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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.
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%%%%%%%%%%%%%%%%%%%%%%%% SECTION 2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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**}[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 $k0$, 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 $\tn-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_{in$ 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.
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
**