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%     Lagrangian flows: The Dynamics of Globally Minimizing Orbits       %
%                                                                        %
%                        Ricardo Mane                                    %
%                                                                        %
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%_________________________________________________________________________


\begin{document}


\title[LAGRANGIAN FLOWS]{\large \bf Lagrangian Flows:   The Dynamics of
                         \\  Globally Minimizing Orbits}

\author[RICARDO MA\~N\'E]{Ricardo Ma\~n\'e}

\address{I.M.P.A. \\ Estrada Dona Castorina 110 \\  Jardim Bot\^anico 22.460 \\
Rio de Janeiro \\ RJ \\
Brazil.}

\maketitle

%______________________________________________________________________

The objective of this note is to present some results, to be proved in a
forthcoming paper, about certain special solutions of the Euler-Lagrange
equations on closed manifolds. Our main results extend to time dependent
periodic Lagrangians with minor modifications.

We have chosen the autonomous case because this formally simpler framework
allows to reach more easily the core of our concepts and results. Moreover
the autonomous case exhibits certain special features involving the energy
as a first integral that deserve special attention. They are closely related
to the link found by Carneiro [C] between the energy and Mather's action
function [Ma].

\vspace{13pt}

{\bf I. The Autonomous Case.}

\vspace{13pt}

Let $L$ be a Lagrangian on a closed manifold $M$, i.e. $L:TM\rightarrow\R$
is a $C^\infty $ function and has positive definite Hessian on the fibers.
The Euler-Lagrange equation,
\begin{equation}\label{eq:E-L}
\frac d{dt}\left(\frac{\partial L}{\partial v}(x,\dot x)\right)-\frac{\partial L}{%
\partial v}(x,\dot x)=0 \tag{E-L}
\end{equation}
generates a smooth flow $f_t:TM\rightarrow TM$ defined as follows. Given
$w=(p,v)\in TM,$ denote
$$
x_w:\R\rightarrow M
$$
the solution of (E-L)
%(\ref{eq:E-L})
with initial condition:

\begin{eqnarray*}
x_w(0)&=&p \\
\dot x_w(0)&=&v.
\end{eqnarray*}
%
Now define $f_t:TM\rightarrow TM$ by
$$
f_t(w)=(x_w(t),\dot x_w(t))
$$
Hence every orbit $\gamma (t)$ of this flow can be uniquely written as
$\gamma (t)=(x(t),\dot x(t))$, where $x(t)$ is a solution of (E-L).
%(\ref{eq:E-L}).

It is well known that solutions of (E-L),
%(\ref{eq:E-L}),
and trough them, orbits of the
flow $f_t:TM\rightarrow TM$, are characterized by local variational
properties.

Here we shall revisit an old subject: orbits of the flow $f_t$, selected in the
intrincate phase portrait of $f_t$, by requiring of them to satisfy global
variational properties instead of the local ones that every orbit satisfies.
Research on these special orbits goes back to Morse ([Mo], 1924) and Hedlund
([H], 1932), and has recently reappeared in the works of Bangert ([B]) and
Mather ([Ma1], [Ma2]). Our approach while visibly conceptually indebted to
those works, will be independent and selfcontained leading to new results
and also to stronger forms of already known ones, like Mather's Graph
Theorem ([Ma]) or the coboundary property ([M]).

Recall that the action of the Lagrangian $L$ on an absolutely continuous
curve $x:[a,b]\rightarrow M$ is defined by
$$
S_L(x)=\int_a^b\!L(x(t),\dot x(t))\,dt.
$$
Given two points $p_i\in M$, $i=1,2$ denote $Ac(p_1,p_2)$ the set of
absolutely continuous curves $x:[0,T]\rightarrow M$ , with
$x(0)=p_1$, $x(T)=p_2$. For each $k\in\R$ we define the {\it action potential}
$\Phi_k:M\times M\rightarrow\R$ by
$$
\Phi _k(p_1,p_2)=\inf \{S_{L+k}(x)\mid x\in Ac(p_1,p_2)\}.
$$

\vspace{13pt}

\noindent{\bf Theorem I. }
{\it
There exists $c(L)\in R$ (called the {\it critical value} of $L$)
such that:

%\begin{enumerate}
%\renewcommand{\theenumi}{\alph{enumi}}
\begin{itemize}
\item[a)] If $k<c(L)$, then $\Phi_k(p_1,p_2)=-\infty$, $\forall p_1,p_2\in M$.

\item[b)] If $k\geq c(L)$, then $\Phi_k(p_1,p_2)>-\infty$, $\forall p_1,p_2\in M$,
and $\Phi_k$ is a Lipschitz function.

\item[c)] If $k\geq c(L)$, then
\begin{alignat*}{2}
&\Phi_k(p_1,p_3)\leq \Phi_k(p_1,p_2)+\Phi_k(p_2,p_3),&\qquad &\forall  p_1,p_2,p_3\in M,
\\
&\Phi_k(p_1,p_2)+\Phi_k(p_2,p_1)\geq 0,&\quad &\forall p_1,p_2\in M.
\end{alignat*}
\item[d)] If $k>c(L)$, then $\Phi_k(p_1,p_2)+\Phi_k(p_2,p_1)>0$, $\forall
p_1\neq p_2$ in $M$.
\end{itemize}
}

\vspace{13pt}

Defining $d_k:M\times M\rightarrow\R$ by $d_k(p_1,p_2)=\Phi_k(p_1,p_2)+\Phi
_k(p_2,p_1),$ the properties above say that $d_k(\cdot,\cdot)$ is a metric
for $k>c(L)$ and a pseudometric for $k=c(L)$.


Denote ${\mathcal M}(L)$ the set of invariant probabilities of the flow $f_t$.

\vspace{13pt}

\noindent{\bf Theorem II. (Ergodic Determination of $c(L)$). }
$$
c(L)=-\min \{\int\!L\,d\mu\mid\mu\in{\mathcal M}(L)\}.
$$

\vspace{13pt}

\noindent{\bf Definition. }{\sl We say that $\mu\in{\mathcal M}(L)$ is a {\it minimizing measure} if
$$
\int\!L\,d\mu =-c(L).
$$
We will denote by $\widehat{\mathcal M}(L)$ the set of the minimizing measures in
${\mathcal M}(L)$
}

\vspace{13pt}

Generically, in the sense of [M], the structure of $\widehat{\mathcal M}(L)$ is simple.

\vspace{13pt}

\noindent{\bf Theorem III. }
{\it
For a generic $L$, $\widehat{\mathcal M}(L)$ contains a single measure
and this measure is uniquely ergodic. When this measure is supported by a
periodic orbit, this orbit is hyperbolic.
}

\vspace{13pt}

But we can hope even more:

\vspace{13pt}

\noindent{\bf Conjecture.  }
{\it For a {\it generic }$L$, $\widehat{\mathcal M}(L)$ consist of a single
measure {\it supported in a periodic orbit}
}

\vspace{13pt}

The prerequisite of the next definition is this remark: For every absolutely
continuous $x:[a,b]\rightarrow M$ and all $k\geq c(L)$:
\begin{equation}\label{eq:*}
S_{L+k}(x)\geq\Phi_k(x(a),x(b))\geq -\Phi_k(x(b),x(a)). \tag{$*$}
\end{equation}

\vspace{13pt}

\noindent{\bf Definition. }{\sl
Set $c=c(L)$. We say that $x:[a,b]\rightarrow M$ is a {\it %
semistatic} curve if it is absolutely continuous and:
\begin{equation}\label{eq:1}
S_{L+c}(x|_{[t_0,t_1]})=\Phi_c(x(t_0),x(t_1)),
\end{equation}
for all $a<t_0\leq t_1<b$; and that is a {\it static} curve if
\begin{equation}\label{eq:2}
S_{L+c}(x|_{[t_0,t_1]})=-\Phi_c(x(t_1),x(t_0))
\end{equation}
for all $a<t_0\leq t_1<b.$
}

\vspace{13pt}

By ($*$),
%(\ref{eq:*}),
 equality (\ref{eq:2}) implies (\ref{eq:1}). Hence static curves
are semistatic.
Semistatic curves are solutions of (E-L).
%(\ref{eq:E-L}).
This follows from classic
results that grant that absolutely continuous curves with much weaker
variational properties are solutions.

\vspace{13pt}

\noindent{\bf Definition.}
\begin{align*}
\Sigma (L)&=\{w\in TM\mid x_w:\R\rightarrow M\text{ is semistatic}\}, \\
\widehat\Sigma (L)&=\{w\in TM\mid x_w:\R\rightarrow M\text{ is static}\}, \\
\Sigma^+(L)&=\{w\in TM\mid x_w|_{[0,\infty )}\text{ is semistatic }\}.
\end{align*}

\vspace{13pt}

\noindent{\bf Remarks.}
%\begin{enumerate}
%\renewcommand{\theenumi}{\alph{enumi})}
\begin{itemize}
\item[a)] Replacing $c$ by any other real number in the definition of semistatic
solution the set $\Sigma^+(L)$ (and then $\widehat \Sigma (L)\subset \Sigma
(L)\subset \Sigma^+(L)$) becomes empty.

\item[b)] In [Ma2], $\Sigma$ denotes what in our setting would be the closure of
the union of the supports of the minimizing measures. This set is in general
much smaller than the set called $\widehat\Sigma (L)$.
\end{itemize}
%\end{enumerate}

\vspace{13pt}

\noindent{\bf Theorem IV. (Characterization of Minimizing Measures).}
{\it

A measure $\mu\in{\mathcal M}(L)$ is minimizing if and only if
$\supp(\mu)\subset\widehat\Sigma (L)$.
}

\vspace{13pt}

\noindent{\bf Theorem V. (Recurrence Properties).}
{\it
%\begin{enumerate}
%\renewcommand{\theenumi}{\alph{enumi}}
\begin{itemize}
\item[a)] $\Sigma (L)$ is chain transitive.
\item[b)] $\widehat\Sigma (L)$ is chain recurrent.
\item[c)] The $\omega$-limit set of a semistatic orbit is contained in
$\widehat\Sigma (L)$.
\end{itemize}
%\end{enumerate}
}

\vspace{13pt}

\noindent{\bf Theorem VI. (Graph Properties).}
{\it
\begin{itemize}
\item[a)] If $\gamma(t)$, $t\geq 0$ is an orbit in $\Sigma^+(L)$ then denoting
          $\pi:TM\rightarrow M$ the canonical projection, the map
          $\pi\mid_{\{\gamma(t)\mid t\geq 0\}}$ is 
          {\it injective with Lipschitz inverse}.
\item[b)] Denoting $\Sigma_0\subset M$ the projection of 
          $\widehat\Sigma (L)$, for
          every $p\in\Sigma_0(L)$ there exists a {\it unique } 
          $\zeta(p)\in T_pM$ such that
          $$
          (p,\zeta (p))\in\Sigma^+(L).
          $$
          Moreover
          $$
          (p,\zeta (p))\in\widehat\Sigma (L),
          $$
          and the vectorfield $\zeta$ is Lipschitz. Obviously
          $$
          \widehat\Sigma (L)=\graph(\zeta).
          $$
\end{itemize}
}

\vspace{13pt}

The following result will imply the {\it covering property}:
$$
\pi\Sigma^+(L)=M,
$$
while also dealing with the {\it injectivity of} $\pi$ {\it on certain subsets}
of $\Sigma^+(L)$.

Define the pseudometric $d_c(\cdot,\cdot)$ on $M$ by
$$
d_c(a,b)=\Phi_c(a,b)+\Phi_c(b,a).
$$
Denote by ${\mathcal G}$ these set of equivalence classes of the equivalence
relation $d_c(a,b)=0$ in $\widehat\Sigma(L)$. If $\Gamma\in{\mathcal G}$ set
$$
\Gamma^+=\{w\in\Sigma^+(L)\mid\omega(w)\subset\Gamma\}.
$$
Obviously $\Gamma^+$ is forward invariant. Set:
$$
\Gamma_0^+=\bigcup_{t>0}\pi f_t\Gamma^+.
$$

\vspace{13pt}

\noindent{\bf Theorem VII. (Covering Property). }
{\it If $\Gamma \in {\mathcal G}$
\begin{itemize}
\item[a)] $\pi\Gamma^+=M$.
\item[b)] For all $p\in\Gamma_0^+$, there exists a {\it unique}
          $\zeta_\Gamma(p)\in T_pM$ such that
          $$
          (p,\zeta_\Gamma(p))\in\Gamma^+.
          $$
          Moreover $\Gamma_0^+$ is an open and dense subset of $M$ and
          $\zeta_\Gamma$ is Lipschitz.
\end{itemize}
}

\vspace{13pt}

%\newpage

\noindent{\bf Remarks.}
\begin{itemize}
\item[a)] The solutions of
$$
\dot x=\zeta_\Gamma(x),
$$
are defined in $[0,\infty)$ and are semistatic curves.
\item[b)] On $\widehat\Sigma (L)\cap\Gamma$, we have $\zeta_\Gamma=\zeta$.
\end{itemize}

\vspace{13pt}

The next result is an stronger form of Theorem III.

\vspace{13pt}

\noindent{\bf Theorem VIII. (Generic Structure of $\Sigma(L)$).}
{\it

For a generic Lagrangian $L$, $\widehat\Sigma(L)$ is a uniquely ergodic set,
if it is a periodic orbit, it is a hyperbolic periodic orbit.
}

\vspace{13pt}

Now we can state the extension to all $\widehat\Sigma(L)$ of a property proved
in [M] for supports of ergodic minimizing measures.

\vspace{13pt}

\noindent{\bf Theorem IX. (Coboundary Property).}{\it

If $c=c(L)$, then $(L+c)\mid_{\widehat\Sigma(L)}$ is a Lipschitz 
coboundary. More precisely, taking any $p\in M$ and defining 
$G:\widehat\Sigma(L)\rightarrow\R$ by
$$
G(w)=\Phi_c(p,\pi(w)),
$$
then
$$
(L+c)\mid_{\widehat\Sigma(L)}=\frac{dG}{df},
$$
where
$$
\frac{dG}{df}(w):=\lim_{h\rightarrow 0}\frac 1h(G(f_h(w))-g(w)).
$$
}

\vspace{13pt}

Exploiting that the energy, $E:TM\rightarrow\R$, defined as usual by
$$
E(x,v)=\frac{\partial L}{\partial v}(x,v)v-L(x,v)
$$
is a first integral of the flow generated
by $L$, leads to the {\it information on the position of} $\Sigma^+(L)$. First
observe that it is easy to check that a semistatic curve
$x:[a,b]\rightarrow M$ satisfies:
\begin{equation}\label{eq:**}
E(x(t),\dot x(t))=c. \tag{**}
\end{equation}
This follows from calculating the derivate at $\lambda=1$ of the function
$F:\R\rightarrow\R$ given by
$$
F(\lambda)=\int_a^b(L+c)(x_\lambda(t),\dot x_\lambda(t))\,dt,
$$
where $x_\lambda:[a,b]\rightarrow M$ is given by
$x_\lambda(t)=x(\lambda t)$.

\,    From ($**$)
%(\ref{eq:**})
follows that
$$
\Sigma(L)\subset E^{-1}(c),
$$
that together with $\pi\Sigma(L)=M$ implies:
$$
\pi E^{-1}(c)=M.
$$
Hence,
$$
c\geq\max_qE(q,0).
$$
Moreover $\widehat\Sigma(L)\subset E^{-1}(c)$ implies:

\vspace{13pt}

\noindent{\bf Corollary. }{\it
$\mu\in{\mathcal M}(L)$ is minimizing if and only if
$$
\int\frac{\partial L}{\partial v}(x,v)\,v\,d\mu=0,
$$
$$\supp(\mu)\subset E^{-1}(c(L)).
$$
}

\vspace{13pt}

\,  From the view point of the variational calculus, the relevance of the
critical value appears in the following results.

\vspace{13pt}

\noindent{\bf Theorem X. }{\it

If $k>c(L)$, for all $a,b\in M$, there exists a solution of (E-L)
%(\ref{eq:E-L})
such
that $x(0)=a$, $x(T)=b$ for some $T\geq 0$, and
$$
S_{L+k}(x\mid_{[0,T]})=\min S_{L+k}(y),
$$
where the minimum is taken over all the absolutely continuous
$y:[0,T_1]\rightarrow M$, $T_1\geq 0$, $y(0)=a$, $y(T_1)=b$. Moreover, the
solution is contained in the energy level $E^{-1}(k)$.
}

\vspace{13pt}

Using that
$$(L+k)(x,v)=\frac{\partial L}{\partial v}(x,v)v,$$
on $E^{-1}(k)$, it follows that:

\vspace{13pt}

\noindent{\bf Corollary.}
{\it
\begin{itemize}
\item[a)] If $k>c(L)$ and $a,b\in M$, there exists a solution of (E-L)
%(\ref{eq:E-L})
such that $x(0)=a$, $x(T)=b$ for some $T\geq 0$, $E(x(t),\dot x(t))=k$
for all $t\in\R$, and
\begin{equation}\label{eq:***}
\int_0^T\frac{\partial L}{\partial v}(x(t),\dot x(t))\dot x\,dt=
\min\int_0^T\frac{\partial L}{\partial v}(y(t),\dot y(t))\dot y\,dt,
\end{equation}
where the minimum is taken over all the absolutely continuous
$y:[0,T_1]\rightarrow M$, $T_1\geq 0$, $y(0)=a$, $y(T_1)=b$, and
$E(y(t),\dot y(t))=k$ for a.e. $t\in [0,T_1]$.
%
\item[b)] Conversely, if given $k>c(L)$, and $a,b\in M$, 
          there exists an absolutely continuous $x:[0,T]\rightarrow M$, 
          with $x(0)=a$, $x(T)=b$, $E(x(t),\dot x(t))=k$ and satisfying 
          the minimization property (\ref{eq:***}), then $x(t)$ is a 
          solution of (\ref{eq:E-L}).
\end{itemize}
}

\vspace{13pt}

An interesting characterization of the critical value, in terms of an
analogous to Tonelli's Theorem ([Ma]) in a prescribed energy level is given
by the following result.

\vspace{13pt}

\noindent{\bf Theorem XI. }{\it

Suppose that $k\in\R$ has the following property: for all $a,b\in\pi E^{-1}(k)$
there exists an absolutely continuous curve $x:[0,T]\rightarrow M$, $T\geq 0$,
such that:
\begin{itemize}
\item[a)] $E(x(t),\dot x(t))=k$ for a.e. $t\in [0,T]$,
\item[b)] $x(0)=a$, $x(T)=b$,
\item[c)] $\displaystyle{\int_0^T
          \frac{\partial L}{\partial v}(x(t),\dot x(t))\dot x(t)
          \,dt=\min \int_0^T\frac{\partial L}{\partial v}
          (y(t),\dot y(t))\dot y(t)\,dt,}$\newline
     where the minimum is taken over all the absolutely continuous
     $y:[0,T_1]\rightarrow M$, $T_1\geq 0$, $y(0)=a$, $y(T_1)=b$, and
     $E(y(t),\dot y(t))=k$ for a.e. $t\in [0,T_1]$.
\end{itemize}
Then $k>c(L)$ and $x(t)$ is a solution of (E-L).
%(\ref{eq:E-L}).
}

\vspace{13pt}

Now let us prove the characterization of the minimizing measures given in
Theorem IV.

Assume $c(L)=0$, define $A_n:TM\rightarrow\R$, $F_i:TM\rightarrow\R$, 
$i=1,2$ by:
\begin{align*}
A_n(\theta)&=S_L(x|_{[0,n]}), \\
F_1(\theta)&=\Phi_0(\pi(\theta),\pi(f_1(\theta))), \\
F_2(\theta)&=\Phi_0(\pi(f_1(\theta)),\pi(\theta)),
\end{align*}
where $x:R\rightarrow M$ is the solution of (E-L)
%(\ref{eq:E-L})
with
$(x(0),\dot x(0))=\theta$.

Then by Birkhoff's Theorem, for all $\mu\in{\mathcal M}(L)$, we have:
$$
\int L\,d\mu =\int A_n\,d\mu,\text{ for all }n,
$$
and
$$
\int F_1\,d\mu\geq 0,
$$
this last property because
\begin{eqnarray*}
\int F_1\,d\mu &=& \int \lim_{n\rightarrow\infty}
         \frac 1n\sum_{j=0}^{n-1}F_1(f_j(\theta))\,d\mu \\
&=&\int\lim_{n\rightarrow\infty}\frac 1n\sum_{j=0}^{n-1}
         \Phi_0(\pi(f_j\theta),\pi(f_{j+1}(\theta))\,d\mu \\
&\geq& \int\lim_{n\rightarrow\infty}\frac 1n\Phi_0(\pi(\theta),\pi
(f_n(\theta))\,d\mu=0.
\end{eqnarray*}
Moreover from the definition of $\Phi_0$
\begin{equation}\label{eq3}
A_1\geq F_1.
\end{equation}

Now suppose that $\mu$ is minimizing, then:
$$
c(L)=0=\int L\,d\mu=\int A_1\,d\mu\geq\int F_1\,d\mu\geq 0.
$$
Hence
$$
\int A_1\,d\mu=\int F_1\,d\mu,
$$
that by (\ref{eq3}), implies $A_1=F_1$ $\mu$-a.e.. By the continuity of
both functions we get
\begin{equation}\label{eq4}
A_1(\theta)=F_1(\theta),\text{ for all } \theta\in\supp(\mu).
\end{equation}
Then, if we prove:
\begin{equation}\label{eq5}
A_1(\theta )=-F_2(\theta),\text{ for all } \theta\in\supp{\mu},
\end{equation}
the prove of $\supp(\mu)\subset\widehat\Sigma (L)$ will be complete. To prove
(\ref{eq5}) we use that
$$
\int L\,d\mu = 0,
$$
implies that for a.e. $\theta$, there exists a sequence $n_j\rightarrow\infty$
such that:
\begin{equation}\label{eq6}
A_{n_j}(\theta)\rightarrow 0,
\end{equation}
\begin{equation}\label{eq7}
d(\theta,f_{n_j}(\theta))\rightarrow 0.
\end{equation}
\, From (\ref{eq7}) follows that
$$
F_2(\theta)\leq\lim_{j\rightarrow +\infty}A_{n_j-1}(f_{n_j-1}(\theta)).
$$
Since $A_{n_j}(\theta)=A_1(\theta)+A_{n_j-1}(f_{n_j-1}(\theta))$ 
and~(\ref{eq6}) we obtain $F_2(\theta)\leq -A_1(\theta)$. 
Using~(\ref{eq4}), we get $F_2(\theta)\leq -F_1(\theta)$. 
Using that $F_2+F_1\geq 0$ (because $d_0(\cdot ,\cdot)$ 
is a pseudometric), we obtain (\ref{eq5}).

Now suppose $\supp(\mu)\subset\widehat\Sigma(L)$. On $\widehat\Sigma(L)$ we
have $A_1=F_1=-F_2$ by definition of $\widehat\Sigma(L)$. Then
$$
\int L\,d\mu =\int A_1\,d\mu =\int F_1\,d\mu \geq 0.
$$
Moreover the same argument used to prove that the $\mu$-average of $F_1$ is
positive can be used to prove that the $\mu$-average of $F_2$ is positive.
Then:
$$
\int L\,d\mu =\int A_1\,d\mu =-\int F_2\,d\mu \leq 0.
$$

\vspace{15pt}

Finally, let us recall a weaker concept of global minimization (taken from
Bangaert [B]) that, as the next three results will show, has many interesting
connections with the forms of global minimization introduced above.

\vspace{13pt}

\noindent{\bf Definition. }{\sl
We say that a solution $x(t)$ of (E-L)
%(\ref{eq:E-L})
is a {\it minimizer} (resp. {\i forward minimizer}) if
$$
S_L(x\mid_{[t_0,t_1]})\leq S_L(y),
$$
for every $t_0\leq t_1$ (resp. $0<t_0\leq t_1$) and every absolutely
continuous $y:[t_0,t_1]\rightarrow M$ with $y(t_i)=x(t_i)$, $i=1,2$.
}

\vspace{13pt}

Denote $\Lambda(L)$ (resp $\Lambda^+(L)$) the set of $(p,v)\in TM$ such
that the solution $x(t)$ of (E-L)
%(\ref{eq:E-L})
with initial condition $(x(0),\dot
x(0))=(p,v)$ is a minimizer (resp. a forward minimizer).

\vspace{13pt}

\noindent{\bf Theorem XII. }{\it
The $\omega$-limit set of an orbit in $\Lambda^+(L)$ is contained in
$\widehat\Sigma(L)$.
}

\vspace{13pt}

\noindent{\bf Theorem XIII. }{\it
$f_t|_{\Lambda(L)}$ is chain transitive.
}

\vspace{13pt}

\noindent{\bf Theorem XIV. }{\it
\begin{itemize}
\item[a)] There exists $C>0$ such that setting $c=c(L)$,
$$
\mid S_L(x\mid_{[t_0,t_1]})\mid\leq C,
$$
for every forward minimizer $x(t)$ and all $0\leq t_0\leq t_1$.
\item[b)] If $x(t)$ is a forward minimizer and $p\in M$ is such that $p=\lim
_{n\rightarrow +\infty}x(t_n)$, for some sequence $t_n\rightarrow +\infty$,
then the limit
$$
\lim_{n\rightarrow +\infty}S_L(x\mid_{[t_0,t_n]}),
$$
exists and only depends on $p$.
\end{itemize}
}

\vspace{13pt}

Now let us recall a device, exploited at length and in a protagonic role in
[B], [Ma1], [Ma2] to enlarge the scope of the methods presented above. It
consists in observing that the Lagrangians $L$ and $L-\theta$, where $\theta$
is a closed 1-form on $M$, generate the {\it same} flow. Then the set of
minimizing measure of $L-\theta$, to be denoted by 
$\widehat{\mathcal M}^\theta(L)$ is contained in 
$\widehat{\mathcal M}(L)$, and the subsets of $TM$ given by $\Sigma
(L-\theta)$, $\widehat\Sigma(L-\theta)$, $\Lambda(L-\theta)$, $\Lambda
^+(L-\theta)$ are invariant sets of $f_t$. All these sets, as well as
the critical value $c(L-\theta)$ of $L-\theta$ {\it depend only on the
cohomology class} $[\theta]\in H^1(M,\R)$ of $\theta$. It is not difficult
to check the convexity of the function $\beta^*:H^1(M,\R)\rightarrow\R$
defined by%
$$
\beta^*([\theta])=c(L-\theta).
$$
(This is the dual of Mather's action function $\beta:H_1(M,\R)\rightarrow\R$).
Define the {\it strict critical value} $c_0(L)$ by
$$
c_0(L)=\min_\theta c(L-\theta).
$$

Using the concept of homology (or asymptotic cycle) of a 
$\mu\in{\mathcal M}(L)$, it can be proved that:
$$
-c_0(L)=\min\{\int L\,d\mu\mid\mu\in{\mathcal M}(L),\;\rho(\mu)=0\}.
$$
This is part of the duality between $\beta^*$ and Mather's action function.

Observe that in an energy level $E^{-1}(c)$ with $c>c_0(L)$, the results of
Theorem X and its Corollary can be applied replacing $L$ by $L-\theta$
where $\theta$ is a closed form satisfaying:
$$
c>c(L-\theta)>c_0(L).
$$
Moreover, observe that
$$
c_0(L)\geq\max_xE(x,0),
$$
because the energy functions of $L$ and $L-\theta$ coincide. The equality holds
when $L$ is a mechanical Lagrangian, i.e. of the form
$$
L(x,v)=\frac 12 \langle v,v\rangle_x-V(x),
$$
where $\langle\cdot,\cdot\rangle_x$ is a Riemannian structure on $M,$ and
$V:M\rightarrow\R$ is a potential. In fact, in this case $c(L)=\max_xE(x,0)$
and the minimizing measures are linear combinations of the Dirac
probabilities concentrated at the maximums of $V$.

But in general the equality doesn't hold. An example will be given after the
following corollary of Theorem X.

\vspace{13pt}

\noindent{\bf Corollary. }{\it
If $k>c_0(L)$, for every free homotopy class of $M$ there exists a periodic
orbit in $E^{-1}(k)$ such that its projection on $M$ belongs to that free
homotopy class.
}

\vspace{13pt}

Now let us exhibit a Lagrangian on the two dimensional torus $T^2=\R^2/\Z^2$
having energy levels  $E^{-1}(k)$, $k\in [a,b]$ with $b>\max_xE(x,0)$, such
that the corollary above doesn't hold in $E^{-1}(k)$; hence
$c_0(L)>b\max_xE(x,0)$.

The example will be a Lagrangian of the form:
$$
L_\lambda (x,v)=\frac 12 \langle v,v\rangle_x
+\lambda\,\psi_1(x_2)\,v_1-\psi_2(x_2),\qquad\lambda\in\R,
$$
where  $\langle\cdot,\cdot\rangle_x$ is the Euclidean 
inner product and $\psi_1,\psi_2$ are bump functions as in the picture.

\centerline{\epsfxsize=7.5cm\epsfbox{manex.eps}}


Then the energy is
$$
E(x,v)=\frac 12\|v\|^2+\psi_2(x_2),
$$
independentely of $\lambda$. When $\lambda>0$ the effect of the term
$\lambda\psi_1(x_2)v_1$ is that of a magnetic field normal to the plane
supported in the band $[0,1]\times\supp(\psi_1)\cong\R/\Z\times\supp(\psi_1)$.
Denote $c=\max\psi_2$ and $[k_1,k_2]=\psi_2^{-1}(c)$. Then
$0<k_1<\frac 12<k_2<1$. Denote $S_\lambda\subset T^2$ the set of points that
can be reached from a point $(x_1,x_2)$ with $x_2=1$, trough a solution of
(E-L)
%(\ref{eq:E-L})
with $L=L_\lambda$, contained in the level $E=c$ (i.e. the
initial velocity satifies $\frac 12\|v\|^2=c$). For $\lambda=0$,
$S_0=T^2-[k_1,k_2]\times\R/\Z$. When $\lambda >0$, $S_\lambda$ diminishes,
say $S_\lambda\subset [k_1(\lambda),k_2(\lambda)]\times\R/\Z$ with
$0<k_1(\lambda)<k_1<k_2<k_2(\lambda)<1$. Then the level $E^{-1}(c)$, for
$\lambda>0$ doesn't contain an orbit whose projection is in the homotopy
class of $(0,x_2)$, $x_2\in\R$, because such orbits would intersect $x_2=1$.

\vspace{13pt}

Since the minimizing measures of $L-\theta$ share the same basic 
properties, it is covenient, following Mather [Ma1], 
to extend the term minimizing measures to all the measures in
$$
{\mathcal M}^*(L)=\bigcup_\theta{\mathcal M}^\theta (L).
$$

Similary it is also convenient to extend the terms semistatic, static, and
minimizing orbits to all the orbits having the corresponding property for
some $L-\theta$. The sets
$$
\Sigma^*(L)=\bigcup_\theta\Sigma(L-\theta), \qquad
\widehat \Sigma ^{*}(L)=\bigcup_\theta\widehat\Sigma(L-\theta) 
\qquad\text{ and }\qquad \Lambda (L)=\bigcup_\theta\Lambda(L-\theta),
$$
are closed, $\widehat\Sigma(L-\theta)$ and $\Lambda(L-\theta)$ are upper
semicontinuous functions of $[\theta]$. From this property follows that:

\vspace{13pt}

\noindent{\bf Corollary. }{\it
$\Sigma^*(L)$ is chain transitive.
}

%\newpage
\bigskip
\bigskip

\centerline{\bf REFERENCES}

\vspace{13pt}
{\small
\begin{itemize}

\item[{\bf [B]}] V. Bangert, {\it Minimal Geodesics}. 
                 Erg. Th. and Dyn. Sys. {\bf 10} (1990), 263--287.

\item[{\bf [C]}] M. J. Carneiro, {\it On the Minimizing Measures of 
                 the Action of Autonomous Lagrangians}. 
                 Preprint ICEx, UFMG (1994).

\item[{\bf [H]}] G. A. Hedlund, {\it Geodesics on a Two Dimensional 
                 Riemannian Manifold With Periodic Coeficients}. 
                 Ann. of Math. {\bf 33} (1932), 719--739.

\item[{\bf [M]}] R. Ma\~n\'e, {\it Generic Properties and 
                 Problems of Minimizing Measures}. Preprint, IMPA (1993).

\item[{\bf [Ma1]}] J. Mather, {\it Action Minimizing Measures for 
                  Positive Lagrangian Systems}. Math. Z. {\bf 207} 
                  (1993) 169--207.

\item[{\bf [Ma2]}] J. Mather, {\it Variational Construction 
                   of Connecting Orbits}. Ann. Inst. Fourier 
                   {\bf 43} (1993) 1349--1386.

\item[{\bf [Mo]}] M. Morse, {\it A Fundamental Class of Geodesics 
                  on any Closed Surface of Genus Greather than one}. 
                  Trans. Amer. Math. Soc. {\bf 26} (1924), 25--60.
\end{itemize}
}
\end{document}





