry-Mather theory.

\begin{Def}\rm
A locally Lipschitz function $u: M \longrightarrow \R$ is a {\it
subsolution} of $H_{\eta}(x,d_xu)=k$, with $k\in \R$, if $H_{\eta}(x,d_xu)\leq 
k$ for almost every $x\in M$.
\end{Def}

This definition makes sense because, by Rademacher's theorem, $d_xu$ exists 
almost
everywhere.
It is possible to show that there exists $c[\eta]\in \R$, such that
$H_{\eta}(x,d_xu)=k$ admits no subsolutions for $k<c[\eta]$ and has
subsolutions for $k\geq c[\eta]$, see \cite{LPV, Fathibook}. The constant 
$c[\eta]$ is called {\it
Ma\~n\'e's critical value} and coincides with $\a(c)$, where $c=[\eta]$ (see
\cite{ContrerasIturriaga,Fathibook}).

\begin{Def}\rm A function
$u: M \longrightarrow \R$ is an $\eta$-{\it critical subsolution}, if
$H_{\eta}(x,d_xu)\leq \a(c)$ for almost every $x\in M$.\label{etacriticalsub}
\end{Def}

In particular, {Ma\~n\'e's critical energy level} 
$\cE^*_c:=\{(x,p)\in {\rm T}^*M:\; 
H(x,p)=\a(c)\}$ is the only energy level
in which there exist classical or weak solutions of Hamilton-Jacobi 
equation $H_{\eta}(x,du)=\a(c)$.
Denote by $\cS_{\eta}$ the set of critical subsolutions.
This set $\cS_{\eta}$ is non-empty. In fact, it is showed in \cite{Fathibook}
that:

\begin{Prop}
If $u: M \longrightarrow \R$ is an $\eta$-{\it critical subsolution}, then
for every $x,\,y \in M$
$$u(y) - u(x) \leq h_{\eta}(x,y)\,. $$
Moreover, for any $x\in M$, the function $h_{\eta,x} (\cdot) := h_{\eta}(x,\,
\cdot)$ is a
$\eta$-critical subsolution.
\end{Prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There is a more geometric way to define the Aubry set in the cotangent space, 
using directly these critical subsolutions; it is possible
to show that this set is given by the intersection of the 
graph of the differentials of this critical subsolutions (where they are 
defined) plus the $1$-form $\eta$. In particular, it turns out that each 
critical subsolution is 
differentiable on the (projected) Aubry set and their differentials coincide 
on this set. The following can be proven, see \cite{Fathibook, FathiSiconolfi}.

\begin{Teo} Let 
%
\be {\cA}^*_c := \bigcap_{u\in \cS_{\eta}} \left\{(x,\eta_x + d_xu):\; 
u\;\mbox{is 
differentiable at}\; x\right\} \subset {\rm T}^*M\,.\label{2.7}\ee
%
Then ${\cA}^*_c ={\cal L}(\widetilde{\cA}_c)$, where 
%$\widetilde{\cA}_c$ is the Aubry set defined in (\ref{2.5}) and 
${\cal L}$ is the Legendre transform defined in (\ref{2.1}).
\end{Teo}

Using critical subsolutions, one can also obtain a nice representation of 
$h_{\eta}$;
for $x\in \cA_{c}$ and $y\in M$:
$$ h_{\eta}(x,y) = \sup_{u \in \cS_{\eta}} \left\{u(y)-u(x)\right\},$$
where this supremum is actually attained, for fixed $x,y\in \cA_c$. Moreover, 
for $x,y\in \cA_{c}$:
$$\d_c(x,y) = \sup_{u,v\in \cS_{\eta}} \left\{(u-v)(y)-(u-v)(x)\right\}\,.$$


These results can be improved (see \cite{FathiSiconolfi}):

{\Teo\label{astar} 
For any $\eta$-critical subsolution $u: M \longrightarrow \R$  and 
for each $\e>0$, there exists a $C^1$ function $\tu: M \longrightarrow \R$
such that:
\begin{itemize}
\item[{\rm i)}]   $\tu(x)=u(x)$ and $H_{\eta}(x,d_x\tu)=\a(c)$ on $\cA_{c}$;
\item[{\rm ii)}]  $|\tu(x)-u(x)|<\e$ and $H_{\eta}(x,d_x\tu)<\a(c)$ on $M
\setminus \cA_{c}$.
\end{itemize}
}

This implies that $C^1$ $\eta$-critical subsolutions are
dense in $\cS_{\eta}$ with the $C^0$-topology.
This result has been recently extended by Patrick Bernard (see \cite{Bernard}),
showing that every $\eta$-critical subsolution coincides, on the Aubry set, 
with a 
$C^{1,1}$ $\eta$-critical subsolution (that is the best regularity that one 
can generally expect). 
Denoting the set of $C^1$  $\eta$-critical subsolutions by $\cS^1_{\eta}$, 
one can rewrite:
%
\be{{\cA}}^*_c = \bigcap_{u\in \cS^1_{\eta}} {\rm Graph}(\eta + du);
\label{2.8}\ee
%
in particular, there exists a $C^{1}$ $\eta$-critical subsolution $\tilde{u}$
such that:
%
\be{{\cA}}^*_c = {\rm Graph}(\eta + d\tilde{u}) \cap \cE^*_c.
\label{2.9}\ee
%
In the same way, one obtains:
\beqano
h_{\eta}(x,y) &=& \max_{u \in \cS^{1}_{\eta}} \left\{u(y)-u(x)\right\}\\
\d_c(x,y) &=& \max_{u,v\in \cS^{1}_{\eta}} \{(u-v)(y)-(u-v)(x)\}\,,
\eeqano
for any $x,\,y \in \cA_{c}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SECTION 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Minimizing properties of Lagrangian graphs and uniqueness results.}
\label{sec3}

In this section we shall prove the main results announced in Section 
\ref{sec1}. Let us start by proving some action minimizing properties of 
probability measures supported on Lagrangian graphs. 

Given a Lagrangian graph $\Lambda$ with Liouville class $c$, we shall say that
$\Lambda$ is $c$-{\it subcritical}, or simply {\it subcritical}, if 
$\Lambda \subset \{(x,p)
\in {\rm T}^*M:\; H(x,p)\leq \a(c)\}$, where $\a :{\rm H}^1(M;\R) 
\longrightarrow \R $ is Mather's $\a$-function.
Given a subcritical Lagrangian graph $\Lambda$ with Liouville class $c$, 
we shall call $\Lambda_{crit}=\{(x,p)\in \Lambda:\; H(x,p) = \a(c)\}$ its 
{\it critical} part. The key result we need to prove is the following 
characterization of minimizing measures.

\begin{Lem} \label{lemmadellacontesa}
Let $\m$ be an invariant probability measure on ${\rm T}M$ and $\m^*=\cL_*\m$ 
its push-forward to ${\rm T}^*M$, via the Legendre transform $\cL$. Then, 
$\m$ is a Mather's measure if and only if ${\rm supp}\,\m^*$ is contained 
in the critical part of a subcritical Lagrangian graph. In particular, 
any invariant probability measure $\m^*$ on ${\rm T}^*M$, whose support
is contained in an invariant Lagrangian graph with Liouville class $c$,
is the image, via the Legendre transform, of a $c$-action minimizing measure.
\end{Lem}

\begin{Proof}

({\it i})
If $\m$ is a Mather's measure with cohomology class $c$, then the support of 
$\m^*$ is contained in $\cL(\widetilde\cM_c)\subseteq \cA^*_c$. By (\ref{2.8}),
$\cA^*_c$ is an intersection of subcritical Lagrangian graphs,
so ${\rm supp}\,\m^*$ is contained in at least one subcritical Lagrangian 
graph 
$\Lambda$.
In particular ${\rm supp}\,\m^*$ is contained in the critical part of 
$\Lambda$, 
simply because $\cL(\widetilde\cM_c)$ is contained in the energy level 
${\cal E}^*_c = \{(x,p)\in {\rm T}^*M:\; H(x,p) = \a(c)\}$, see Theorem 
\ref{Theograph}.

({\it ii}) Let us fix $\eta$ to be a closed $1$-form with $[\eta]=c$.
Since we are assuming that $\Lambda$ is a $c$-subcritical Lagrangian graph,
we can
write $\Lambda=\{(x,\eta(x)+du(x))\,,\, x\in M\}$, where $u:M\to\RRR$ is $C^1$.
By Theorem \ref{emmeinenne}, in order to show that $\m$ is a $c$-action 
minimizing measure, it is enough to show that ${\rm supp}\,\m \subseteq 
\widetilde{\NN}_c$, \ie that any orbit in ${\rm supp}\,\m$ is a 
$c$-minimizer, see (\ref{maneset}). To this purpose, let us consider
$(x,v)\in{\rm supp}\,\m$ and let $\g(t)\=\p(\Phi_t(x,v))$, where $\Phi_t$
is the Euler-Lagrange flow and $\p$ the canonical projection on $M$. Given any 
interval $[a,b]\subset\RRR$, let us consider the difference 
$u(\g(b))-u(\g(a))$ and rewrite it as:
%
\be u(\g(b))-u(\g(a))=\int_a^b d_{\g(s)} u(\g(s)) \dot\g(s)\, ds=
\int_a^b \big[L_\eta(\g(s),\dot\g(s))+H_\eta(\g(s),d_{\g(s)} u) \big] ds\,,
\label{4.12}\ee
%
where the second equality follows from the definition of the Hamiltonian as 
the 
Legendre-Fenchel transform of the Lagrangian and the fact that $\g(s)$ is an 
orbit
of the Euler-Lagrange flow. Note that along the orbit 
$H_\eta(\g(s),d_{\g(s)} u)=\a(c)$, because ${\rm supp}\,\m$ is invariant and
${\rm supp}\,\m^*$ is in the critical part of $\Lambda$. Then
%
\be \int_a^b L_\eta(\g(s),\dot\g(s)) ds = u(\g(b))-u(\g(a)) -\a(c) (b-a)\;.\ee
%
On the other hand, any other curve $\g_1:[a,b]\to M$ such that $\g_1(a)=\g(a)$
and $\g_1(b)=\g(b)$ satisfies:
%
\be u(\g(b))-u(\g(a))=\int_a^b d_{\g_1(s)} u(\g_1(s)) \dot\g(s)\, ds\le
\int_a^b \big[L_\eta(\g_1(s),\dot\g_1(s))+H_\eta(\g_1(s),d_{\g_1(s)} u) \big] 
ds\label{4.14}\ee
%
where the second inequality follows again by the duality between Hamiltonian 
and Lagrangian. Note that now $H_\eta(\g_1(s),d_{\g_1(s)} u)\le \a(c)$, because
$\Lambda=\{(x,\eta(x)+du(x))\}$ is subcritical. Then
%
\be \int_a^b L_\eta(\g_1(s),\dot\g_1(s)) ds \ge u(\g(b))-u(\g(a)) -\a(c) (b-a)
\ee
%
and this proves that $\g$ is a $c$-minimizer. 

Let us finally observe that the Hamilton function on any invariant Lagrangian 
graph $\Lambda=\{(x,\eta+du)\}$ is a constant: $H(x,\eta+du)=k$ 
(see Appendix \ref{app2}). Then $u$ is a 
classical solution of the Hamilton-Jacobi equation with cohomology class $c$. 
As discussed in Section II, there is only one possible value of $k$ for which 
such solutions can exist, namely $k=\a(c)$, and this shows that $\Lambda$
coincides with its critical part. By the result proved in item ({\it ii}), 
if $\m^*$ is supported on $\Lambda$, then $\m$ is a $c$-action minimizing 
measure and this proves the last claim in the statement of the Lemma.
\end{Proof}

We can now prove the well-known uniqueness result  for 
Lagrangian graphs supporting invariant measures of full support, in a fixed 
cohomology class, stated in the Introduction.

\begin{Teo}\label{uniqcom}
\it If $\Lambda \subset {\rm T}^*M$  is a Lagrangian graph on which the 
Hamiltonian dynamics admits an 
invariant measure $\m^*$ with full support, then 
$\Lambda = \cL \big(\widetilde{\cM}_c\big)= \cA_c^*$, 
where $c$ is the cohomology class of $\Lambda$.\
Therefore, if $\Lambda_1$ and $\Lambda_2$ are two Lagrangian graphs as 
above, with the same cohomology class, then $\Lambda_1=\Lambda_2$.
\end{Teo}

\begin{Proof} 
By Lemma \ref{lemmadellacontesa},  the measure $\mu=\cL^{-1}_*\m^*$ is 
$c$-minimizing. This means 
that $\LL^{-1}(\Lambda)={\rm supp}\,\m\subseteq \widetilde\cM_c\subseteq
\widetilde\cA_c$, where the last inclusion follows from Theorem 
\ref{emmeinenne}. Note however that, by Theorems \ref{Theograph} and 
\ref{Mane}, $\widetilde\cM_c$ and $\widetilde\cA_c$
are graphs over their bases and, since ${\rm supp}\,\m$ is a graph over the 
whole $M$, it follows that
%
\be \LL^{-1}(\Lambda)={\rm supp}\,\m = \widetilde\cM_c = \widetilde\cA_c\,.\ee
%
\end{Proof}

One can deduce something more from the above proof.


\begin{Teo}\label{diffbeta}
If $\Lambda$ and $\mu$ are as in Theorem \ref{uniqcom} and $\r$ is 
the rotation vector of 
$\m=\cL^{-1}\m^*$, then $\Lambda=\cL \big(\widetilde{\cM}^\r\big)$. 
Therefore,  if $\Lambda_1$ and $\Lambda_2$ are two Lagrangian graphs 
supporting measures of full support and the same rotation vector $\rho$, 
then $\Lambda_1 = \Lambda_2$. Moreover, Mather's 
$\beta$-function is differentiable at $\r$ with $\partial \beta(\rho) = c$,
where $c$ is the cohomology class of $\L$.\\
\end{Teo}

\begin{Proof}
The first claim follows from the fact that $\widetilde{\cM}^\rho$ is a graph 
over $M$ and that by definition 
$\widetilde{\cM}^\rho \supseteq {\rm supp}\,\m = \LL^{-1}(\Lambda).$
As far as the differentiability of $\beta$ at $\rho$ is concerned,  suppose 
that $c'\in {\rm H}^1(M;\R)$ is a subderivative of $\beta$ at $\rho$.
Therefore, $\b(\r) = \langle c', \rho \rangle - \a(c')$ and this implies that
$\calM^{\rho} \subseteq \calM_{c'}$; in fact, for any $\m \in \calM^{\rho}$:
$$
\int_{{\rm T}M} (L-\hat{\eta}')\, d\m = \int_{{\rm T}M} L\,d\m - 
\int_{{\rm T}M} \hat{\eta}'\,d\m = 
\b(\rho) - \langle c',\rho\rangle = -\a(c')\,, 
$$
where $\eta'$ is a closed $1$-form of cohomology $c'$.
As a result, $\widetilde{\cM}^{\rho} = \cL^{-1}\left(\Lambda\right) \subseteq 
\widetilde{\cM}_{c'}$.
The graph property of $\widetilde{\cM}_{c'}$ and of $\widetilde{\cA}_{c'}$
implies that  $\widetilde{\cA}_{c'} = \widetilde{\cM}_{c'} = \cL^{-1}
\left( \Lambda \right)$ 
and $\cA^*_{c'} = \Lambda $. As a consequence, $c'=c$. In fact,
by Theorem \ref{astar} and Eq.(\ref{2.8}), 
there exists a $C^1$ function $v:M\longrightarrow \R$, such that 
$\Lambda = \{ (x, \eta' + d v):\; x \in M \}$, where $\eta'$ is a closed 
$1$-form with $[\eta']=c'$. This, by definition,
means that $c'$ is the cohomology class of $\Lambda$ and therefore  
$c'=c$.
\end{Proof}

We are now in the position of proving the main uniqueness result in a 
homology class,  stated in the Introduction.

\begin{Teo}\label{uniqschw}
Let $\Lambda$ be a Schwartzman strictly ergodic invariant Lagrangian graph with
homology class $\rho$. The following properties are satisfied:
\begin{itemize}
\item[{\rm (i)}] if $\Lambda \cap \cA^*_c \neq \emptyset$, then 
$\Lambda=\cA^*_c$ and $c=c_{\Lambda}$, where $c_{\Lambda}$ is the cohomology 
class of $\Lambda$.
% In particular, the set $\Lambda$ does not intersect any other 
%invariant Lagrangian graph.
\item[{\rm (ii)}] the Mather function $\alpha $ is differentiable at 
$c_{\Lambda}$ 
and $\partial \alpha (c_{\Lambda}) = \rho$.
\end{itemize}
Therefore,
\begin{itemize}
\item[{\rm (iii)}] any invariant Lagrangian graph that carries a measure with 
rotation vector $\rho$ is equal to the graph $\Lambda$;
\item[{\rm (iv)}] any invariant Lagrangian graph is either disjoint from 
$\Lambda$ or equal to $\Lambda$.
\end{itemize}
\end{Teo}

We shall need the following Lemma.

\begin{Lem}\label{lemmetto} Let $\rho,c$ be respectively an arbitrary homology 
class in $\rH_1(M;\R)$ and an arbitrary cohomology class $ \rH^1(M;\R)$. We have
$$
{\rm (1)}\; \widetilde{\cM}^{\rho} \cap  \widetilde{\cA_c} \neq \emptyset 
\quad \Longleftrightarrow \quad
{\rm (2)}\; \widetilde{\cM}^{\rho} \subseteq \widetilde{\cA_c} \quad 
\Longleftrightarrow \quad {\rm (3)}\;\rho \in \partial \a(c)\,.
$$
\end{Lem}


{\bf Proof of Lemma \ref{lemmetto}.} The implication
${\rm (2) }\Longrightarrow {\rm (1)}$ is trivial. Let us prove that
 ${\rm  (1)} \Longrightarrow {\rm (3)}$.
If $\widetilde{\cM}^{\rho} \cap  \widetilde{\cA_c}\neq 0$, then there exists a 
$c$-minimizing invariant measure $\m$ with rotation vector $\rho$. 
Let $\eta$ be a closed $1$-form with $[\eta]=c$; 
from the definition of $\a$ and $\b$:
\beqano
-\a(c) = \int_{{\rm T}M} (L-\hat{\eta})\,d\m = \int_{{\rm T}M} L\,d\m - 
\langle c,\rho\rangle = \b(\rho) - \langle c,\rho\rangle\,;
\eeqano
since $\b$ and $\a$ are convex conjugated,  then $\rho$ is a subderivative of 
$\a$ at $c$.\\
Finally, in order to show $(3) \Longrightarrow (2)$, let us prove that any 
action minimizing measure with rotation vector $\rho$ is $c$-minimizing. 
In fact, if $\rho \in \partial \a(c)$ then $\a(c)= \langle c,\rho\rangle - 
\b(\rho)$; therefore for any $\m \in \calM^{\rho}$ and $\eta$ as above:
$$
-\a(c) = \b(\rho) - \langle c,\rho\rangle =  \int_{{\rm T}M}(L-\hat{\eta})\, 
d\m.  
$$
This proves that $\m 
\in \calM_c$ and concludes the proof.
\qed\\

{\bf Proof of Theorem \ref{uniqschw}} 
({\it i}) From Theorem \ref{uniqcom}, it follows that $\Lambda=\cA^*_{c_{\Lambda}}$.
 Let us show that it does not intersect any other Aubry set.
Suppose by contradiction that $\Lambda$ intersects another Aubry set  
$\cA_{c}^*$.
By Theorem \ref{diffbeta}, $\Lambda = \cL^{-1}\left(\widetilde{\cM}^{\rho}
\right)$, then 
$\widetilde{\cM}^{\rho} \cap \widetilde{\cA}_c \neq \emptyset$ and, because 
of the previous lemma and the graph property of $\widetilde{\cA}_c$, we can 
conclude that $\cA^*_c = \Lambda$. The same argument used in the proof of 
Theorem \ref{diffbeta} allows us to conclude that $c=c_{\Lambda}$.\\
({\it ii}) Suppose that $ h \in \partial \a(c_{\Lambda}) $. The previous lemma 
implies that $\widetilde{\cM}^{h} \subseteq \Lambda $; the 
Schwartzman unique ergodicity property of $\Lambda$ implies $h=\rho$. Therefore
$ \alpha $ is differentiable at $c_{\Lambda}$ and $\partial \alpha 
(c_{\Lambda}) = \rho$.

To prove ({\it iv}), let $\Lambda_1$ be an invariant Lagrangian graph, and call
$c_1$ its cohomology class.
If the compact invariant set $\Lambda\cap\Lambda_1$ is not empty, 
then we can find a probability measure $\mu^*$ invariant under the flow
and whose support is contained in this intersection. Since $\mu^*$ is contained
in the Lagrangian graph $\Lambda_1$, by Lemma \ref{lemmadellacontesa}, 
it is $c_1$-minimizing. Hence, the support of $\mu^*$ is contained in 
$\cA^*_{c_1}$.
This shows that the intersection $\Lambda \cap \cA^*_{c_1}$ contains the 
support 
of $\mu^*$ and is therefore not empty. 
By ({\it i}), $\Lambda=\cA^*_{c_1}$. Moreover, note that $\cA^*_{c_1}\subseteq
\L_1$, because, on the one hand,
$\L_1={\rm Graph}(\eta_1+du_1)$, with $[\eta_1]=c_1$ and $u_1$ a classical 
solution to the Hamilton-Jacobi equation (see proof of Lemma 
\ref{lemmadellacontesa}), and, on the other hand, 
$\cA^*_{c_1}=\cap_{u\in\cS^1_\eta} {\rm Graph}
(\eta_1+du_1)$, see (\ref{2.8}). Therefore, $\Lambda=
\Lambda_1$, since they are both graphs over the base.

To prove ({\it iii}), consider an invariant Lagrangian graph $\Lambda_1$, 
with cohomology class $c_1$, which carries an invariant measure 
$\mu^*$ whose rotation vector is $\rho$. By Lemma \ref{lemmadellacontesa},
the measure $\mu^*$  is $c_1$-minimizing. Therefore, we have 
$\widetilde{\cM}^\rho\cap \widetilde\cA_{c_1}\neq \emptyset$. 
By Lemma \ref{lemmetto}, it 
follows that  ${\cal L}^{-1}(\Lambda)=\widetilde{\cM}^\rho\subseteq 
\widetilde\cA_{c_1}\subseteq {\cal L}^{-1}(\Lambda_1)$. Again, this forces the 
equality $\Lambda=\Lambda_1$ by the graph property.\qed
\\

Finally, observe that Lemma \ref{CaracMinim}, 
Theorem \ref{emmeinenne} and Lemma 
\ref{lemmetto} imply the following property.

\begin{Cor} The Mather function
$\a$ is differentiable at $c$ if and only if the restriction of the 
Euler-Lagrange flow to $\widetilde{\cA}_c$ 
is Schwartzman uniquely ergodic, \ie if and only if 
all invariant measures supported on 
$\widetilde{\cA}_c$ have the same rotation vector (see Appendix \ref{app1}
for the definition and a discussion of Schwartzman ergodic flows).
\end{Cor}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% SECTION 4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Global uniqueness of KAM tori}\label{sec4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we motivate more precisely the problem of uniqueness of KAM 
tori and we prove Corollary \ref{corollary1}.
We also show how to generalize Corollary \ref{corollary1} to cover the case of
invariant tori belonging to the closure of the set of KAM tori. 

KAM theory concerns the study of existence of KAM tori (see Definition 
\ref{kamtorus}) in quasi-integrable Hamiltonian systems of the form
$H(x,p)=H_0(p)+\e f(x,p)$, where: $(x,p)$ are local coordinates on $\TTT^n
\times\RRR^n$, $\e$ is a ``small''  parameter and $f(x,p)$ a smooth function.
If $\e=0$ the system is integrable, in the sense that the dynamics
can be explictly solved: in particular each torus $\TTT^n\times\{p_0\}$
is invariant and the motion on it corresponds to a rotation 
with frequency $\r(p_0)=\frac{\partial H_0}{\partial p}(p_0)$. The question 
addressed by KAM theory is whether this foliation of phase space into
invariant tori, on which the motion is (quasi-)periodic, persists even 
if $\e\neq 0$. 
In 1954 Kolmogorov stated (and Arnol'd and Moser proved it later in different 
contexts) that, in spite of the generic disappearence of the invariant 
submanifolds filled by periodic orbits, pointed out by Poincar\'e, 
for small $\e$ it is always 
possible to find KAM tori corresponding to ``{\it strongly non-resonant}'',
\ie {\it Diophantine}, rotation vectors. Let us recall here the 
definition  and some properties of Diophantine vectors: given $C,\t>0$, we say 
that $\r\in\RRR^n$ is 
a $(C,\t)$-Diophantine vector 
if and only if $C|\media{\r,\n}|\ge |\n|^{-\t}$, $\forall \n\in\ZZZ^n\setminus
\{0\}$. The set of 
$(C,\t)$-Diophantine vectors will be denoted by $\DD(C,\t)$.
Note that, if $\t<n-1$, $\DD(C,\t)=\emptyset$, while for $\t>n-1$, the 
Diophantine vectors have full measure in 
$\RRR^n$, that is $\lim_{R\to\io}\m_0(\cup_{C>0}
\DD(C,\t)\cap B_R)/\m_0(B_R)=1$, where $\m_0$ is the Lesbegue measure and 
$B_R$ is the ball of radius $R$ centered at $0$; for $\t=n-1$, $\cup_{C>0}
\DD(C,\t)$ has measure zero but Haussdorf dimension $n$.
The celebrated KAM Theorem (in one of its several versions) not only shows the 
existence of such tori, but also provides an explicit method to construct them.
\\

{\bf Theorem} (Kolmogorov--Arnol'd--Moser) \cite{Salamon}. 
{\it Let $n\ge 2$, $\t>n-1$, $C>0$, $\ell>2\t+2$, $M>0$
and $r>0$ be given. Let $B_r\in\RRR^n$ be the open ball of radius $r$ 
centered at the origin. Let $H\in C^\ell(\TTT^n\times B_r)$ 
be of the form 
%
\be H(x,p)=H_0(p)+\e f(x,p)\label{1.2}\ee
%
with $|H_0|_{C^\ell}\le M$, $|f|_{C^\ell}\le M$, $\left|\frac{\partial^2 H_0}
{\partial p^2}\right|\ge M^{-1}$ and
$\r= \frac{\partial H_0}{\partial p}(0)\in \DD(C,\t)$. Then, for any $s<\ell-
2\t-1$, 
there exists $\e_0>0$ such that for
any $\e\le\e_0$ the Hamiltonian {\rm(\ref{1.2})} admits a $C^{s,s+\t}$ 
KAM torus with rotation vector $\r$, \ie a $C^{s+\t}$ invariant torus such that
the Hamiltonian flow on it is $C^s$-conjugated with a rotation with 
frequency $\r$.} \\

{\bf Remarks.} \\
1) If $H\in C^\io$ then the KAM torus mentioned in the theorem above 
is $C^\io$. If $H$ is real analytic then the KAM torus is real analytic.\\
2) As already mentioned, the proof of this theorem is constructive
and it actually contains much more informations than those summarized in the 
above
statement. For instance, in the analytic case, the proof consists of an
iterative method allowing one to construct order by order the series defining
the conjugation function (and to prove convergence of the formal series). 
In the differentiable case the proof is based on the idea of iteratively 
approximating differentiable functions by analytic ones and then using 
the inductive approximation scheme of the analytic case. In the differentiable 
case the proof provides an explicit construction of a KAM torus 
(however the construction {\it a priori} depends on a number of arbitrary 
choices one
has to make along the proof -- \eg the choice of cutoffs one needs to introduce
in the iterative approximation scheme).\\
3) The invariant torus constructed in the proof of the KAM Theorem is locally
unique, in the sense that for any prescribed (and admissible) $s$ 
there is at most one $C^{s,s+\t}$ KAM torus with rotation 
vector $\r$ within a $C^s$-distance $\d(n,s,C,\t)$ to the one constructed 
in the proof of the KAM Theorem, see \cite{BroerTakens, 
salamon-zehnder, Salamon}. Note that the $C^s$-distance 
$\d$ within which one can prove uniqueness of the KAM torus in a prescribed
regularity class depends both on the irrationality properties of $\r$ and
on the regularity class $s$ itself. It is then {\it a priori} possible that 
even for small $\e$ there exist different KAM tori {\it within a prescribed 
$C^1$-distance} from the one constructed in the proof of the Theorem, possibly 
less regular than that torus. Quite surprisingly, even in the analytic case,
we are not aware of any proof of ``global'' uniqueness of the 
invariant analytic KAM torus with rotation vector $\r$ (of course in the 
analytic case the analytic torus one manages to construct is unique within
the class of analytic tori -- however nothing {\it a priori} guarantees that 
less regular invariant tori with the same rotation vector exist).
\\

The questions arisen in remark (3) is our main motivation for the study of the 
problem of global uniqueness of KAM tori. Our result, in the form stated 
in the Corollary in Section \ref{sec1}, settles the question and shows that,
at least in the case of Tonelli Hamiltonians, it is not possible
to have two different KAM tori with the same rotation vector. Note that
the assumption of strict convexity of the Hamiltonian is necessary
to exclude trivial sources of non-uniqueness: for instance, in 
the context of quasi-integrable Hamiltonians, 
global uniqueness could be lost simply because the unperturbed Hamiltonian 
induces a  map $p\to \partial_p H_0(p)$ from actions to frequencies that is not
one to one. Let us also remark that, 
apparently, the Hamiltonian considered in KAM Theorem is not a Tonelli
Hamiltonian, since the latter, by definition, is defined globally on the whole 
$\TTT^n\times\RRR^n$. However any $C^\ell$ strictly convex Hamiltonian defined
on $\TTT^n\times B_r$ for some $r>0$ can be extended to a global
$C^\ell$ Tonelli Hamiltonian. Then in the statement of the KAM Theorem above 
it is actually enough to assume $H$ to be a $C^\ell$ Tonelli Hamiltonian, 
locally satisfying the (in)equalities listed after (\ref{1.2}).\\

Given the proof of our main results in Section \ref{sec3}, 
the proof of Corollary \ref{corollary1} is very simple.\\

{\bf Proof of Corollary \ref{corollary1}.}
Since the Lagrangian KAM torus $\TT$ admits an invariant measure $\m^*$ of full
support, which is the image via the conjugation $\f$ of the uniform 
measure on $\TTT^n$, then the claims follow from Theorem \ref{diffbeta}. 
Note that for rationally independent rotation vectors,
a classical remark, generally attributed to Herman \cite{Herman} 
(see Appendix \ref{app2} for a proof), implies that $\TT$ is automatically 
Lagrangian.\qed\\

An interesting generalization of the result of Corollary \ref{corollary1}
concerns the invariant tori belonging to the $C^0$-closure $\overline\Upsilon$ 
of the set $\Upsilon$ of all Lagrangian KAM tori. 
Note that, for
quasi-integrable systems, $\Upsilon$ is not empty.
The set $\Upsilon$ can be seen as a subset of ${\rm Lip}(\TTT^n,\RRR^n)$.
This follows from Theorem \ref{diffbeta},
and from Mather's graph theorem, see 
Theorems \ref{Theograph}, \ref{matherrotation}, and the 
results in \cite{Mather91}. Moreover, any family of invariant Lagrangians 
graphs on which the function $\alpha$ (or $H$) is bounded gives rise to a 
family of functions in ${\rm Lip}(\TTT^n,\RRR^n)$ with uniformly bounded 
Lipschitz constant. This is because, given $\L$ in such a family and 
denoting by $(\eta+du)$ its graph, for any pair 
of points $x,y\in\TTT^n$ and any smooth curve $\g(t)$ on $\L$ 
connecting $x$ to $y$ with unit speed, we have that $u(x)-u(y)\le
\int_0^{|x-y|}L_\eta(\g(t),\dot\g(t))+\a(c)|x-y|$, where $c=[\eta]$, 
see (\ref{4.14}).
By Ascoli-Arzel\`a theorem, it follows  that $\overline\Upsilon$
is also a subset of ${\rm Lip}(\TTT^n,\RRR^n)$, consisting of  functions whose 
graphs are invariant $C^0$-Lagrangian tori.
Herman \cite{HermanTori}
showed that, for a generic Hamiltonian $H$ close enough to an 
integrable Hamiltonian $H_0$, the dynamics on the generic  tori in  $\overline
\Upsilon$ is not conjugated to a 
rotation. These  ``new'' tori therefore represent the majority, in the sense 
of topology, and hence most invariant tori cannot be obtained by the KAM 
algorithm. 
More precisely, Herman showed that in $\overline\Upsilon$ there exists a dense 
$G_\d$ set (\ie
a dense countable intersection of open sets) of invariant Lagrangian graphs  
on which the dynamics is strictly ergodic and weakly mixing, and for which the 
rotation vector, in the sense of Section \ref{sec2}, is not Diophantine. These 
invariant graphs are therefore not obtained by the KAM theorem, however our 
uniqueness result do still apply to these graphs since strict ergodicity 
implies Schwartzman strict ergodicity.

More generally, given any Tonelli Lagrangian on $\T^n$, we consider the set 
$\tilde \Upsilon$ of invariant Lagrangian graphs on which the dynamics 
of the flow is topologically conjugated to an {\it ergodic} linear flow on 
$\T^n$ 
(of course, far from the canonical integrable Lagrangian the set $\tilde 
\Upsilon$ may 
be empty). The dynamics on anyone of the invariant graph in $\tilde \Upsilon$ 
is strictly ergodic. Since the set of strictly ergodic flows on a compact set 
is a 
$G_\delta$ in the $C^0$ topology, see for example \cite[Corollaire 4.5]
{FathiHerman}, 
it follows that there exists a dense $G_\delta$ subset ${\cal G}$ of the $C^0$ 
closure   
of $\tilde \Upsilon$ in  ${\rm Lip}(\TTT^n,\RRR^n)$, such that the dynamics on 
any $\Lambda\in {\cal G}$ is strictly ergodic. 
Therefore we have the following proposition.

\begin{Prop}
There exists a dense $G_\d$ set $\cal G$ in the $C^0$ closure of $\tilde
\Upsilon$ consisting of 
strictly ergodic invariant Lagrangian graphs.  Any $\Lambda\in {\cal G}$ 
satisfies the following properties:
\begin{itemize}
\item[{\rm (i)}] the invariant graph $\Lambda$ has a well-defined rotation 
vector $\rho(\Lambda)$.
\item[{\rm (ii)}]  Any invariant Lagrangian graph that intersects $\Lambda$ 
coincides with $\Lambda$.
\item[{\rm (iii)}] Any Lagrangian invariant graph that carries an invariant 
measure whose rotation is $\rho(\Lambda)$ coincides with $\Lambda$. 
\end{itemize}
\end{Prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\appendixpage
\section{Schwartzman unique and strict ergodicity}\label{app1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Section \ref{sec2} we have introduced the concept of rotation vector of a 
measure. This is closely related to the notion of Schwartzman asymptotic cycle 
of a flow,
introduced by Sol Schwartzman in \cite{Schwartzman}, as a first attempt to 
develop 
an algebraic topological approach to the study of dynamics. In particular,
we would like to  provide some examples and investigate some properties of what
we call {\it Schwartzman uniquely ergodic flows}.

One can give a different description of the Schwartzman asymptotic cycle of a 
flow.
This is also known as the flux homomorphism in volume preserving and symplectic
geometry, see \cite{Banyaga}[Chapter 3]. We will use the description given in 
\cite{Fathi80}[pages 67-70]. This definition has the technical advantadge of 
not relying on the Krylov-Bogolioubov theory of generic orbits in a dynamical 
system,  although a more geometrical definition showing that ``averaged'' 
pieces of long orbits converge almost everywhere in the first homology group 
for any invariant measure is certainly more heuristic and intuitive.

Let us start with some standard facts. As usual we set $\T=\R/\Z$. The space 
$\T$ is a topological group for the addition. An important feature of $\T$ is 
that the canonical projection $\pi:\R\to\T$ is a covering map. Therefore given 
any continuous path
$\gamma:[a,b]\to\T$, with $a\leq b$, we can find a continuous lift $\bar\gamma:
[a,b]\to\T$ such that
$\gamma=\pi\bar\gamma$. Any two such lifts differ by an integer. It follows
that  the quantity
$\bar\gamma(b)-\bar\gamma(a)$ does not depend on the lift. We will set
$${\cal V}(\gamma)=\bar\gamma(b)-\bar\gamma(a)\in\R.$$
This quantity remains constant on the homotopy class, with fixed end points, 
of the path
$\gamma$. Morever, if $\gamma$ is a closed path, \ie we have $\gamma(a)=
\gamma(b)$
then ${\cal V}(\gamma)\in\Z$, and, if such a closed path is homotopic to $0$ 
(with fixed endpoint) then ${\cal V}(\gamma)=0$. It is also clear that for 
$c\in[a,b]$, we have
$${\cal V}(\gamma|[a,b])= {\cal V}(\gamma|[a,c])+{\cal V}(\gamma|[c,b]).$$
Note also that two continuous paths $\gamma_1,\gamma_2:[a,b]\to \T$ can be 
added by the formula
$$(\gamma_1+\gamma_2)(t)= \gamma_1(t)+\gamma_2(t).$$
For this addition we have
$${\cal V}(\gamma_1+\gamma_2)={\cal V}(\gamma_1)+{\cal V}(\gamma_2).$$
Another important property of the map ${\cal V}$ is its continuity
on the functional space $C^0([a,b],\T)$, endowed with the topology of uniform 
convergence.
If we call $\theta$ the closed 1-form on $\T$ whose lift to $\R$ is the usual 
differential form $dt$ on $\R$, where $dt$ is the differential of the identity 
map $\R\to\R,t\mapsto t$.
It is well-known that when $\gamma:[a,b]\to \T$ is $C^1$, we have
$${\cal V}(\gamma)=\int_\gamma\,\theta=
\int_a^b\theta_{\gamma(t)}(\dot\gamma(t))\,dt.$$

If $X$ is a topological space, and  $F:X\times [a,b]\to \T$ is a given map, 
we will define 
${\cal V}(F):X\to \R$ by 
$$\forall x\in X,\ {\cal V}(F)(x)={\cal V}(F_x),$$
where $F_x:[a,b]\to\T$ is defined by $F_x(t)=F(x,t)$.
The continuity of ${\cal V}$ on $C^0([a,b],\T)$ implies that ${\cal V}(F)$ is 
continuous.
Furthermore, the  continuity of ${\cal V}$ on $C^0([a,b],\T)$ also implies 
that the map $C^0(X\times[a,b],\T)\to C^0(X,\T),F\mapsto {\cal V}(F)$ is 
continuous , when we provide spaces of continuous maps with the compact open 
topology.

If $F$ can be lifted to a continuous map $\bar F:X\times [a,b]\to \R$ with 
$F=\pi\bar F$, then 
$${\cal V}(F)(x)=\bar F(x,b)-\bar F(x,a).$$

Suppose now that $X$ is a topological space, and that $(\phi_t)_{t\in\R}$ is a 
continuous flow on $X$. We will define $\Phi:X\times [0,1]\to X$ by $\Phi(x,t)
=\phi_t(x)$.
If $f:X\to \T$ is continuous, we set
$${\cal V}(f,\phi_t)={\cal V}(f\circ \Phi):X\to\R.$$
There is another another way to define ${\cal V}(f,\phi_t)$ which is used in 
\cite{Fathi80}. The function $F(f,\Phi):X\times[0,1]\to\T$, defined by 
$$F(f,\Phi)(x,t)= f(\phi_t(x))-f(x)$$
is continuous and identically $0$ on $X\times\{0\}$, it is therefore homotopic 
to a constant and can be lifted to a continuous map $F(f,\Phi):X\times[0,1]\to
\R$, with
$F(f,\Phi)|X\times\{0\}$ identically $0$. We have 
$${\cal V}(f,\phi_t)(x)=F(f,\Phi)(x,1).$$
Note that is $f$ is homotopic to $0$ then it can be lifted continuously to 
$\bar f:X\to \R$.
In that case $\bar F(f,\Phi)=\bar f\Phi-\bar f$, and
$${\cal V}(f,\phi_t)(x)=\bar f(\phi_1(x))-\bar f.$$
If $\mu$ is a measure with compact support invariant under the flow $\phi_t$, 
for a continuous 
$f:X\to\T$, we define
${\cal S}(\mu,\phi_t)(f)$, or simply ${\cal S}(\mu)(f)$ when $\phi_t$ is fixed,
by $${\cal S}(\mu)(f)=\int_X{\cal V}(f,\phi_t)(x)\,d\mu(x).$$
It is not difficult to verify that for $f_1,f_2:X\to\T$, then
$${\cal S}(\mu)(f_1+f_2)={\cal S}(\mu)(f_1)+{\cal S}(\mu)(f_2).$$
Moreover, if $f:X\to\T$ is homotopic to $0$ it can be lifted to $\bar f:X\to\R$
and
$${\cal S}(\mu)(f)=\int_X[\bar f(\ph_1(x))-\bar f(x)]\,d\mu(x)=\int_X
\bar f(\ph_1(x))\,d\mu(x)-\int_X\bar f(x)\,d\mu(x)=0$$
since $\mu$ is invariant by $\phi_1$.
Therefore, if we denote by $[X,\T]$ the set of homotopy classes of continuous 
maps from $X$ to $\T$, which is an additive group, the map
 ${\cal S}(\mu)$ is a  well-defined additive homomorphism from the additive 
group
$[X,\T]$ to $\R$.

When $X$ is a good space (like a manifold or a locally finite polyhedron), it 
is well-known that $[X,\T]$ is canonically identified with the first cohomology
group $\rH^1(X;\Z)$.
In that case ${\cal S}(\mu)$ is in $\operatorname{Hom}(\rH^1(X;\Z),\R)$. Since 
the first cohomology group with real coefficients $\rH^1(X;\R)$ is $\rH^1(X;\Z)
\otimes \R$,
we can view ${\cal S}(\mu)$ as an element of the dual $\rH^1(X;\R)^*$ of the 
$\R$-vector space $\rH^1(X;\R)$. When $\rH^1(X;\R)$ is finite-dimensional then
$\rH^1(X;\R)^*$ is in fact equal to the first homology group $\rH_1(X;\R)$, and
therefore ${\cal S}(\mu)$ defines an element of $\rH_1(X;\R)$, \ie a 1-cycle. 
This 1-cycle ${\cal S}(\mu)$ is called the {\it Schwartzman asymptotic cycle} 
of $\mu$. Note that $\rH^1(X;\R)$ is finite dimensional when 
$X$ is a finite poilyhedron or a compact manifold. It should be also noted 
that for a manifold $M$ the projection $TM\to M$ is a homotopy equivalence. 
Therefore
$\rH^1(TM;\R)=\rH^1(M;\R)$ is finite dimensional when $M$ is a compact manifold.

We now study the behavior of Schwartzman asymptotic cycles under 
semi-conjugacy. 
\begin{Prop}\label{SchSemiConj}
 Suppose $\phi^i_t:X_i\to X_i, i=1,2$ are two continuous flows. Suppose also 
 that $\psi:X_1\to X_2$ is a continuous semi-conjugation between the flows, \ie 
 $\psi\circ \phi^1_t=\phi^2_t\circ \psi$, for every $t\in\R$. Given a 
probability measure
  $\mu$ with compact support on $N_1$ invariant under $\phi^1_t$, then, 
  for every continuous map $f:X_2\to\T$, we have
 $${\cal S}(\psi_*\mu,\phi^2_t)([f])={\cal S}(\mu,\phi^1_t)([f\circ \psi]),$$
where  $\psi_*\mu$ is the image of $\mu$ under $\psi$. In particular, if we 
are in the situation where 
$\operatorname{Hom}([X_i,\T])\equiv \rH_1(X_i;\R),
i=1,2$, we obtain
$${\cal S}(\psi_*\mu,\phi^2_t)=H_1(\psi)({\cal S}(\mu,\phi^1_t)).$$
\end{Prop}
\begin{Proof} Notice that $f\psi\phi^1_t(x)-f\psi(x)=f\phi^2_t(\psi(x))-
f(\psi(x))$. Therefore by uniqueness of liftings ${\cal V}(f\psi,\phi^1_t)(x)
={\cal V}(f,\phi^2_t)(\psi(x))$.
An integration with respect to $\mu$ finishes the proof.
\end{Proof}

We would like now to relate the Schwartzman asymptotic cycles to the rotation 
vectors
$\rho(\mu)$ defined for Lagrangian flows. We first consider the case of a 
$C^1$ flow $\phi_t$ on the manifold $N$. We call $X$ the continuous vector 
field on $N$ 
generating $\phi_t$, \ie
$$\forall x\in N, \quad X(x)={\frac {d\phi_t(x)}{dt}}\Big|_{t=0}.$$
By the flow property $\phi_{t+t'}=\phi_t\circ\phi_{t'}$, this implies 
$$\forall x\in N,\ \forall t\in \R, \quad \frac {d\phi_t(x)}{dt}=X(\phi_t(x)).$$

In the case of a manifold $N$, the identification of $[N,\T]$ with 
$\rH^1(N;\Z)$ is best described with the de Rham cohomology. We consider the 
natural map 
$I_N:[X,\T]\to \rH^1(X;\R)$ defined by 
$$I_N([f])=[f^*\theta],$$
where $[f]$ on the left hand side denotes the homotopy class of the 
$C^\infty$ map $f:N\to\T$, and $[f^*\theta]$ on the right hand side  
is the cohomology class of the pullback by $f$ of the closed 1-form on $\T$ 
whose lift to $\R$ is $dt$. Note that any homotopy class in $[N,\T]$ contains 
smooth maps because $C^\infty$ maps are dense in $C^0$ maps (for the Whitney 
topology). Therefore the map $I_N$ is indeed defined on 
the whole of $[N,\T]$.
As it is well-known, this map $I_N$ induces an isomorphism of $[N,\T]$ on 
$\rH^1(N;\Z)\subset \rH^1(N;\R)=\rH^1(N;\Z)\otimes \R$.

Given a $C^\infty$ map $f:N\to \T$, the $C^1$ flow $\phi_t$ on $N$, and 
$x\in N$, we compute ${\cal V}(f,\phi_t)(x)$. If $\gamma_x:[0,1]\to N$ is the 
path
$t\mapsto\phi_t(x)$, by definition, we have ${\cal V}(f,\phi_t)(x)={\cal V}
(f\circ\gamma_x)$.
Since $\gamma_x$ is $C^1$, we get
$${\cal V}(f,\phi_t)(x)=\int_{f\circ \gamma_x}\theta=\int_{\gamma_x}f^*\theta.$$
Since $\gamma_x(t)=\phi_t(x)$, we have $\dot\gamma_x(t)=X(\phi_t(x))$.
It follows that 
$${\cal V}(f,\phi_t)(x)=\int_0^1(f^*\theta)_{\phi_t(x)}(X[\phi_t(x)])\,dt.$$
Recall that the interior product $i_X\omega$ of a differential form $\omega$ 
with $X$
is given by
$$(i_X\omega)_x(\cdots)=\omega_x(X(x),\cdots).$$
When $\omega$ is a differential 1-form then $i_X\omega$ is a function.
With this notation, we get
$${\cal V}(f,\phi_t)(x)=\int_0^1(i_Xf^*\theta)(\phi_t(x))\,dt.$$
Therefore if $\mu$ is an invariant measure for $\phi_t$, which we will assume 
to have a compact support, we obtain
$${\cal S}(\mu)=\int_N\int_0^1(i_Xf^*\theta)(\phi_t(x))\,dtd\mu(x).$$
Since $i_Xf^*\theta$ is continuous, and  we are assuming that $\mu$ has a 
compact support, we have
$${\cal S}(\mu)=\int_0^1\int_N(i_Xf^*\theta)(\phi_t(x))\,d\mu(x)dt.$$
By invariance of $\mu$ under $\phi_t$, we get $\int_N(i_Xf^*\theta)(\phi_t(x))
\,d\mu(x)=
\int_N(i_Xf^*\theta)(x)\, d\mu(x)$, and therefore
$${\cal S}(\mu)=\int_0^1\int_N(i_Xf^*\theta)(x)\,d\mu(x)dt=\int_N(i_Xf^*\theta)
\,d\mu.$$
This shows that as an element of $\rH^1(M;\R)^*$, the Schwarztman asymptotic 
cycle 
${\cal S}(\mu)$ is given by
$${\cal S}(\mu)([\omega])=\int_Ni_X\omega\,d\mu.$$
We can now easily compute Schwartzman asypmtotic cycles for linear flows on 
$\T^n$. Such a flow is determined by a constant vector field $\alpha\in \R^n$ 
on $\T^n$ (here we use the canonical trivialisation of the tangent bundle of 
$\T^n$), the associated flow $R^\alpha_t:
\T^n\to\T^n$ is defined by $R^\alpha_t(x)=x+[t\alpha]$, where $[t\alpha]$ is 
the class in $\T^n=\R^n/\Z^n$ of the vector $t\alpha\in\R^n$. If $\omega$ is a 
1-form with constant coefficients, \ie $\omega=\sum_{i=1}^na_idx_i$, with 
$a_i\in\R$, the interior product
$i_\alpha\omega$ is the constant function $\sum_{i=1}^n\alpha_ia_i$. Therefore, 
it follows that ${\cal S}(\mu)=\alpha\in \R^n\equiv \rH_1(\T^n;\R)$. 

We now compute Schwartzman asymptotic cycles for Euler-Lagrange flows. In  
this case $N=TM$ and $\phi_t$ is an Euler-Lagrange flow
$\phi_t^L$ of some Lagrangian $L$. If we call $X_L$ the vector field generating
$\phi_t^L$, since this flow is obtained from a second order ODE on $M$, we get
$$\forall x\in M, \forall v\in T_xM, \quad T\pi(X_L(x,v))=v,$$
where $T\pi:T(TM)\to TM$ denotes the canonical projection.
Since this projection $\pi$ is a homotopy equivalence, to compute 
${\cal S}(\mu)$ we only need to consider forms of the type $\pi^*\omega$ where 
$\omega$ is a closed 1-form on the base $M$. In this case $(i_{X_L}\pi^*\omega)
(x,v)=\omega_x(T\pi(X_L(x,v))=\omega_x(v)$. Therefore, for any measure 
probability measure $\mu$ on $TM$ with compact support and invariant under 
$\phi_t^L$,  we obtain
$${\cal S }(\mu)[\pi^*\omega]=\int_{TM}\omega_x(v)\,d\mu(x,v)
=\int_{TM}\hat\omega\,d\mu.$$
This is precisely $\rho(\mu)$ as it was defined above in section \ref{sec2}. 
Note that the only property we have used is the fact that $\phi_t$ is the flow 
of a second 
order ODE on the base $M$.


To simplify things, in the remainder of this appendix, we will assume that $X$ 
is a compact space, for which we have $[X,\T]=\rH^1(X;\Z)$, and $\rH_1(X;\Z)$ 
is finitely generated. In that case,
the dual space $\rH^1(X;\R)^*$ is $\rH_1(X;\R)$, and for every flow $\phi_t$ 
on $X$ and every probability measure $\mu$ on $X$ invariant under $\phi_t$, 
the Schwartzman asymptotic cycle is an element of the finite dimensional-vector
space $\rH_1(X;\R)$.

Suppose that $x$ is a periodic point of $\phi_t$ or period $T>0$. One can 
define an invariant  probability measure $\mu_{x,t_0}$ for $\phi_t$ by
$$\int_Xg(x)\,d\mu_{x,t_0}=\frac1{t_0}\int_0^{t_0}g((\phi_t(x))\,dt,$$
where $g:X\to \R$ is a measurable function. We let the reader verify that
${\cal S}(\mu_{x,t_0})$ is equal in $\rH_1(X;\R)$ to the homology class 
$[\gamma_{x,t_0}]/t_0$, where $\gamma_{x,t_0}$ is the loop $t\mapsto \phi_t(x), 
t\in[0,t_0]$. When $x$ is a fixed point of $\phi_t$, then the Dirac mass 
$\delta_x$ at $x$ is invariant under $\phi_t$, and in that case ${\cal S}
(\delta_x)=0$.

\begin{Def}\rm For a flow $\phi_t$ on $X$, we denote by ${\cal S}(\phi_t)$ 
the set of all 
Scwartzman asymptotic cycles ${\cal S}(\mu)$, where $\mu$ is an arbitrary 
probability measure on $X$ invariant under $\phi_t$.
\end{Def}
Since $X$ is compact, note that for the weak topology the set $\calM(X)$ of 
probability Borel measures on $X$ is compact and convex. It is even metrizable,
since we are assuming $X$ metrizable.
Furthermore the subset ${\calM}(X,\phi_t)\subseteq {\calM}(X)$ of probability 
measures invariant under $\phi_t$ is, as it is well-known, compact convex and 
non empty. Therefore
 ${\cal S}(\phi_t)$ is a compact convex non-empty subset of $\rH_1(X;\R)$.
 
For the case of a linear flow $R^\alpha$ on $\T^n$, we have shown above that
${\cal S}(R^\alpha_t)=\{\alpha\}\subset \R^n\equiv \rH_1(\T^n;\R)$.
 
The following corollary is an easy consequence of Proposition \ref{SchSemiConj}.
\begin{Cor}\label{SchCon} For $i=1,2$, suppose that $\phi^i_t$ is a continous 
flow on the compact space $X_i$, which satisfies $\operatorname{Hom}([X_i,\T],
\R)\equiv \rH_1(X_i;\R)$.
If $\psi:X_1\to X_2$ is a topological conjugacy between $\ph^1_t$ and $\phi^2_t$
(\ie the map $\psi$ is a homeomorphism that satisfies $\psi\phi^1_t=\phi^2_t 
\psi$, for all $t\in \R$), then we have
$${\cal S}(\phi^2_t)=H_1(\psi)[{\cal S}(\phi^1_t)].$$
\end{Cor}  

We denote by ${\mathfrak F}(X)$ the set of continuous flows on $X$. We can 
embed ${\mathfrak F}(X)$ in $C^0(X\times [0,1],X)$ by the map $\phi_t\mapsto 
F^{\phi_t}\in 
C^0(X\times [0,1],X)$, where
$$F^{\phi_t}(x,t)=\phi_t(x).$$
The topology on $C^0(X\times [0,1],X)$ is the compact open (or uniform) 
topology, and
we endow ${\mathfrak F}(X)$ with the topology inherited from the embedding 
given above.

\begin{Lem}The map $\phi_t\mapsto {\cal S}(\phi_t)$ is upper semi-continuous on 
${\mathfrak F}(X)$. This means that for each open
 subset $U\subseteq \rH_1(X;\R)$, the set 
 $\{\phi_t\in {\mathfrak F}(X)\mid {\cal S}(\phi_t)\subset U\}$ is open in 
${\mathfrak F}(X)$.
 \end{Lem}
 \begin{Proof} Since the topology on $C^0(X\times [0,1],X)$ is metrizable, 
if this where not true we could find an open set $U\subset \rH_1(X;\R)$ and a 
sequence  $\phi^n_t$ of continuous flows on $X$ converging uniformly to a flow
$\phi_t$, with ${\cal S}(\phi_t)\subset U$, and 
$ {\cal S}(\phi^n_t)$ is not contained in $ U$. this means that for each $n$ we
can find a probability measure $\mu_n$ on $X$ invariant under $\phi^n_t$ and 
such that
its Schwartzman asymptotic cycle ${\cal S}(\mu_n,\phi^n_t)$ for $\phi_n^t$ is 
not in the open set $U$. Since ${\calM}(X)$ is compact for the weak topology, 
extracting a subsequence if necessary, we can assume that $\mu_n\to\mu$. It is 
not difficult to show that
$\mu$ is invariant under the flow $\phi_t$. We now show that ${\cal S}(\mu_n,
\phi^n_t)\to {\cal S}(\mu,\phi_t)$. This will yield a contradiction and finish 
the proof because ${\cal S}(\mu_n,\phi^n_t)$ is in the closed set $\rH_1(X;\R)
\setminus U$, for every $n$, and ${\cal S}(\mu,\phi_t)\in U$.

To show that the linear maps ${\cal S}(\mu_n,\phi^n_t)\in \rH_1(X;\R)=
\rH^1(X;\R)^*$ converge to the linear map ${\cal S}(\mu,\phi_t)$, it suffices 
to show that 
${\cal S}(\mu_n,\phi^n_t)([f])\to {\cal S}(\mu,\phi_t)([f])$, 
for every $[f]\in[X,\T]=\rH^1(X;\Z)\subset \rH^1(X;\R)=\rH^1(X;\Z)\otimes\R$.
Fix now a continuous map $f:X\to \T$. Denote by $F_n, F:X\times [0,1]\to \T$ 
the maps
defined by 
$$F_n(x,t)=f(\phi^n_t(x))-f(x)\text { and}\; F(x,t)=f(\phi_t(x))-f(x).$$
By the uniform continuity of $f$ on the compact metric space $X$,
the sequence $F_n$ converges uniformly to $F$. Since 
$F_n|X\times \{0\}\equiv 0$, if we call $\tilde F_n:X\times [0,1]\to \R$ 
the lift of $F_n$ such that $\tilde F_n|X\times \{0\}\equiv 0$, then the 
sequence 
$\tilde F_n$ also converges uniformly to $\tilde F$, that is the lift of $F$ 
such that  $\tilde F |X\times \{0\}\equiv 0$. Since the $\mu_n$ are probability
measures, we have 
$$\left| \int_X\tilde F_n(x,1)\mu_n(x)-\int_X\tilde F(x,1)\mu_n(x)\right|
\leq \lVert \tilde F_n-\tilde F\rVert_\infty\longrightarrow 0.$$
Since $\mu_n\to \m$ weakly, we also have 
$$\left|\int_X\tilde F(x,1)\mu_n(x)-\int_X\tilde F(x,1)\mu(x)\right|
\longrightarrow 0.$$
Therefore ${\cal S}(\mu_n,\phi^n_t)([f])=\int_X\tilde F_n(x,1)\mu_n(x)\to 
{\cal S}(\mu,\phi_t)([f])=\int_X\tilde F(x,1)\mu_0(x)$.
\end{Proof}
\begin{Def}{\bf[Schwartzman unique ergodicity]} \rm We say that a flow $\phi_t$
is Schwartzman uniquely ergodic if ${\cal S}(\phi_t)$ is reduced to one point.
\end{Def}
By the computation done above linear flows on the torus $\T^n$ are Schwartzman 
uniquely ergodic. Of course, all uniquely ergodic flows (\ie flows having 
exactly one invariant probability measure) are also Schwartzman uniquely 
ergodic. Moreover, by Corollary \ref{SchCon}, any flow topologically conjugate
to a Schwartzman uniquely ergodic flow is itself Schwartzman uniquely ergodic.

\begin{Teo} The set ${\mathfrak S}(N)$ of Schwartzman uniquely ergodic flows 
is a $G_\delta$ in ${\mathfrak F}(X)$.
\end{Teo}
\begin{Proof}. Fix some norm on $\rH^1(X;\R)$. We will measure diameters of 
subsets of 
$\rH^1(X;\R)$. with respect to that norm. Fix $\epsilon>0$. Call 
${\cal U}_\epsilon$ the set of flows $\phi_t$ such that the diameter of 
${\cal S}(\phi_t)\subset \rH_1(X;\R)$ is 
$<\epsilon$. If $\phi^0_t\in {\cal U}_\epsilon$, we can find $U$ an open subset
of $\rH_1(X;\R)$ of diameter $<\epsilon$ and containing ${\cal S}(\phi^0_t)$. 
By the lemma above the set $\{\phi_t\in {\mathfrak F}(X)\mid {\cal S}(\phi_t)
\subset U\}$ is open in ${\mathfrak F}(X)$ contains $\phi^0_t$ and is 
contained in ${\cal U}_\epsilon$.
The set of Schwartzman uniquely ergodic flows is $\cap_{n\geq 1}{\cal U}_{1/n}$.
\end{Proof}

\begin{Prop}
Let $\f_t:X\longrightarrow X$ be a continuous flow on the compact path 
connected space $X$. Suppose that there exist 
$t_i\uparrow +\infty$ such that 
$\f_{t_i}{\longrightarrow} \f$ in $C(N,N)$ (with the $C^0$-topology). Then, 
$\f_t$ is Schwartzman uniquely ergodic. 
In particular, periodic flows and (uniformly) recurrent flows are Schwartzman 
uniquely 
ergodic (in both cases $\f={\rm Id}$).
\end{Prop}
\begin{Proof} Fix a continuous map $f:X\to\T$. Consider the function 
$F:X\times [0,+\infty)\to \T, (x,t)\mapsto f(\phi_t(x))-f(x)$. We have 
$F(x,0)=0$, for every $x\in X$. 
Call $\bar F:X\times [0,+\infty)\to \R$ the (unique) continuous lift of 
$F$ such that
$\bar F(x,0)=0$, for every $x\in X$. The definition of the Schwartzman 
asymptotic cycle gives
$${\cal S}(\mu)([f])=\int_X\bar F(x,1)\,d\mu(x),$$
for every probability measure invariant under $\phi_t$.
We claim that we have
$$\forall t,t'\geq 0,\forall x\in X, \quad \bar F(x,t+t')=\bar F(\phi_t(x),t')
+F(x,t).$$
In fact, if we fix $t$ and we consider each side of the equality above as a 
(continuous) function of $(x,t')$ with values in $\R$, we see that the two 
sides are equal for $t'=0$, and that they both lift the function
$$(x,t')\mapsto f(\phi_{t+t'}(x))-f(x)=f(\phi_t'(\phi_t(x))-f(\phi_t(x))+
f(\phi_t(x))-f(x)$$
with values in $\T$.
By induction, it follows easily that 
$$\forall k\in\N, \quad \bar F(x, k)=\sum_{j=0}^{k-1}\bar F(\phi_j(x),1).$$
Therefore, if $t\geq 0$ and $[t]$ is its integer part, we also obtain
\begin{equation*}
\bar F(x,t)=\bar F(\phi_{[t]}(x),t-[t])+\sum_{j=0}^{[t]-1}\bar F(\phi_j(x),1).
\tag{$*$}
\end{equation*}
It follows that
\begin{equation*}
\forall t\geq 0,\forall x\in X, \quad \lvert \bar F(x,t)\rvert\leq 
([t]+1)\lVert \bar F|X\times [0,1]\rVert_\infty.\tag{$**$}
\end{equation*}
By compactness $\lVert \bar F|X\times [0,1]\rVert_\infty$ is finite.
If we integrate equality ($*$) with respect to a probability measure $\mu$ on 
$X$ invariant under the flow $\phi_t$,  we obtain
$$\int_X\bar F(x,t)\,d\mu(x)=\int_X\bar F(x,t-[t])\,d\mu(x)+[t]\int_X\bar 
F(x,1)\,d\mu(x).$$
Therefore we have
\begin{equation*}
{\cal S}(\mu)([f])=\lim_{t\to+\infty}\int_X\frac{\bar F(x,t)}t\,d\mu(x).
\tag{$***$}
\end{equation*}
Suppose now that we set $\gamma_x(s)=\phi_s(x)$; we have 
$\bar F(x,t)={\cal V}(f\gamma_x|[0,t])$.
Fix now some point $x_0\in X$, and consider $t_i\to +\infty$ such that 
$\phi_{t_i}\to \phi$
in the $C^0$ topology. Since $\bar F(x_0,t)/t$ is bounded in absolute value by
$2 \lVert \bar F|X\times [0,1]\rVert_\infty$, for $t\geq 1$, extracting a 
subsequence if necessary, we can assume that $\bar F(x_0,t_i)/t_i\to c\in \R$.
If $x\in X$, we can find a continuous path $\gamma:[0,1]\to M$ with 
$\gamma(0)=x_0$
and $\gamma(1)=x$. The map $\Gamma:[0,1]\times [0,t]\to \T,(s,s')\to 
\phi_{s'}(\gamma(s))$ is continuous, therefore we can lift it to a continuous 
function with values in $\R$, and this implies the equality
$${\cal V}(\Gamma|[0,1]\times\{0\})+{\cal V}(\Gamma|\{1\}\times[0,t])
-{\cal V}(\Gamma|[0,1]\times\{1\})-{\cal V}(\Gamma|\{0\}\times[0,t])=0.$$
This can be rewritten as
$${\cal V}(f\gamma_x|[0,t])-{\cal V}(f\gamma_{x_0}|[0,t])={\cal V}
(f\phi_t\gamma)-{\cal V}(f\gamma),$$
which translates to
$$\bar F(x,t)-\bar F(x_0,t)={\cal V}(f\phi_t\gamma)-{\cal V}(f\gamma).$$
Since $\phi_{t_i}\to \phi$ uniformly, by continuity of ${\cal V}$, the left 
hand-side remains bounded as $t=t_i\to +\infty$. It follows that 
$(\bar F(x,t_i)-\bar F(x_0,t_i))/t_i\to 0$.
Hence for every $x\in X$, we also have that $\bar F(x,t_i)/t_i$ tends to the 
same limit $c$ as $\bar F(x_0,t_i)/t_i$. Since $\bar F(x,t)/t$ is uniformly 
bounded for $t\geq 1$,
by $(**)$, by Lebesgue's dominated convergence we obtain from ($***$) that
${\cal S}(\mu)([f])=c$, where $c$ is independent of the invariant measure $\mu$.
This is of course true for any $f:X\to \T$. Therefore ${\cal S}(\mu)$ does 
not depend on the invariant measure $\mu$.
 \end{Proof}
An interesting property of Schwartzman uniquely ergodic flows (which also 
shows that they have some kind of rigidity) is the following 
proposition, that follows immediately from the definition of Schwartzman unique 
ergodicity and what we remarked above about the asymptotic cycles of fixed and 
periodic points (see also \cite{Schwartzman}).
 
\begin{Prop}
Suppose that $\phi_t $ is a Schwartzman uniquely ergodic flow on $X$. If there 
exists either a fixed point or a closed orbit homologous to zero, then all 
closed orbits are homologous to zero. In the remaining case, if $C_1$ and 
$C_2$  are closed orbits with periods $\t_1$ and $\t_2$, then $\frac{C_1}
{\t_1}$ and $\frac{C_2}{\t_2}$ are homologous. Since $[C_1]$ and $[C_2]$ are 
in $\rH_1(X;\Z)$, it follows in this case that the ratio of the periods of any 
two closed orbits must be rational. Consequently, for any continuous family of 
periodic orbits of $\f_t$, all orbits have the same period.
\end{Prop}  

\begin{Def}{\bf[Schwartzman strict ergodicity]} \rm We say that a flow 
$\phi_t$ is Schwartzman strictly ergodic if it is  Schwartzman uniquely 
ergodic and it has an invariant measure $\mu$ of full support (\ie $\mu(U)>0$ 
for every non-empty open subset $U$ of $X$).
\end{Def}
Linear flows on the torus $\T^n$ are Schwartzman strictly ergodic (they 
preserve Lebesgue measure). Of course, all strictly ergodic flows (\ie flows 
having exactly one invariant probability measure, and the support of this 
measure is full) are also Schwartzman strictly ergodic. A minimal flow which 
is  Schwartzman uniquely ergodic is in fact Schwartzman strictly ergodic 
(because all invariant measures have full support). Moreover, any flow 
topologically conjugate 
to a Schwartzman strictly ergodic flow is also  Schwartzman strictly ergodic.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Lagrangian graphs and KAM tori}\label{app2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this Appendix we recall some very well-known properties of Lagrangian 
graphs and KAM tori used in the previous sections. We provide the standard 
proofs for the convenience of the reader.
First of all, let us observe that, as remarked in Section \ref{sec1}, the 
canonical symplectic structure is intrinsically defined, 
\ie it does not depend on the choice of the local coordinates. In fact, it is 
easy to check that the Liouville form $\l$ can 
be equivalently defined as $$ \lambda {(x,p)} = (d\pi{(x,p)})^*p \;\in 
{\rm T}^*_{(x,p)}{\rm T}^*M,$$
where $\pi: {\rm T}^*M \,\longrightarrow\, M$ is the canonical projection on 
$M$.
Now, let us prove that smooth Lagrangian graphs correspond to closed $1$-forms.

\begin{Prop} Let $\Lambda=\{(x,\eta(x)),\; x\in M\}$ a smooth section of 
${\rm T}^*M$. 
$\Lambda$ is Lagrangian if and only if $\eta$ is a closed $1$-form.
\end{Prop}

\begin{Proof}
Let us consider:
\beqano
s_{\eta}: M &\longrightarrow & {\rm T}^*M\\
x &\longmapsto& (x,\eta(x))\,.
\eeqano
We want to prove first that $s_{\eta}^*\l = \eta$, where $\l$ is the 
tautological form introduced above and $s_{\eta}^*\l$ denotes its pull-back. 
Recalling that $\lambda{(x,p)} = (d\pi {(x,p)})^*p$, we get:
\beqano
(s_\eta^*\lambda){(x)} &=& (ds_{\eta}(x))^*\l{(x,p)} = (ds_{\eta}(x))^*(d
\pi{(x,p)})^*\eta(x) = \\
&=& d(\pi \circ s_{\eta}{(x,p)})^* \eta(x) = \eta(x)\,,
\eeqano
where in the last equality we used that $\pi \circ s_{\eta}$ is the identity 
map.
Using this property, the claim follows immediately. In fact:
\beqano
\Lambda\;\mbox{is Lagrangian} &\Longleftrightarrow& 
\omega \big|_{{\rm T}\Lambda} = 0 \quad \Longleftrightarrow \quad 
s_{\eta}^* \omega = 0 \quad
\Longleftrightarrow \quad s_{\eta}^* d\l = 0 \\
&\Longleftrightarrow& d s_{\eta}^* \l = 0 \quad \Longleftrightarrow \quad 
d\eta  = 0 \quad
\Longleftrightarrow \quad {\eta}\; \mbox{is closed}.
\eeqano
\end{Proof}

Let us now consider a Hamiltonian $H: {\rm T}^*M \longrightarrow \R$.
%and see the relation betweenthe property of being Lagrangian and the 
%dynamics. \\

\begin{Prop}
Let $\Lambda$ be a Lagrangian submanifold. Then $\L$ is invariant if and only if
$H\big|_{\Lambda} \equiv {\rm const.}$
\end{Prop}

\begin{Proof} ({\it i}) The Hamiltonian vector field $X_{H}$ is
defined by $\omega(X_H,\cdot)=dH$. Since $\Lambda$ is 
invariant, $X_{H}\big|_{\Lambda}$ is tangent to $\Lambda$. But $\Lambda$ is 
Lagrangian, therefore $0=\omega(X_H,V)=dH\cdot V$ for any $V \in T\Lambda$, 
and this implies that $H$ is constant on $\Lambda$. ({\it ii}) Since $H$ is 
constant on $\L$, we have that, for every $V\in {\rm T}\L$, $0=dH\cdot V=
\omega(X_H,V)$. 
Since $\L$ is Lagrangian, $X_H$ belongs to ${\rm T}\L$ itself and therefore 
$\L$ is invariant.
\end{Proof}

\begin{Rem}
As observed in the proof of Lemma \ref{lemmadellacontesa}, if $\Lambda$ is an 
invariant 
Lagrangian graph, then the value of the constant is given by $\a(c_{\Lambda})$,
where 
$\a$ is the $\a$-function associated to $H$ and $c_{\Lambda}$ the cohomology 
class of $\Lambda$.
\end{Rem}

To conclude, let us show that in the case of KAM tori with rationally 
independent rotation vectors, 
the condition of being Lagrangian is automatically satisfied. Actually, the 
same result holds in a slightly more general setting, as was observed in 
\cite{Herman}[page 52, Proposition 3.2], from where we borrowed the proof.

\begin{Prop} \label{Lagrangiantori}
Given a Hamiltonian $H$, let $\TT\subset\TTT^n\times\RRR^n$ be an invariant 
graph over 
$\TTT^n$ such that the Hamiltonian flow on $\TT$ is conjugated to a 
flow $R_t$ on $\TTT^n$, which is transitive, \ie with a dense orbit. 
Then $\TT$ is Lagrangian.
\end{Prop}

%$\f: \TTT^n\to \TT$
%such that $\f^{-1}\circ\Phi_t\circ\f= R_\r^t$, $\forall t\in\RRR$,
%where $R_\r^t: x \to x+\r t$.

\begin{Proof}
Let $\f: \TTT^n\to \TT$ be the conjugation: $\f^{-1}\circ\Phi_t\circ\f= R_t$, 
$\forall t\in\RRR$. Consider the inclusion $i_{\TT}$ of $\TT$ into 
$\T^n\times \R^n$. We want to prove that
$\omega\big|_{\TT} = i_{\TT}^*\omega \= 0$.
Let us start by proving that the restriction of the symplectic form 
$i_{\TT}^*\omega$
is invariant under the Hamiltonian flow $\Phi_t$. In fact, 
\beqano
\Phi_t^*\left(i_{\TT}^*\omega\right) =  \left(i_{\TT}\circ\Phi_t\right)^*
\omega =
\left(\Phi_t \circ i_{\TT}\right)^*\omega = i_{\TT}^* \left(\Phi_t^*\omega
\right) = i_{\TT}^*\omega\,,
\eeqano
where we used that $\TT$ is invariant ($\Phi_t \circ i_{\TT} = i_{\TT}\circ
\Phi_t $) and
$\Phi_t$ is a symplectomorphism for any $t\in \R$ (\ie $\Phi_t^*\omega = 
\omega$).
Consider now the $1$-form on $\T^n$ given by $\omega_1 = \f^*
\left(i_{\TT}^*\omega\right)$. Let us show that $\omega_1$ is invariant under 
$R_t$; in fact,
\beqano
(R_t)^*\omega_1 &=&  ({R_t})^* 
\left(\f^*\left(i_{\TT}^*\omega\right)
\right) =  \left( \f \circ R_t \right)^*i_{\TT}^*\omega = 
\left( \Phi_t \circ \f \right)^*i_{\TT}^*\omega = \\
&=&\f^* \left( \Phi_t^* \left(i_{\TT}^*\omega \right)\right) = 
\f^*\left(i_{\TT}^*\omega \right) = \omega_1\,,
\eeqano
where we used that $\f^{-1}\circ\Phi_t\circ\f= R_t$. Since 
$R_t$ is transitive, then $\omega_1$ 
invariant implies $\omega_1$ constant: $\omega_1 = \sum_{i<j} a_{ij}\,dx_i
\wedge dx_j.$ 
But $\omega_1$ is exact (since $\omega = -d\l$ is exact), therefore 
$\omega_1 = \f^*\left(i_{\TT}^*\omega \right) \equiv 0$ and (using that $\f$ 
is invertible) the result is proved.
\end{Proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%APPENDIX 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem \ref{emmeinenne}}\label{app3}

In this Appendix we shall give the proof of the second equivalence in 
(\ref{numero}) and this will conclude the proof of Theorem \ref{emmeinenne},
as already discussed after its statement. We want to show that
%
\be\m\in{\calM}(L)\ {\rm and}\ {\rm supp}\ \m\subseteq\widetilde{\cN}_c
\ \Longrightarrow\ {\rm supp}\ \m\subseteq\widetilde{\cA}_c\,.\label{app.1}\ee
%
Let us first recall the definition of 
{\it non-wandering point} for a flow $\Phi_t: X \longrightarrow X$.

\begin{Def}
A point $x\in X$ is called {\it non-wandering} if for each neighborhood 
$\cal{U}$ and each positive integer $n$, there exists $t>n$ such that 
$f^t(\cal{U})\cap \cal{U} \neq \emptyset$.
\end{Def}

We will denote the set of {\it non-wandering} points for $\Phi_t$ by 
$\Omega(\Phi_t)$. Note that, if $\m$ is an invariant measure, then ${\rm supp}\
\m\subseteq \Omega(\Phi_t)$. In fact, by the ergodic decomposition theorem,
every point $x\in{\rm supp}\ \m$ is in the support of an ergodic invariant 
measure $\m_1$: therefore, $x$ is non-wandering, by the ergodicity of $\m_1$. 
Given this remark, (\ref{app.1}) is a simple consequence of the following 
Proposition.

\begin{Prop}\label{propnonwandering} If $M$ is a compact manifold and $L$ 
a Tonelli Lagrangian on ${\rm T} M$, then  
$\Omega \left( \Phi^L_t\big|\widetilde{\cN}_c\right) \subseteq 
\widetilde{\cA}_c$ 
for each $c\in \rH^1(M;\R)$.
\end{Prop}

\noindent{\bf{Remarks.}}\\
1) Proposition \ref{propnonwandering} also shows that the Aubry set is 
non-empty. In fact, any 
continuous flow on a compact space possesses non-wandering points.\\
2) As remarked above, every point in the support of an invariant 
measure $\m$ is non-wandering. Therefore, if $x\in{\rm supp}\ \m\subseteq
\widetilde{\cN}_c$,
then $x\in \Omega \left( \Phi^L_t\big|\widetilde{\cN}_c
\right)$ and, by the above proposition, $x\in \widetilde{\cA}_c$. This proves
(\ref{app.1}).\\

Before proving the above statement, we need some preliminary results. Let 
$u:M\longrightarrow \R$ be an $\eta$-critical subsolution, with $[\eta]=c$
(see Definition \ref{etacriticalsub}),  
and $\g:[a,b]\longrightarrow M$ a curve.
It is easy to check (see for instance \cite{Fathibook}), that for all 
$a\leq t \leq t'\leq b$:
\beqa{numero3}
u(\g(t')) - u(\g(t)) \leq \int_t^{t'} L(\g(s),\dot{\g}(s))\,ds + \a(c)(t'-t)\,,
\eeqa
%
see also (\ref{4.12}) and (\ref{4.14}).
We will say that $\g:[a,b] \longrightarrow M$ is $(u,L,\a(c))$-{\it calibrated}
on $[a,b]$ if for all $a\leq t \leq t' \leq b$ the above inequality is an 
equality:

$$u(\g(t')) - u(\g(t)) = \int_t^{t'} L(\g(s),\dot{\g}(s))\,ds + \a(c)(t'-t)\,.$$

In \cite{Fathibook} many properties of such curves have been studied. In 
particular, they provide a useful characterization of the Aubry and Ma\~n\'e 
sets in terms of critical subsolutions. Let us denote by $\widetilde{
{\cal I}}_c(u)$ the set of points $(x,v)\in {\rm T}M$ such that the curve 
$\g(t)=\pi \Phi^L_t(x,v)$ is $(u,L,\a(c))$-calibrated on $\R$. In 
\cite{Fathibook} it is proven that:
\beqa{numero2}
\widetilde{\cA}_c = \bigcap_{u\in\cS^1_{\eta}} \widetilde{{\cal I}}_c(u)
\quad {\rm and} \quad \widetilde{\cN}_c = \bigcup_{u\in\cS^1_{\eta}}
\widetilde{{\cal I}}_c(u).
\eeqa



\begin{Proof}[{\bf Proposition \ref{propnonwandering}}]
Without any loss of generality we can assume that $c=0$ and $\a(0)=0$.
Let $(x,v) \in \Omega \left( \Phi^L_t\big|\widetilde{\cN}_0\right)$. By the 
definition of non-wandering point, there exist a sequence $(x_k,v_k) \in 
\widetilde{\cN_0}$ and $t_k \rightarrow +\infty$, such that $(x_k,v_k) 
\rightarrow (x,v)$  and $\Phi^L_{t_k}(x_k,v_k) \rightarrow (x,v)$ as 
$k\rightarrow +\infty$.
From (\ref{numero2}),  for each $(x_k,v_k)$ there exists a critical 
subsolution $u_k$, such that the curve $\g_k (t) = \pi \Phi^L_{t} (x_k,v_k)$ is 
$(u_k,L,0)$-calibrated. Moreover, up to extracting a subsequence, we can 
assume that, on any compact interval, $\g_k$ converge in the $C^1$-topology
to $\g(t)=\Phi^L_t(x,v)$.

Pick now any critical subsolution $u$. If we show that $\g$ is 
$(u,L,0)$-calibrated, using (\ref{numero2}) we can conclude that $(x,v)\in 
\widetilde{\cA}_0$. First of all, observe that, by the continuity of $u$,
$$u(\g_k(t_k)) - u(x_k)\stackrel{k\rightarrow \infty}{\longrightarrow} 0\;.$$
Using that $\eta$-critical 
subsolutions are equi-Lipschitz \cite{Fathibook}, we can also conclude that
$$u_k(\g_k(t_k)) - u_k(x_k)\stackrel{k\rightarrow \infty}{\longrightarrow} 0$$
%
and, therefore, 
%
\be
\int_0^{t_k} L(\g_k(s),\dot{\g}_k(s))\,ds = 
u_k(\g_k(t_k)) - u_k(x_k) \stackrel{k\rightarrow \infty}{\longrightarrow} 0\;. 
\ee
%
Let $0\leq a\leq b$ and choose $t_k \geq b$. Observe now that
\beqano
u(\g_k(b)) - u(\g_k(a)) &=& u(\g_k(t_k)) - u(x_k) - \left[u(\g_k(t_k))- 
u(\g_k(b)) \right] -
\left[u(\g_k(a))- u(x_k) \right] \geq\\
&\geq &u(\g_k(t_k)) - u(x_k) -  \int_b^{t_k} L(\g_k(s),\dot{\g}_k(s))\,ds - 
\int_0^{a} L(\g_k(s),\dot{\g}_k(s))\,ds = \\
&=& u(\g_k(t_k)) - u(x_k) +  \int_a^b L(\g_k(s),\dot{\g}_k(s))\,ds 
- \int_0^{t_k} L(\g_k(s),\dot{\g}_k(s))\,ds\,;
\eeqano
taking the limit as $k\rightarrow \infty$ on both sides,
one can conclude:
$$
u(\g(b)) - u(\g(a)) \geq \int_a^b L(\g(s),\dot{\g}(s))\,ds
$$
and therefore, from (\ref{numero3}), it follows the equality. This shows 
that $\g$ is $(u,L,0)$-calibrated on $[0,\infty)$. To show that it is indeed 
calibrated on all $\R$,  one can make a symmetric argument, letting $(y_k,w_k)= 
\Phi^L_{t_k}(x_k,v_k)$ play the role of $(x_k,v_k)$ in the previous argument.
In fact, one has $(y_k,w_k) \rightarrow (x,v)$  and $\Phi^L_{-t_k}(y_k,w_k) 
\rightarrow (x,v)$ as $k\rightarrow +\infty$ and the very same argument works.

\end{Proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vspace{1.truecm}

{\sc Albert Fathi}\\
{\it\'Ecole normale sup\'erieure de Lyon,\\
Unit\'e de Math\'ematiques Pures et Appliqu\'ees, UMR CNRS 5669,\\
46, all\'ee d'Italie,\\
69364 Lyon Cedex 07 France}\\

{\sc Alessandro Giuliani}\\
{\it Dipartimento di Matematica,\\
Universit\`a degli Studi di Roma Tre,\\
L.go S. Leonardo Murialdo 1\\
00146 Roma Italy}\\

{\sc Alfonso Sorrentino}\\
{\it Department of Mathematics,\\
Princeton University,\\
Washington Road,\\
Princeton NJ 08544 USA}

\end{document}

\begin{Rem}
From a geometric point of view, if we consider the space ${\rm T}^*M$ equipped 
with the canonical symplectic form, 
the graph of the differential of a $C^1$ $\eta$-critical subsolution 
(plus the $1$-form $\eta$)
is nothing else than a $c$-Lagrangian graph (\ie a Lagrangian graph with 
cohomology class $c$, as introduced in Section \ref{sec1}). In 
particular, Ma\~n\'e's $c$-critical energy level $\cE^*_{c}$ corresponds to a 
$(2n-1)$-dimensional hypersurface, such that the region it bounds is convex 
and does not 
contain in its interior any $c$-Lagrangian graph. In other words, all 
$c$-Lagrangian graphs that are contained in the region bounded by $\cE^*_c$ 
must intersect $\cE^*_c$ and all these intersections contain the Aubry set 
$\cA^*_c$ (that constitutes a sort of ``{\it non-removable intersection set}'' 
- see \cite{Paternain-Siburg}).
It follows that the Aubry set (and also Mather set) 
only depends on the geometry of the critical energy level (that encodes
all information about dynamics) and it follows easily from this definition that
they are invariant by exact symplectomorphisms (see also \cite{Bernardsymp}).
\end{Rem}

\bibitem{Paternain-Siburg}
Gabriel~P. Paternain, Leonid Polterovich, and Karl~Friedrich Siburg.
\newblock Boundary rigidity for {L}agrangian submanifolds, non-removable
  intersections, and {A}ubry-{M}ather theory.
\newblock {\em Mosc. Math. J.}, 3(2):593--619, 745, 2003.
\newblock Dedicated to Vladimir I. Arnol'd on the occasion of his 65th birthday.
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