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% ***********************************************************************
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\begin{document}

\title[Hamiltonians without conjugate points]
      {\large \bf Convex Hamiltonians  \\ without Conjugate Points}
\author[G. Contreras]{Gonzalo Contreras}
\address{Depto. de Matem\'atica. PUC-Rio\\
         R. Marqu\^es de S\~ao Vicente, 225 \\
         22453-900 Rio de Janeiro \\
         Brasil}
\email{gonzalo@@mat.puc-rio.br}


\author[R. Iturriaga]{Renato Iturriaga}
\address{CIMAT  \\
         A.P. 402, 3600 \\
         Guanajuato. Gto. \\
         M\'exico.}
\email{renato@@fractal.cimat.mx}


\thanks{G. Contreras was partially supported by CNPq-Brazil.}
\thanks{R. Iturriaga was partially supported by Conacyt-Mexico,
 grant 0324P-E}


\date{December, 1996.}
%\date{\today}







%\centerline{\Large \bf Convex Hamiltonians without Conjugate Points.}
\maketitle


\bigskip


\noindent
{\large \bf Introduction.}
\bigskip



Let $M$ be a closed  connected riemannian manifold
and $T^*M$ its cotangent bundle. By a convex Hamiltonian
on $T^*M$ we shall mean a $C^3$ function $H:T^*M\to\re$
satisfying the following conditions:
\begin{itemize}
\item[(a)] {\it Convexity:\/} For all $q\in M$, $p\in T^*_q M$, the
           Hessian matrix
           $\frac{\partial^2 H}{\partial p_i\, \partial p_j}(q,p)$
           (calculated with respect to linear coordinates on $T^*_qM$)
           is positive definite.

\item[(b)] {\it Superlinearity:\/}
           $$
           \lim_{\lv p\rv\to\infty} \frac{H(q,p)}{\lv p\rv} = +\infty\,.
           $$
\end{itemize}

The {\it Hamiltonian equation\/} for $H$ is defined as
           \begin{equation}\label{Ham}
           \begin{aligned}
           q' &= H_p \, ,\\
           p' &= -H_q \\
           \end{aligned}
           \end{equation}
           where $H_q$ and $H_p$ are the partial derivatives
           with respect to $q$ and $p$.
Observe that the Hamiltonian function $H$ is constant along
the solutions of~\eqref{Ham}. Its level sets $\Sigma =H^{-1}\{e\}$ are called
{\it energy levels\/} of $H$. Then the compacity
of $M$ and the superlinearity hypothesis imply that the energy
levels are compact. Since the Hamiltonian
vectorfield~\eqref{Ham} is Lipschitz,  the solutions of~\eqref{Ham}
are defined on all $\re$. Denote by $\psi_t$ the corresponding
{\it Hamiltonian flow\/} on $T^*M$.



Our main interest in this paper are {\it disconjugate\/} orbits of the
Hamiltonian flow, i.e. orbits without conjugate points.
Let $\pi: T^*M\to M$ be the canonical projection and define the
{\it vertical subspace\/} on $\theta\in T^*M$ by $V(\theta)= \ker(d\pi)$.
Two points $\theta_1$, $\theta_2\in T^*M$ are said {\it conjugate\/} if
$\theta_2= \psi_{\tau}(\theta_1)$ for some $\tau\ne 0$ and
$d\psi_\tau (V(\theta_1)) \cap V(\theta_2) \ne \{0\}$.

In this work we generalize some results of the theory of geodesics
without conjugate points. A first step is to construct the {\it Green
subspaces} for disconjugate orbits:

\bigskip

\newpage
\noindent{\bf Proposition A.}

{\it Suppose that the orbit of $\theta\in T^*M$ does not contain conjugate
points and $H(\theta)=e$ is a regular value of $H$.
Then there exist two $\vr$-invariant Lagrangian subbundles
$\E$, $\F \subset T(T^*M)$  along the orbit of $\theta$ given by
\begin{align*}
\E(\theta) &= \lim_{t\to +\infty} d\psi_{-t}\bigl(V(\psi_t(\theta))\bigr)\; ,\\
\F(\theta) &= \lim_{t\to +\infty} d\psi_{t} \bigl(V(\psi_{-t}(\theta))\bigr)\;.
\end{align*}
Moreover, $\E(\theta)\cup\F(\theta)\subset T_\theta\Sigma$,
$\E(\theta)\cap V(\theta) = \F(\theta) \cap V(\theta) = \{0\}$,
$\langle X(\theta) \rangle \subseteq \E(\theta)\cap \F(\theta)$
and $\dim \E(\theta) = \dim \F(\theta) = \dim M$, 
where $X(\theta)=(H_p,-H_q)$ is the
Hamiltonian vectorfield and $\Sigma = H^{-1}\{e\}$.
}

\bigskip

These bundles where constructed  for disconjugate geodesics of
riemannian metrics by Green \cite{Green} and of Finsler metrics by
Foulon \cite{Foulon}. In the Finsler case this is done by defining
a new connection $\widetilde{\nabla}$ such that the solutions $\ga$ of
the Euler-Lagrange equation satisfy
$\widetilde{\nabla}_{\dot{\ga}} (\dot{\ga}) \equiv 0$.
This approach leads to extensions of many important theorems in Riemannian
geometry to Finsler geometry.
See Chern \cite{Chern} and the references therein for a survey and an
introduction to this field.

\bigskip

If the energy level $\Sigma$ does not have conjugate points,
the Green bundles $\E$ and $\F$ can be defined on all $\theta\in \Sigma$.
Nevertheless, in general they are neither continuous (cf. \cite{BBB})
nor transversal in
$T\Sigma$ (i.e. $\dim \E\cap\F =1$).

An example of the relationship between the transversality of the Green
subspaces and hyperbolicity appears in the following

\bigskip

\noindent
{\bf Proposition B.}

{\it Let $\Ga$ be a periodic orbit of $\psi_t$ without conjugate points. Then
$\Ga$ is hyperbolic (on its energy level) if and only if $\E(\theta)\cap
\F(\theta)=\langle X(\theta) \rangle$ for some $\theta\in \Ga$, where
$\langle X(\theta)\rangle$ is the 1-dimensional subspace generated
by the Hamiltonian vectorfield $X(\theta)$. In this case $\E$ and $\F$ are
its stable and unstable subspaces.}

\bigskip


We can compare proposition A with the following theorem by
G. Paternain and M. Paternain \cite{Pat1}.

\bigskip

\begin{Theorem}\label{T:Pat} {\bf [Paternain] \cite{Pat1}}

Suppose that $e$ is a regular value of $H$,  and
that $\psi_t\vert_{\Sigma}$ admits a continuous invariant Lagrangian
subbundle $E$. Then,
\begin{itemize}
\item[(a)] $E(\theta)\cap V(\theta)= \{ 0 \} \qquad \forall \theta\in
           \Sigma.$
\item[(b)] $\pi(\Sigma)=M$.
\item[(c)] $\Sigma$ contains no conjugate points.
\end{itemize}
\end{Theorem}

\bigskip

It is not difficult  to see (e.g. \cite{ManheK}) that when the flow
$\psi_t\vert_{\Sigma}$ is Anosov, the stable and unstable
subbundles of $T \Sigma$ are Lagrangian. Paternain's theorem
then implies that Anosov Lagrangian flows do not have conjugate points.
This result was proven for geodesic flows by Klingenberg \cite{Kli}.
An extension of Klingenberg result appears in Ma\~n\'e \cite{ManheK}.


Define
$$
B(\theta) = \bigl\{\xi\in T_\theta\Sigma\,\vert\,
             \sup_{t\in\re} \lv d\psi_t(\theta)\,\xi\rv<+\infty\,\bigr\}\,.
$$
An important property (cf. proposition~\ref{P:B.1}) in the proof of
proposition B is that if the orbit of $\theta$ is disconjugate then
$B(\theta)\subseteq\E(\theta)\cap\F(\theta)$. This in turn implies that
$\langle X(\theta)\rangle\subseteq\E(\theta)\cap\F(\theta)$ and
$\E(\theta)\cup\F(\theta)\subseteq T_\theta\Sigma$ (cf. corollary
\ref{C:B.2}).
The global version of proposition B is given by


\bigskip

\noindent
{\bf Theorem C.}

{\it Let $\Sigma=H^{-1}\{e\}$ be a regular energy level without conjugate
points.

Then the following statements are equivalent:
\begin{itemize}
\item[(a)] $\psi_t\vert_{\Sigma}$ is Anosov.
\item[(b)] For all $\theta\in\Sigma$, $\E(\theta)$ and $\F(\theta)$ are
           transversal in $T_\theta\Sigma$.
\item[(c)] For all $\theta\in \Sigma$, $\E(\theta)\cap \F(\theta)
           =\langle X(\theta)\rangle$.
\item[(d)] If $\theta\in \Sigma$, $v\in T_\theta \Sigma$, $v\notin \langle
           X(\theta)\rangle$ then \quad
           $\sup\limits_{t\in\re} 
           \lv d \psi_t(\theta)\cdot v \rv = + \infty\;.$
\end{itemize}
Where $\langle X(\theta)\rangle \subset T_\theta \Sigma$
is the 1-dimensional subspace generated by the vectorfield
$X(\theta)$ of $\psi$.
}

\bigskip

This theorem was proven by Eberlein (cf. \cite{Eb}) for geodesic flows.
An interesting ingredient on its proof is the characterization of recurrent
quasi-hyperbolic $\re$-actions of vector bundles.
Given a vector bundle $\pi:\EE\to B$ consider a continuous $\re$-action
$\La:\re\to\text{Isom}(\EE,\EE)$ where $\La_t:\EE\to\EE$ is a bundle map which
is a linear isomorphism on each fiber and $\La_{s+t}=\La_s\circ \La_t$. The
action $\La$ induces a continuous flow $\phi_t:B\hookleftarrow$ such that
$\phi_t\circ\pi=\pi\circ\La_t$. We say that $\La$ is {\it quasi-hyperbolic} if
$$
\sup_{t\in\re}\lv \La_t(\xi)\rv = + \infty
\qquad \text{ for all } \xi\in\EE,\; \xi\ne 0\,,
$$
and we say that it is {\it hyperbolic} if there exists a
continuous splitting $\EE=E^s\oplus E^u$ and $C>0$, $\la>0$ such that
\begin{itemize}
\begin{itemize}
\item[(i)] $\lv \La_t(\xi)\rv\le C\, \ee^{-\la t}$ for all
           $t>0$, $\xi\in E^s$.
\item[(ii)] $\lv \La_{-t}(\xi)\rv \le C\, \ee^{-\la t}$ for all
           $t>0$, $\xi\in E^u$.
\end{itemize}
\end{itemize}

It is easy to see that hyperbolic actions are quasi-hyperbolic.
The converse is false by counterexamples in Robinson \cite{Rob}
and Franks \& Robinson \cite{Fra-Ro}.
But it is true with an additional recurrency hypothesis, (that in our
case is always true because the Lagrangian flow preserves the Liouville
measure).

Define the {\it non-wandering} set $\Om(\phi)$ of $\phi$ as the set
of points $b\in B$ such that for every neighbourhood $U$ of
$b$ there exists $T>0$ such that $\phi_T(U)\cap U\ne 0$.
The following theorem is a slight modification of a theorem in
Freire \cite{Freire}, written there for quasi-Anosov flows
(i.e when $\La_t=d\phi_t$ and $\EE$ is a continuous invariant subbundle
of $TB$ transversal to the vectorfield $X$, see theorem~\ref{T:QA}).
For completeness of the
exposition we present a proof of the following theorem in section \S\QHYPERB.

\bigskip

\begin{Theorem}\label{T:Q-A}

Let $\pi:\EE\to B$ be a continuous vector bundle
and $\La:\re\to\text{\rm Isom}(\EE,\EE)$ a
continuous $\re$-action with induced flow $\phi_t$:
$\pi\circ\La_t=\phi_t\circ\pi$.
If $\La$ is quasi-hyperbolic and $\Om(\phi\vert_B)=B$, then $\La$ is
hyperbolic.
\end{Theorem}

\bigskip

The reason to use this version of Freire's result is that
our Hamiltonian flows are not of contact type and a priori
they may not preserve any continuous bundle transversal to
the vectorfield $X(\theta)$. Indeed, in apendix~\EXAMPLES\/
we show an example of a convex Hamiltonian without conjugate
points where this phenomenon occurs. The solution is to study
the behaviour of an (orbitwise) reparametrization of the flow
which preserves a continuous tranversal bundle $\EE\subset T(T^*M)$.
This reparametrization can not be made global but instead we use
the quasi-hyperbolicity of the action
$\La_t(\theta)= P\bigl(\psi_t(\theta)\bigr)\circ d\psi_t(\theta)
\circ P(\theta)$ where $P(\theta):T_\theta T^*M\to \EE(\theta)$ is the
projection along the direction of the vectorfield $X(\theta)$.



\bigskip
\bigskip

\noindent{\bf Generic Lagrangians.}

\bigskip

A $C^4$ function  $L:TM\to\re$ is called a (convex autonomous) {\it Lagrangian} if
it satisfies the following properties:
\begin{itemize}
\item[(a)] {\it Convexity:\/} For all $x\in M$, $v\in T_x M$, the
           Hessian matrix
           $\frac{\partial^2 L}{\partial v_i\, \partial v_j}(x,v)$
           (calculated with respect to linear coordinates on $T_x M$)
           is positive definite.
\item[(b)] {\it Superlinearity:\/}
           $$
           \lim_{\lv v \rv\to \infty} \frac{L(x,v)}{\lv v\rv} = +\infty\,,
           $$
           uniformly on $x\in\re$.
\end{itemize}


Observe that by the compactness of $M$, the {\it Euler-Lagrange\/}
equation, that in local coordinates is:
           \begin{equation}\label{E-L}
           \frac{d\,}{dt}\, \frac{\partial L}{\partial v}(x,\dot{x})
           = \frac{\partial L}{\partial x}(x,\dot{x})\, ,
           \end{equation}
generates a complete flow $\vr:TM\times\re\to TM$ defined by
$$
\vr_t(x_0,v_0) = \bigl( x(t), \dot{x}(t)\bigr),
$$
where $x:\re\to M$ is the maximal solution of~\eqref{E-L} with initial
condition $\dot{x}(0)=v_0$.

A Lagrangian flow is conjugated to a Hamiltonian flow by the Legendre
transform ${\cal F}_L:TM\to T^*M$ defined by
$$
{\cal F}_L(x,v)=\left(x ,\frac{\partial L}{\partial
v}(x,v)\right).
$$
 The corresponding Hamiltonian is given by
$H(q,p)=E({\cal F}_L^{-1}(q,p))$, 
where $E:TM\to\re$ is the {\it energy \/} function defined by
\begin{equation}\label{E:Energy}
E(x,v) := \left.\frac{\partial L}{\partial v}
\right\vert_{(x,v)}\cdot v - L(x,v)\;.
\end{equation}
Conversely, given a convex Hamiltonian $H$ on $T^*M$, the corresponding
convex Lagrangian is given by the Legendre transform of the
Hamiltonian ${\cal F}_H:T^*M\to T^{**}M\approx TM$ which can be seen as
$$
L(x,v) = \max_p \left\{ \, p(v) - H(x,p)\,\right\}
       = (p\, H_p - H)\circ {\cal F}_H \, .
$$





Observe that item (b) in theorem~\ref{T:Pat} implies that the energy
$e$ of an Anosov level for the Lagrangian flow satisfies
$$
e > e_0 := - \max_{p\in M} L(p,0)\, ,
$$
because $e_0$ is the minimal energy level $e$ for which
$\pi(\Sigma)=M$. Define
$$
c(L) := - \min\left\{ \int_{TM} L\, d\mu \; \Big\vert\;
\mu \text{ is a } \vr-\text{invariant probability}\right\}\, .
$$
It is proven in  \cite{Ma3}, \cite{CDI},  that $c(L)\ge e_0$.

We say that a $\vr$-invariant probability $\mu$ is {\it minimizing}
if
$$
c(L) = - \int L \; d\mu\, ,
$$
and we say that it is {\it uniquely ergodic} if it is
the only invariant probability measure supported inside
$\text{supp}(\mu)$.

Action minimizing curves do not contain conjugate points.
This fact is known since Morse earlier works (cf. \cite{Morse}).
We present a simple proof of this fact in section~\S\INDEX\/ and
we use proposition B to prove the following theorem, which was
stated as theorem III in Ma\~n\'e's unfinished work \cite{Ma3}.


\bigskip


\noindent{\bf Theorem D.}

{\it
  There exists a generic set ${\cal O}\subseteq C^\infty(M,\re)$
such that for all $\phi\in{\cal O}$ the Lagrangian $L+\phi$ has a
unique  minimizing measure and this measure is uniquely ergodic.
When this measure is supported on a periodic orbit or a fixed point,
this orbit (point) $\Ga$ is hyperbolic and its stable and unstable manifolds
intersect transversally $W^s(\Ga)\pitchfork W^u(\Ga)$.
}

\bigskip

On Ma\~n\'e \cite{Ma3}, it is conjectured that there exists a
generic set ${\cal O}$ such that this unique minimizing measure
is supported on a periodic orbit or an equilibrium point.

The first statement of theorem D was proved by Ma\~n\'e in \cite{Ma2}. Here we
prove the generic hyperbolicity of the minimizing periodic orbit and
the transversality property $W^s(\Ga)\pitchfork W^u(\Ga)$. The hyperbolicity
of $\Ga$ was announced in Ma\~n\'e \cite{Ma3} and the transversality
$W^s(\Ga)\pitchfork W^u(\Ga)$ was conjectured in Ma\~n\'e \cite{Ma4}.
 We use the Clebsch transformation \cite{Clebsch} to derive formulas
for the index form on an orbit without conjugate points.
Then we use similar arguments to the Rauch comparison theorem
to obtain the transversality of the Green bundles of the
periodic orbit. Finally, proposition B implies the hyperbolicity of the
periodic orbit.


\bigskip
\bigskip

\noindent{\bf The metric entropy.}

\bigskip

The machinery developed for the proof of theorem C can be used to
obtain formulas for the metric entropy of a Hamiltonian
flow without conjugate points. Oseledec's theory \cite{Os}
gives a splitting
$$
T_\theta\Sigma = E^s(\theta) \oplus E^c(\theta) \oplus E^u(\theta)
$$
for a set of points $\theta\in\La\subseteq\Sigma$ of total measure,
such that
\begin{align*}
\lim_{t\to\pm\infty} \tfrac 1t \log\lv d\psi_t(\theta)\,\xi\rv <0
\quad & \text{ for } \quad \xi\in E^s(\theta)\, , \\
\lim_{t\to\pm\infty} \tfrac 1t \log\lv d\psi_t(\theta)\,\xi\rv =0
\quad & \text{ for } \quad \xi\in E^c(\theta)\, , \\
\lim_{t\to\pm\infty} \tfrac 1t \log\lv d\psi_t(\theta)\,\xi\rv >0
\quad & \text{ for } \quad \xi\in E^u(\theta)\, .
\end{align*}
Following Freire and Ma\~n\'e \cite{Fre-Ma}, we see in proposition
\ref{P:G.5} that
\begin{align*}
E^u(\theta)\subseteq\F(\theta)
&\subseteq E^u(\theta) \oplus E^c(\theta)\,,\\
E^s(\theta)\subseteq\E(\theta)
&\subseteq E^s(\theta)@,@,@,@,@,\oplus E^c(\theta).
\end{align*}


Given a riemannian metric on $M$, define the {\it horizontal subspace}
as the kernel of the connection map. Then there exists symmetric linear
isomorphisms $\U :H(\theta)\to V(\theta)$ such that
$\F(\theta)=\graph(\U(\theta))$ for all $\theta\in\Sigma$. It satisfies the
Ricatti equation~\eqref{E:Ricatti}
$$
\dot{\U} + \U\, H_{pp} \,\U + \U \, H_{pq}
+ H_{qp}\, \U + H_{qq} = 0,
$$
where $H_{pp}$, $H_{qp}$, $H_{pq}$ and $H_{qq}$ are linear maps which
coincide with the second derivatives of $H$ in local coordinates,
and $\dot{\U}$ is the covariant derivative of $\U$ defined as
$$
\dot{\U}(\theta)\, v
= \lim_{h\to 0} \frac 1h 
\bigl[ \tau_h\,\U(\psi_h(\theta))\, v - \U(\theta)\, v\bigr]
\in V(\theta) \approx T_\theta^* M,
$$
where $\tau_h:T^*_{\pi\psi_h(\theta)}M\to T^*_{\pi(\theta)} M$ is the
parallel transport.

For $\theta\in\La$ define the unstable Jacobian by
$$
\chi(\theta) := \lim_{T\to +\infty} \tfrac 1T
\log \,\bigl| \det \, d\psi_T\vert_{E^u(\theta)}\bigr|\, .
$$
We prove in proposition~\ref{P:G.5} that
$$
\chi(\theta) = \lim_{T\to + \infty} \frac 1T
\int_0^T \text{\rm tr}\left[ H_{pq}+H_{pp}\,\U\right](\psi_t(\theta))\, dt\,,
$$
where $\text{tr\,}(\;)$ is trace. From Ruelle's inequality
\cite{Ruelle}, for any invariant measure $\mu$ for $\psi_t\vert_\Sigma$, we
obtain that the metric entropy satisfies
$$
h_\mu(\psi\vert_\Sigma) \le \int_\Sigma \chi\, d\mu
\le \int_\Sigma \text{tr}[H_{pq}+H_{pp}\,\U]\, d\mu\, .
$$
Pesin's formula states that for a smooth invariant measure $\mu$ of a
$C^{1+\a}$ diffeomorphism $f:N\hookleftarrow$, we have that
$$
h_\mu(f) = \int_N \chi\, d\mu\, .
$$
In our case we obtain

\medskip
\noindent{\bf Theorem E.}

{\it Let $m$ be the Liouville measure on an energy level $\Sigma$ without
conjugate points. Then
$$
h_m(\psi\vert_\Sigma) = \int_\Sigma \text{tr}[H_{pq}+H_{pp}\,\U]\; dm\,.
$$
}

\smallskip
This theorem was proved for geodesic flows by Freire and Ma\~n\'e
\cite{Fre-Ma}, Finsler metrics by Foulon \cite{Foulon} and
mechanical Lagrangians by Innami \cite{Innami}.

\bigskip
\bigskip

\noindent{\bf The exponential map.}

\bigskip

The quasi-hyperbolicity in the case $\E(\theta)\pitchfork\F(\theta)$ is
obtained from proposition~\ref{P:G.4} which states the horizontal growth of
vertical vectors, i.e.
$$
\lim_{t\to+\infty}\lV d\pi\circ d\psi_t\vert_{V(\theta)}\rV = + \infty.
$$
The bounds obtained in proposition~\ref{P:G.4} are not uniform,
by an example in Ballman, Brin \& Burns \cite{BBB}.
In section~\S\EXP\/ we give a uniform lower bound for
$\lV d\pi\circ d\psi_t\vert_{V(\theta)}\rV$, $\theta\in\Sigma$,
$\lv t\rv \ge 1$. This bound was given by Freire and Ma\~n\'e \cite{Fre-Ma}
in the case of geodesic flows.

Define the exponential map of an energy level $\Sigma$ as
$\exp_q:T^*_q M\to M$, 
$\exp_q(t\theta) = \pi(\psi_t(\theta))$, where 
$\theta\in \Sigma\cap T^*_q M$. 
This map is not differentiable at $\0\in T^*_q M$, but its
Lagrangian version is differentiable with $d\exp_q(\0)=I$. An important
question is wether the lift of this map to the universal cover, 
$\exp_q:T^*_q M\to \tM$,
is a diffeomorphism in an energy level without
conjugate points and, in particular, if the universal cover $\tM$ is
homeomorphic to $\re^n$. We show in corollary~\ref{C:G.8} that if the energy
level has no conjugate points then $\exp_q$ is a
local diffeomorphism.


Let $\Theta=p\, dq$ be the Liouville's 1-form \eqref{E:Theta} in $T^*M$.
Given a regular energy level $\Sigma$ and $q\in\pi(\Sigma)$, let
$\Sigma_q:=(\pi\vert_\Sigma)^{-1}\{q\}$.
Since $\tM$ is simply connected, in order to obtain a global diffeomorphism
$\exp_q:T^*_q M\to\tM$ it is sufficient to show that  $\exp_q$ is a covering
map. For this it is enough the condition $\Theta(X)>0$. Indeed, in
section~\S\EXP\/ we prove the following

\bigskip

\noindent{\bf Theorem F.}

{\it
Let $H:T^*M\to\re$ be convex and $\Sigma=H^{-1}\{e\}$ a regular energy level.
Suppose that $q\in\pi(\Sigma)$ satisfies
\begin{itemize}
\item[(a)] The positive orbit of all $\theta\in\Sigma_q$ has no conjugate
points.
\item[(b)] $\inf\bigl\{\,\lv\Theta\bigl(X(\vartheta)\bigr)\rv \,\big\vert\,
           {\vartheta\in \psi_{\re^+}(\Sigma_q)} \,\bigr\}> 0$.
\end{itemize}
Then the (Lagrangian) exponential map $\exp_q:T^*_q\tM\to\tM$ associated to
$(\psi_t,\Sigma)$ is a diffeomorphism.
}


\bigskip




We give an example (cf. ex.~\EXAMPLES.4)  in apendix~\EXAMPLES\/
in which $\exp_q$ is not surjective and
where all the orbits in $\Sigma$ of $\psi_t$ starting at $\pi^{-1}\{q\}$ have
no conjugate points.
This shows that condition (a) is not sufficient in theorem F.
By theorem X in Ma\~n\'e \cite{Ma3} (see also \cite{CDI}),
this can not happen for high energy levels. Example~1
in apendix~\EXAMPLES\/ also shows that the norm of the
derivative $\Vert d_\theta(\exp_q)\Vert$
may not be bounded below on an energy level without conjugate points.
We don't know if the exponential map is a diffeomorphism when the
whole energy level has  no conjugate points.



The condition $\Theta(X)=p\, H_p >0$ is written
$\Theta(X) = v\, L_v >0$ in Lagrangian form. It is satisfied by
Finsler Lagrangians, $L(x,v) = \frac 12 \lV v\rV_x^2$,
mechanical Lagrangians $L(x,v)=\frac 12 \langle v ,v\rangle_x - \phi(x)$
(with $\Theta(X)=1$, $H(q,p) =\frac 12 \langle p,p \rangle_q + \phi(q)$)
and all convex Hamiltonians on sufficiently high energy levels. It fails
for magnetic Lagrangians $L(x,v)=\frac 12 \langle v , v\rangle_x - \phi(x)
+\eta_x(v)$ (with $d\eta\ne 0$) on lower energy levels. If one adds a
closed 1-form to a Lagrangian, the Euler-Lagrange equation and hence the
Lagrangian flow does not change. This may help to make a given Lagrangian
flow satisfy condition (b).




The obstruction to obtain a bound on
$\Vert d_{\psi_t(\theta)}(\exp_q)\Vert$ is the angle
\linebreak
$\varangle\bigl(X(\psi_t(\theta)), d\psi_t(V(\theta)\cap
T_\theta\Sigma)\bigr)$. If the orbit of $\theta$ is disconjugate this angle is
always nonzero by proposition~\ref{P:G.7}. On Example 3 in apendix~\EXAMPLES\/
this angle reaches zero {\it before} the first conjugate point appear.
Moreover, there are crossings of solutions on $\Sigma$ starting at
$\pi(\theta)$ before the first conjugate point.


\bigskip

On section \S\GREEN\/ we prove proposition A, theorem E and we show the proof
of item (c) in theorem~\ref{T:Pat}.  In section \S\TRANS\/
we study the transversal behaviour of the Hamiltonian flow and prove
proposition B. On section \S\QHYPERB\/ we show the proof of 
theorem~\ref{T:Q-A} and we prove theorem C. On section \S\INDEX\/ 
we give a characterization of the index form and on section~\S\TEOD\/ 
we prove theorem D. On section~\S\EXP\/ we give a uniform lower 
bound for $d\pi\circ d\psi_t\vert_{V(\theta)}$ and prove theorem F.
Finally, we add some examples in the apendix.

We want to thank G. Paternain, M. Paternain and  R.O. Ruggiero for usefull
conversations and J. Sotomayor for pointing us reference \cite{H}.
We also want to thank the hospitality of CMAT, Uruguay; IMPA, Brazil;  
CIMAT, Mexico and PUC-Rio, Brazil.













\stepcounter{section}

%\label{GREEN}

\bigskip
\bigskip


\centerline{\large \bf \thesection. The Green Bundles.}


\bigskip
\medskip

For proofs of the basic results of this section see Abraham \& Marsden
\cite{A-M}.
There is a canonical symplectic structure on $T^*M$ given by a closed
non-degenerate 2-form $\om=d\Theta$, where $\Theta$ is the Liouville's
1-form, defined by
\begin{equation}\label{E:Theta}
\Theta_\theta(\xi) = \theta(d\pi(\theta)\, \xi)\, ,
\end{equation}
where $\pi:T^*M\to M$ is the canonical projection. The Hamiltonian
vectorfield $X$ of a function $H:T^*M\to\re$ is defined by
$\om(X,\cdot)= -d H$.   The Hamiltonian flow $\psi_t$ preserves the function
$H$ and the 2-form $\om$. The level sets of $H$ are called
{\it energy levels}. Given a local chart $q:U\subseteq M\to\re^n$ it
induces a {\it natural chart} $(q,p):T^*U\to\re^n\times\re^n$ of $T^*M$.
In natural charts the Liouville's 1-form is written as $\Theta = p\, dq$.
Hence in natural charts $\om = dp \wedge dq$ and the Hamiltonian equations
are given by
\begin{equation} \label{E: Hamilton}
\dot{q} = H_p
\quad ,\quad
\dot{p} = - H_q \, .
\end{equation}
In general, a set of local coordinates $(q,p)$ of $T^*M$ is called {\it
symplectic} if the canonical 2-form is written as $\om = dp \wedge dq$.
A subspace $E\subset T(T^*M)$ is said {\it Lagrangian} if  $\dim E= n
= \dim M$ and $\om(x,y)=0$ for all $x,\, y \in E$.

Fix a riemannian metric on $M$ and the corresponding induced metric on $T^*M$.
Then $T_\theta T^*M$ splits as a direct sum of two Lagrangian subspaces:
the vertical subspace $V(\theta)= \text{\rm ker\,}(d\pi(\theta))$ and the
horizontal subspace $H(\theta)$ given by the kernel of the connection map.
Using the isomorphism $K:T_{\theta}T^*M\to T_{\pi(\theta)}M\times
T_{\pi(\theta)}^*M$, $\xi\mapsto (d\pi(\theta)\, \xi\,,\,
\nabla_\theta(\pi\, \xi))$, we can identify $H(\theta) \approx
T_{\pi(\theta)}M\times\{0\}$ and $V(\theta)\approx \{0\}\times
T^*_{\pi(\theta)}M \approx T_{\pi(\theta)}M$. If we choose local coordinates
along $t\mapsto \pi\,\psi_t(\theta)$ such that $t\mapsto
\frac{\partial\,}{\partial q_i} (\pi\,\psi_t(\theta))$ are parallel
vectorfields, then this identification becomes
$\xi\leftrightarrow (dq(\xi),\, dp(\xi))$.
Let $E\subset T_\theta T^*M$ be an $n$-dimensional subspace
such that $E\cap V(\theta)= \{ 0\}$. Then $E$ is a graph of some linear map
$S:H(\theta)\to V(\theta)$. It can be checked that $E$ is Lagrangian if and
only if in symplectic coordinates $S$ is symmetric.






\bigskip

\noindent{\bf Proof of Proposition A.}

\medskip

Take $\theta\in T^*M$ and $\xi=(h,v)\in T_\theta T^*M = H(\theta) \oplus
V(\theta) \approx T_{\bpi(\theta)}M\oplus T_{\bpi(\theta)}M$. Consider a
variation
$$
\a_s(t) = \bigl( q_s(t),p_s(t)\bigr)
$$
such that for each $s\in ]-\e,\e[$, $\a_s$ is a solution of the Hamiltonian $H$
such that $\a_0(0)=\theta$ and $\frac{d}{ds}\a_s(\theta)\vert_{s=0}=\xi$. 
Writing $d\phi_t(\xi) = \bigl( h(t), v(t)\bigr)$, we obtain the Hamiltonian
Jacobi equations
\begin{alignat}{2} \label{E:G.Jacobi}
&\dot{h}\, &= \quad H_{pq}\, h &+ H_{pp}\, v \, ,\notag\\
&\dot{v}\, &= - H_{qq}\, h     &- H_{qp}\, v \, ,
\end{alignat}
where the covariant derivatives are evaluated along $\bpi(\a_o(t))$, and
$H_{qq}$, $H_{qp}$, $H_{pp}$ and $H_{qq}$ are linear operators on
$T_{\bpi(\theta)}M$, that in local coordinates coincide with the matrices of
partial derivatives
$\left( \frac{\partial^2 H}{\partial q_i \partial q_j}\right)$,
$\left( \frac{\partial^2 H}{\partial q_i\, \partial p_j}\right)$,
$\left( \frac{\partial^2 H}{\partial p_i\, \partial p_j}\right)$
and
$\left(\frac{ \partial ^2 H}{\partial p_i\,\partial q_j}\right)$.
Moreover, since the Hamiltonian $H$ is convex, then $H_{pp}$ is positive
definite.

We derive now the Ricatti equation. Let $E$ be a Lagrangian subspace of
$T_\theta T^*M$. Suppose that for $t$ in some interval $]-\e,\e[$ we have that
$d\psi_t(E)\cap V(\psi_t(\theta))=\{0\}$. Then we can write
$d\psi_t(E)=\text{graph }S(t)$, where $S(t): H(\psi_t \,\theta) \to V(\psi_t
\,\theta)$ is a symmetric map. That is, if $\xi\in E$ then
$$
d \psi_t(\xi) = \bigl( h(t), S(t) \, h(t) \bigr)\, .
$$
Using equation~\eqref{E:G.Jacobi} we have that
$$
\dot{S} h  + S(H_{pq}h + H_{pp}Sh) = - H_{qq}h - H_{qp}Sh \, .
$$
Since this holds for all $h\in H(\psi_t(\theta))$ 
we obtain the Ricatti equation:
\begin{equation}\label{E:Ricatti}
\dot{S} + S H_{pp}S + S H_{pq} + H_{qp}S + H_{qq} = 0\, .
\end{equation}

Take the symmetric map $S(t)$ for which $d\psi_t(V(\theta)) =
\text{graph}(S(t))$, $t>0$. Let $Y(t) : T_{\bpi(\theta)}M\approx V(\theta)\to
H(\psi_t \theta)\approx T_{\bpi\psi_t \,\theta}M$, $t\in\re$, 
be the family  of isomorphisms 
$Y(t) w = d\bpi\cdot d\psi_t(\theta)\,(0,w)$, where
$(0,w)\in H(\theta)\oplus V(\theta)$. Then for $t>0$, $Y(t)$ satisfies
\begin{align}\label{E:G.JacobiFields}
\dot{Y} = (H_{pq} &+ H_{pp}\, S)\,\, Y \label{E:G.Y}  \\
Y(0)=0\quad &,\quad \dot{Y}(0) = H_{pp} \, , \notag
\end{align}
where $\dot{Y}(0)$ is calculated from~\eqref{E:G.Y} using 
that
$$
\lim_{t\to 0} S(t), Y(t) = I,.
$$ 

Given $u\in T_{\bpi(\theta)}M$, define
\begin{align*}
h_1(t) &= Y(t)\, u \, , \\
v_1(t) &= S(t)\, Y(t)\, u \, , \quad t> 0\, .
\end{align*}
Then $(h_1,v_1)$ is a solution of equation~\eqref{E:G.Jacobi} with
$h_1(0)=0$, $v_1(0) = u$. Let $\xi\in T_\theta
T^*M$ and let $(h(t),v(t))=d\psi_t(\xi)\in H\oplus V$. Since 
$d\psi_t$ preserves the symplectic form, we have that
$$
\langle h(t),v_1(t)\rangle - \langle v(t),h_1(t)\rangle
=\langle h(0), u\rangle
$$
Hence
$$
\left\langle Y^*(t) \, S(t)\, h(t) \, ,\, u\right\rangle
- \left\langle Y^*(t)\, v(t)\, , \, u \right\rangle
=\langle h(0)\, , \, u\rangle
$$
for all $u\in T_{\bpi (\theta)}M$. Therefore
\begin{equation} \label{E:G.5}
v(t) = S(t) \, h(t) - (Y^*(t))^{-1}\, h(0)\, . \label{E:v(t)}
\end{equation}
Since
\begin{equation*}
\dot{h} = H_{pq}\, h + H_{pp}\, v
\end{equation*}
we get
\begin{equation}\label{E:G.6}
\dot{h} = (H_{pq} + H_{pp}S)h - H_{pp} (Y^*)^{-1} h(0)  \label{E:dot(h)}
\end{equation}
Differentiating $Y^{-1}\, h$ and using equation~\eqref{E:G.Y}, we obtain
\begin{equation}\label{E:ache}
h(t) = Y(t)\, Y(c)^{-1}\, h(c)
+ Y(t)\, \int_t^c Y^{-1}(s)\, H_{pp} \,( Y^*(s))^{-1}\, h(0)\,\, ds \,
,\text{ for } 0<t<c.
\end{equation}

Now consider the map $d\psi_t$ restricted to the subspace
$d\psi_{-c}(V(\psi_c\,\theta))$, $c>0$. Its projection $Z_c(t)$ to the
horizontal subspace is given by the solutions of~\eqref{E:G.Jacobi} with
$h(c)=0$. Then $Z_c(t)$ is the family of isomorphisms given by
\begin{equation}\label{E:G.JacobiField}
Z_c(t) = Y(t)\, \int_t^c Y^{-1}(s)\, H_{pp}\, (Y^*(s))^{-1}\,\, ds
\,\, , \,\, \, 0<t<c\, ,
\end{equation}
because $Z_c(t)$ satisfies~\eqref{E:G.Jacobi} for $0<t<c$, $Z_c(c)=0$ and
$$
Z_c(0) = \lim_{t\to 0^+} Z_c(t) = I\, .
$$
Observe that from its definition, the linear map $Z_c(t)$ is defined for all
$t\in\re$ and the no-conjugate points condition implies that $Z_c(t)$ is an
isomorphism for all $t\ne c$. 
Moreover, since $Z_c(t)$ is a matrix solution of
the Jacobi equation~\eqref{E:G.Jacobi}, then  $\dot{Z}_c(0)$ exists, 
and taking the limit in~\eqref{E:G.JacobiField}, we have that
the linear isomorphism
\begin{align}
\dot{Z}_d(0) - \dot{Z}_c(0) &= H_{pp}(\theta)\,
                 \int_c^d Y(s)^{-1} \, H_{pp}\,\, (Y^*(s))^{-1}
       \,\, ds    \label{E:ZetaPunto} \\
\intertext{is symmetric and the second factor on the right is
positive definite for all $0<c<d$. In particular}
Z_d(t) - Z_c(t) &= Y(t)\, H_{pp}(\theta)^{-1}\,
                \left[ \dot{Z}_d(0) - \dot{Z}_c(0) \right]\,\, ,
                 \quad  0<c< d\, .  \label{E:DeltaZeta}  
\end{align}





For $\e>0$ define
\begin{equation}\label{E:B(t)}
Z_{-\e}(t):= Y(t)\, N_c + Z_c(t)\quad , \quad t\in\re\, ,
\end{equation}
where $N_c:=- Y(-\e)^{-1}\, Z_c(-\e)$. We have that
$Z_{-\e}(t)$ is the solution of the Jacobi
equation~\eqref{E:G.Jacobi} satisfying
$Z_{-\e}(-\e)=0$, $Z_{-\e}(0)=I$ and
\begin{equation}\label{E:Bdot}
\dot{Z}_{-\e}(0) - \dot{Z}_c(0)=H_{pp}(\theta)\; N_c\, .
\end{equation}

\begin{Claim}\label{C:NSymmetric}
$Y^{-1}(t)\, Z_c(t)$ is symmetric for all $t\ne 0$. In particular
$N_c$ is symmetric for all $c>0$.
\end{Claim}

\noindent {\it Proof:}
Using~\eqref{E:dot(h)},  we have that
\begin{alignat*}{3}
&\dot{Z}_c &= A\, Z_c \,&+\, &H_{pp} \, (Y^*)^{-1} \, &, \\
&\dot{Y} &= A\, Y \,&+ &0 \;\,\qquad &, \quad t\ne 0 \, , \\
\intertext{where $A= H_{pq}+ H_{pp} \, S$. And then}
&\dot{Z}_{-\e} &= A\, Z_{-\e}\, &+ &H_{pp}(Y^*)^{-1}\, &.
\end{alignat*}
Using~\eqref{E:v(t)} we have that the functions
\begin{equation}\label{E:G.12.5}
\begin{aligned}
h_2(t) &= Z_{-\e}(t)\, u \, ,   \\
v_2(t) &= S(t)\, h_2(t) - \bigl(Y^*(t)\bigr)^{-1}\, u \, ,
\end{aligned}
\end{equation}
are solutions of~\eqref{E:G.Jacobi} with $h_2(-\e) = 0$, $h_2(0)=u$.
 Similarly, the functions
\begin{equation}\label{E:H3}
\begin{aligned}
h_3(t) &= Z_c(t)\, w\, ,  \\
v_3(t) &= S(t)\, h_3(t) - \bigl( Y^*(t)\bigr)^{-1}\, w\, ,
\end{aligned}
\end{equation}
are solutions of~\eqref{E:G.Jacobi} with $h_3(0)=w$, $h_3(c)=0$.

Since the flow preserves the symplectic form, we have that the following
expressions do not depend on $t$ for all $u$, $w\in T_{\bpi(\theta)}M$:
\begin{align*}
\langle\, h_2(t),v_3(t)\,\rangle
&-
\langle\, v_2(t),h_3(t)\,\rangle
\\
\langle\, Z_{-\e} \, u \, , \, (S Z_c - (Y^*)^{-1})\, w\, \rangle
&-
\langle\, (S Z_{-\e} - (Y^*)^{-1})\, u \, ,\, Z_c\, w\, \rangle
\\
Z_c^* S Z_{-\e} - Y^{-1} Z_{-\e} - Z^*_c S  Z_{-\e}
&+ Z^*_c (Y^*)^{-1} = Z^*_c  (Y^*)^{-1} - Y^{-1} Z_{-\e} \, .
\end{align*}
Using formula~\eqref{E:B(t)}, this is equal to
\begin{equation}\label{E:Symmetry}
(Y^{-1}Z_c)^* - Y^{-1}Z_c - N_c = - N_c^*\, ,
\end{equation}
where the right hand side corresponds to its value when $t=-\e$. Since
$Z_c(c)=0$, evaluating the equation at $t=c$, we get that $N_c$ is symmetric
for all $c>0$. Using~\eqref{E:Symmetry} again, we obtain that $Y^{-1}Z_c$ is
symmetric for all $t\ne 0$.

\begin{Claim}\label{C:NPositiveDef} $N_c$ is positive definite for all $c>0$.
\end{Claim}

\noindent {\it Proof:}
It is enough to show it for $N_c^{-1}$. We have that
\begin{equation*}
\frac{d\,\,}{dt} - Z_c(t)^{-1}\, Y(t)\Big\vert_{t=0}
= Z_c^{-1}\, \dot{Z}_c\, Z_c^{-1} \, Y - Z_c^{-1}\,\dot{Y}\big\vert_{t=0}
= - H_{pp} \, .
\end{equation*}
Therefore $-Z_c(t)^{-1} Y(t)$ is positive definite for small $t<0$.
By claim~\ref{C:NSymmetric}, this matrix is symmetric for all $t<0$.
Since for $t<0$ its determinant never vanishes, it is positive definite for all
$t<0$, in particular for $t=-\e$, $N_c^{-1}$ is positive definite.

\qed

\bigskip


Define a partial order on the symmetric isomorphisms of $T_{\bpi(\theta)}M$
by writing
\linebreak
$A\rhd B$ if $H_{pp}(\theta)^{-1}\, (A-B)$ is positive definite.
\, From~\eqref{E:ZetaPunto}
we have that the family ${\cal F} = \{ \dot{Z}_c(0) - \dot{Z}_1(0)\,\vert\,
c>1\,\}$ is monotone increasing. Moreover, by~\eqref{E:Bdot} and claim
\ref{C:NPositiveDef}, we have that
\begin{align*}
\dot{Z}_c(0) - \dot{Z}_1(0)
&\lhd H_{pp}\, N_c + \left( \dot{Z}_c(0) - \dot{Z}_1(0)\right)
\\
&\lhd \dot{Z}_{-\e}(0) - \dot{Z}_1(0)\, .
\end{align*}
So that the family ${\cal F}$ is bounded above. It can be seen that this
implies the existence of a lower upper bound which is the limit of the family.
We obtain the first part of

\begin{Claim}\label{LimitSolution}
\end{Claim}
{\it
\begin{enumerate}

\item[(a)] $\lim_{c\to+\infty} \dot{Z}_c(0) - \dot{Z}_1(0) = Q$
exists and is symmetric.

\item[(b)] $\lim_{c\to+\infty} Z_c(t) = D(t)$ exists (uniformly for bounded $t$
intervals).
\end{enumerate}
Moreover,
\begin{align*}
{\bold h}(t) &= D(t)\, , \\
{\bold v}(t) &= S(t)\, D(t) + \bigl( Y^*(t) \bigr)^{-1}\, , \, t\ne 0
\end{align*}
is a matrix solution of~\eqref{E:G.Jacobi} such that $D(0)=I$, 
$\dot{D}(0) = Q + \dot{Z}_1(0)$ and $\det D(t)\ne 0$ for all $t\in \re$.
}

Part (b) is a consequence of~\eqref{E:DeltaZeta} and 
the continuous dependence of the solutions of 
equation~\eqref{E:G.Jacobi} on the initial data.

We have that the subspace
$$
\E_\theta(0) := \left\{\, \bigl( u,{\bold v}(0)\, u\bigr)\, \big\vert\, u\in
T_{\bpi(\theta)}M\, \right\} \subset H(\theta)\oplus V(\theta)
$$
is the limit of the subspaces
$$
E_c(\theta) := d\psi_{-c}\bigl(V(\psi_c(\theta))\bigr)
= \left\{\, \bigl(h_3(\theta), v_3(\theta)\bigr)\, \big\vert\, u\in
T_{\bpi(\theta)}M\,\right\}
$$
where $h_3(t)$ and $v_3(t)$ are given by~\eqref{E:H3}. Moreover
$$
\E_\theta(t) := d\psi_t\bigl( \E_\theta(0)\bigr)
= \left\{\, \bigl( {\bold h}(t)\, u , {\bold v}(t) \, u\bigr) \, \big\vert\,
u\in T_{\bpi(\theta)}M\, \right\}
$$
satisfies $\E_\theta(t)\cap V(\psi_t\theta)=\{ 0\}$ and
$$
\E_\theta(t) = \lim_{c\to +\infty} d\psi_{t-c}\bigl( V(\psi_c(\theta))\bigr)
=\lim_{d\to +\infty} d \psi_{-d}\bigl( V(\psi_d(\psi_t \theta))\bigr)
= \E_{\psi_t\theta}(0)\, .
$$
Since the vertical subspace is Lagrangian, then $K_\theta(c)$ is Lagrangian.
By the continuity of the symplectic form, the subspaces $\E_\theta(t)$ are
Lagrangian.


To obtain the  ``unstable'' Green bundle $\F(\theta)$, observe that the 
flow of the Hamiltonian $\ov{H}(\theta) = - H(\theta)$ 
is the flow  $\psi_t$ of $H$ with the time reversed. 
In this case $\ov{H}_{pp}$ is negative definite,
but similar arguments apply to obtain the subbundle $\F$.
Applying the lemma to $\ov{H}$, we obtain the subbundle
$\F$. Moreover, in this case the family ${\cal G}=\{\,
\dot{Z}_{-c}(0)-\dot{Z}_{-1}(0)\,\vert\, c>1\,\}$ is
monotone decreasing and bounded below by
$\dot{Z}_\e(0)-\dot{Z}_{-1}(0)$. This finishes the proof of the first part of
proposition A, we defer the proof of the second statement to corollary
\ref{C:B.2}.


\qed




\bigskip
\bigskip

Let
\begin{equation}\label{E:G.U(t)}
\begin{aligned}
\SS(t) : &= \boldsymbol{v}(t)\; \boldsymbol{h}(t)^{-1} \\
       &= S(t) + Y^*(t)^{-1}\, D(t)^{-1}.
\end{aligned}
\end{equation}
Then $\SS(t)$ is a symmetric solution of the Ricatti equation
\eqref{E:Ricatti}, which is defined on all $t\in \re$. 
The symmetry of $\SS(t)$ can be checked either because 
$\E(t)=\text{graph}\,( \SS(t))$ is a Lagrangian subspace or because
$$
Y(t)^{-1}\, D(t) = \lim_{c\to\infty} Y(t)^{-1}\, Z_c(t)
$$
is symmetric and this implies that $(Y^*)^{-1}\, D^{-1}$ is symmetric.


\bigskip


Let $\SS(\theta)$, $\U(\theta)$ be the the symmetric solutions
of the Ricatti equation~\eqref{E:Ricatti} corresponding to the
green bundles $\E(\theta)=\text{Graph}(\SS(\theta))$ and
$\F(\theta)=\text{Graph}(\U(\theta))$. Let
$K_c(\theta):H(\theta)\to V(\theta)$ be the symmetric linear map
such that $\text{Graph}(K_c(\theta))=d\psi_{-c}(V(\psi_c(\theta)))$.
Define a partial order on the symmetric isomorphisms
of $T_{\pi(\theta)}M$ by writing $A\succ B$ if $A-B$ is positive definite.

\bigskip

\begin{Proposition}\label{P:G.1.5}

For all $\e>0$,
\begin{itemize}
\item[(a)] If $d>c>0$ then $K_{-\e} \succ  K_d  \succ  K_c$.
\item[(b)] If $d<c<0$ then $K_\e \prec K_d  \prec K_c$.
\item[(c)] $\lim\limits_{d\to +\infty} K_d = \SS$, \;
           $\lim\limits_{d\to -\infty} K_d = \U$.
\item[(d)] $\SS \preccurlyeq \U.$
\end{itemize}
\end{Proposition}

\medskip

\noindent{\bf Proof:}

Let $(Z_c(t),\, V_c(t))$ be the matrix solution of
the Jacobi equation~\eqref{E:G.Jacobi} with $Z_c(0)=I$, $Z_c(c)=0$, $c>0$.
\, From~\eqref{E:v(t)} we have that
\begin{equation}\label{E:G.P.1}
V_c(t) = S(t) \, Z_c(t) - (Y^*(t))^{-1}.
\end{equation}
Hence, using~\eqref{E:DeltaZeta} and~\eqref{E:ZetaPunto} we have that
\begin{align}
V_d(t) - V_c(t) &= S(t)\, \left[ Z_d(t) - Z_c(t)\right]
\label{E:G.P.1.5} \\
                &= S(t)\, Y(t)\, \int_c^d Y^{-1}\, H_{pp} \, (Y^*)^{-1}\, ds.
\label{E:G.P.2}
\end{align}
When $t\to 0$ we have that $S(t)\, Y(t)\to I$. Therefore
\begin{equation}\label{E:G.021}
V_d(0) - V_c(0) = \int_c^d \bigl( Y^{-1}\, H_{pp}\, (Y^*)^{-1}\bigr)\, ds.
\end{equation}
Let $K_c(t)=V_c(t)\, Z_c(t)^{-1}$ be the corresponding solution of the
Ricatti equation~\eqref{E:Ricatti}. Since $Z_c(0)=I$, we have that
$K_c(0)=V_c(0)$. By~\eqref{E:G.021}, $K_d(0)-K_c(0)$ is positive definite
if $d>c>0$, hence the sequence $K_d(0)$, $d>0$ is strictly increasing.

\, From~\eqref{E:G.12.5} we have that equation~\eqref{E:G.P.1.5} also holds for
$c=-\e$. Using~\eqref{E:B(t)}, we have that
\begin{align*}
V_d(t) - V_{-\e}(t) &= S(t)\, \left( Z_d(t) - Z_{-\e}(t)\right)\\
                   &= S(t)\, (-Y(t)\, N_d)\,.
\end{align*}
Hence $V_d(0)-V_{-\e}(0)=-N_d$ is negative definite.
Since $Z_{-\e}(0)=I$, then $K_{-\e}(0)=V_{-\e}(0)$. Therefore
$K_{-\e}\succ K_d$ for all $d>0$.

This completes the proof of item (a). Item (c) has already been proven
on claim~\ref{LimitSolution} above. Item (b) is proven similarly using the
concave Hamiltonian $\ov{H} = -H$ and item (d) is a corollary of items (a),
(b) and (c).

\qed

\medskip
\begin{Remark}\label{R:G.1.a}
If a bounded open segment $\{\,\psi_t(\theta)\,\vert\, a^-<t<a^+\,\}$ has no
conjugate points, the arguments above apply to obtain limit solutions
$\lim_{c\to a^\pm} Z_c(t) = D^\pm(t)$, $D^\pm(0)=I$, $\det D(t)\ne 0$,
Lagrangian subspaces $E^\pm(\theta)=\lim_{t\to
a^\pm}d\psi_{-t}(V(\psi_t\,\theta))=
d\psi_{a^\pm}\bigl(V(\psi_{a^\pm}(\theta))\bigr)$
and the monotonicity properties of proposition~\ref{P:G.1.5}.
Nevertheless, in general $E^\pm(\theta)\not\subset T_\theta\Sigma$, 
see remark~\ref{R:G.9}
\end{Remark}

\medskip

\begin{Remark}\label{R:G.1.b}
The existence of the Green bundles, the solutions $\boldsymbol{h}$,
$\boldsymbol{v}$, $\U$, $\SS$ and
the monotonicity properties in proposition~\ref{P:G.1.5} do not need the
uniform boundedness of the operators $H_{pp}$, $H_{pq}$, $H_{qq}$. But in the
unbounded case we can not guarantee neither 
that $X(\theta)\in\E(\theta)\cap\F(\theta)$
nor $\E(\theta)\cup\F(\theta)\subset T_\theta\Sigma$.
\end{Remark}

\bigskip

\begin{Proposition}\label{P:G.2}
\end{Proposition}

{\it
\begin{itemize}
\item[(a)] Any symmetric solution 
           $S_\theta(t):H(\psi_t(\theta))\to V(\psi_t(\theta))$
           of the Ricatti equation~\eqref{E:Ricatti} which is defined
           on $t>0$ is uniformly bounded on $t\ge 1$, i.e. there exists $A>0$
           such that
           $$
           \lV S_\theta (t)\rV < A \quad\text{ for all }
           \quad \theta\in\Sigma, \; t>1.
           $$
\item[(b)] Any measurable symmetric solution of the Ricatti equation 
           defined on the energy level $\Sigma$ is uniformly bounded.
\end{itemize}
}

\medskip

An inmediate consequence is the following

\begin{Corollary}\label{C:G.3}
\end{Corollary}
{\it\begin{itemize}
\item[(a)] The solution $S_\theta(t)$ of equation~\eqref{E:G.Y}
           given by $\text{\rm Image}(Y_\theta(t)\,,S_\theta(t)\, Y_\theta(t))
           = d\psi_t\, V(\theta)$ is uniformly bounded for $\lv t\rv>1$.
\item[(b)] The solutions $\SS_\theta(t)$, $\U_\theta(t)$ corresponding to the
           Green bundles are uniformly bounded on $\Sigma$.
\end{itemize}
}




\medskip

We shall need the following lemma, whose proof is elementary:

\begin{Lemma}
The function $w(r)=R\, \coth \, (R\, t - d)$, $R>0$, satisfies
\begin{itemize}
\item[(i)]    $\dot{w} + w^2 - R^2  = 0$.
\item[(ii)]   $\dot{w}<0$,
              $w\left(-t +\scriptstyle{\frac dR}\right)=
               - w\left(t+\scriptstyle{\frac dR}\right)$.
\item[(iii)]  $\lim_{t\to +\infty}w(t)=R$, $\lim_{t\to -\infty}w(t)= -R$.
\item[(iv)]   $\lim_{t\to \left(\frac dR\right)^+} w(t) = +\infty$,
              $\lim\limits_{t\to \left(\frac dR\right)^-} w(t) = -\infty$.
\end{itemize}
\end{Lemma}

\medskip

\noindent
{\bf Proof of proposition~\ref{P:G.2}:}

We first prove (a).
By the spectral theorem, it is enough to prove that $x^* \,S_\theta(t)\, x$
is uniformly bounded for $\lV x\rV=1$ and $t>1$. Since $H_{pq}^* = H_{qp}$ and
$S^*=S$, we have that
$$
\dot{S} + S^*\, H_{pp}\, S + H_{pq}^* \, S + S^*\, H_{pq} + H_{qq} = 0.
$$
Let $C:= (H_{pp})^{-1}\, H_{pq}$,\; $D:= H_{qq} - C^*  H_{pp}  C - \dot{C}$
and $V_\theta(t):= S_\theta(t) + \, C(\psi_t(\theta))$. Then
$$
\dot{V} + V^*\, H_{pp}\, V + D = 0 \, .
$$

Since $C(\theta)$ and $D(\theta)$ are continuous, then they are uniformly
bounded on the energy level. Hence it is enough to prove that
$x^* V_\theta(t) \, x$ is uniformly bounded on $t>1$ and $\theta\in\Sigma$.
Observe that $V_\theta(t)$ is not necessarily symmetric. Let
$M>0$ be such that
\begin{equation}\label{E:G.2.1}
y^* H_{pp}(\theta)\, y \ge \frac 1M\, \lV y \rV^2
\qquad \text{for all}\quad \theta\in\Sigma.
\end{equation}
Let $R>0$ be such that
\begin{equation}\label{E:G.2.2}
\lv \, y^* D(\theta)\, y \,\rv < M\, R^2
\qquad \text{for all}\quad \theta\in \Sigma,
\quad \lV y\rV=1.
\end{equation}
We claim that
$$
\lv\,y^* V_\theta(t)\, y \,\rv \le M\, R\, \coth(R)
\qquad
\text{for all}\quad \lV y \rV = 1,
\quad
t>1,
\quad
\theta\in\Sigma.
$$
For, let $\theta_0\in\Sigma$, $\lV x \rV=1$.
Write $V(t):=V_\theta(t)$,
$H_{pp}(t):= H_{pp}(\psi_t\,\theta)$.
Suppose that there exists $t_o\in\re$ such that
$$
x^* V(t_0)\, x =: M\,\a > M\, R\, \coth(R).
$$
There exists $d_0\in\re$ such that $R\, \coth\, (R\, t_0 - d_0) =\a$.
Observe that $d_0>0$ and $t_0> \frac {d_0}R>0$. Write
$w(t):= R\, \coth\, (R\, t - d_0)$. Then $w(t)$ is a
solution of $\dot{w}+w^2-R^2=0$ for $t>\frac{d_0}R$.
In particular,
$$
M\, \dot{w} + M\, w^2 - M\, R^2 = 0.
$$

Let $f(t):= x^*\, V(t)\, x - M\, w(t)$. Then $f(t_0)=0$ and
\begin{equation}\label{E:G.2.3}
f'(t) + \left(x^*\, V(t)^* H_{pp}(t)\, V(t)\, x - M\, w(t)^2\right)
+ \left( x^* D(t)\, x + M\, R^2\right) = 0\, .
\end{equation}
Using Schwartz inequality and~\eqref{E:G.2.1}, we have that
\begin{align*}
M\, w(t_0)^2 &= M\, \a^2 = \frac 1M \, (M\, \a)^2
                = \frac 1M\, \langle x , V(t_0)\, x \rangle^2
\\
             &\le \frac 1M\, \lv V(t_0)\, x\rv^2
                \le \bigl(V(t_0)\,x\bigr)^* 
                H_{pp}(t_0)\, \bigl( V(t_0)\,x\bigr)
\end{align*}
\begin{equation}\label{E:G.2.4}
M\, w(t_0)^2 \le x^*\, V(t_0)^* H_{pp}(t_0)\, V(t_0)\, x
\end{equation}
Then~\eqref{E:G.2.4}, \eqref{E:G.2.2} and~\eqref{E:G.2.3} imply that
$f'(t_0)<0$. The same argument can be applied each time that $f(t)=0$.
Therefore $x^*\, V(t)\, x \le M\, w(t)\text{ for all } t> t_0$ and
$$
x^*\, V(t)\, x \ge M\, w(t)\quad\text{for all } \tfrac{d_0}R<t< t_0.
$$
Then
$$
\lim_{t\to\left(\frac{d_0}R\right)^+} x^* \, V(t)\, x
\ge
\lim_{t\to\left(\frac{d_0}R\right)^+} M\, w(t)
= +\infty\, .
$$
Since $\frac{d_o}R>0$, this contradicts the existence of $V(t)$ for $t>0$.
Hence such $t_0$ does not exists and
$$
x^* \, V(t)\, x \le M \, R\, \coth(R)\qquad \text{ for all } t>0.
$$


Now suppose that there exists $t_1\in\re$, $\lV z\rV=1$ such that
$$
x^*\, V(t_1)\, x < - M\, R\, .
$$
We compare $v(t)= x^*\, V(t)\, x$ with $M\, w_1(t)$, where
$w_1(t)= R\, \coth\, (R\, t- c_0)$ is such that $M\, w_1(t_1)=v(t_1)$.
Observe that $w_1(t)$ is defined for $t<\frac{c_0}{R}$ and in
this case $c_0>0$ and $\frac{c_0}{R}>t_1>0$. The same argument
as above shows that
$v(t)\le M\, w_1(t)$ for $t_1\le t<\frac{c_0}R$. Since
$\lim_{t\to\left(\frac{c_0}R\right)^-}w_1(t)=-\infty$,
this contradicts the fact that $V(t)$ is defined for all $t\in\re$.

The prove of (b) is similar to that of (a). In this case we
can change the $M R \coth(R)$ bound by $M  R$. We allow $t_0\in\re$
and $d_0\in\re$ to be nonpositive  if necessary. The second inequality
has also the same proof, but here we allow $t_1\in\re$ and $c_0\in\re$.
\qed

\bigskip

The following proposition states that
$\lim_{t\to \pm\infty}
\lV d\pi\circ d\psi_t(\theta)\vert_{V(\theta)}\rV = +\infty$
for all $\theta\in\Sigma$. This limit is not uniform in $\theta\in \Sigma$.
A uniform lower bound for this norm is given in section~\S\EXP.

\begin{Proposition}\label{P:G.4}\phantom{nada}

For all $R>0$ there exists $T>T(R,\theta)>0$ such that
$\lv Y_\theta(t)\, v\rv > R\, \lv v\rv$ for all $\lv t\rv>T$
and all $v\in T_\theta M \setminus\{0\}$.
\end{Proposition}

\medskip

\noindent{\bf Proof:}

Let
$$
M(t):=\int_t^\infty Y^{-1}(s)\, H_{pp}(s)\, \left( Y^*(s)\right)^{-1}\; ds\, .
$$
\, From~\eqref{E:G.JacobiField}, using that $D(t)=\lim_{c\to\infty} Z_c(t)$,
we have that
\begin{equation}\label{E:G.20}
D(s) = Y(s) \, M(s) \qquad \text{for } \quad s>0\, .
\end{equation}
Consider the solutions $S(t)$, $\SS(t)$ of the Ricatti equation given on
\eqref{E:G.Y} and~\eqref{E:G.U(t)}. From~\eqref{E:G.U(t)} 
and~\eqref{E:G.20} we have that
$$
\SS(t) - S(t) = Y^*(t)^{-1}\, M(t)^{-1}\, Y(t)^{-1}
\quad \text{for } \quad t>0.
$$
Since the solutions $S(t)$, $\SS(t)$ are defined on all $t>0$, by 
proposition~\ref{P:G.2}, there exist $t_0>0$ 
and $k>0$ such that  $\lV S(t) \rV<k$ and
$\lV \SS(t)\rV<k$ for all $t>t_0$. Then for $\lv x\rv=1$ and $t>t_0$,
$$
\lv\left\langle \, M(t)^{-1}\, Y(t)^{-1}\, x\,,\, Y(t)^{-1}\, x\,
\right\rangle\rv
= \left\vert\left\langle\, S(t)\, x \,,\, x\, \right\rangle\right\vert
+   \left\vert\left\langle\, \SS(t)\, x \,,\, x\, \right\rangle\right\vert
\le 2\, k \, .
$$

Let $\la(t)$ be the largest eigenvalue of $M(t)$, $t>0$.
Since $M(t)$ is positive definite
then $\la(t)>0$ and $\lV M(t)\rV=\la(t)$. Moreover,
$$
2\, k \ge
\left\vert\left\langle\, M(t)^{-1}\, Y(t)^{-1}\, x \,,\,
           Y(t)^{-1}\,x\,\right\rangle\right\vert
\ge \frac 1{\la(t)}\; \left\vert \, Y(t)^{-1}\, x\, \right\vert^2 \; ,
$$
$$
\lv\, Y(t)^{-1}\, x\, \rv \le \sqrt{2\, k}\; \lV M(t)\rV^{\frac 12}
\qquad \text{for all }\quad \lv x\rv =1, \; t>t_0\,.
$$
Then if $\lv v \rv=1$, we have that
$$
\lv Y(t)\, v\rv \ge \frac 1{\lV Y(t)^{-1}\rV}
\ge \frac 1{\sqrt{2\, k}\, \lV M(t)\rV^{\frac 12}}
\qquad \text{for} \quad t>t_0\, .
$$
Since $M(t)\to 0$ when $t\to + \infty$, given $R>0$ there exists
$T>t_0$ such that 
$\left(2\, k\, \lV M(t)\rV\right)^{\frac 12}<\frac 1R$ for all
$t>T$ and hence $\lv Y(t)\, v\,\rv>R$ for all $t>T$ and $\lv v \rv =1$.

 Using the Hamiltonian $\ov{H}:=-H$, we get the result for $t< - T$.
\qed



\bigskip
\bigskip





For $\theta\in TM$, define
$$
B(\theta):= \Big\{ \xi \in T_\theta T^*M\, \Big\vert
\, \sup_{t\in\re}\lv d\psi_t(\theta)\cdot \xi \rv < +\infty\,\Big\}
.
$$

\begin{Proposition}\label{P:B.1}
If the $\psi$-orbit of $\theta\in TM$ does not contain conjugate
points, then $B(\theta)\subseteq \E(\theta)\cap \F(\theta)$.
\end{Proposition}

An example showing that in general
$\E(\theta)\cap \F(\theta) \not\subset B(\theta)$ is given in Ballmann,
Brin \& Burns \cite{BBB}.

\medskip
\noindent{\bf Proof:}
Let $\xi\in B(\theta)$ and let $\zeta_\Gamma\in E_T(\theta):=
d\psi_{-T} \left( {V}\left(\psi_T (\theta) \right) \right)$
be such that $d\pi(\theta)\cdot \zeta_T = d\pi(\theta)\cdot\xi$,
in particular $\xi-\zeta_T\in{V(\theta)}$.
Since $\lim_{T\to+\infty} E_T(\theta) = \E(\theta)$ and
$\E(\theta)\pitchfork {V}(\theta)$ then there exists
$\zeta := \lim_{T\to +\infty} \zeta_T \in \E(\theta)$. Moreover,
$d\pi\scirc d\psi_T(\xi -\zeta_T)= d\pi\scirc d\psi_T(\xi)$.
Since $d\pi\scirc d\psi_T(\xi)$ is bounded on $T>0$, by  proposition
\ref{P:G.4},
$\lim_{T\to +\infty} (\xi - \zeta_T)=0$. 
Since $\zeta_T \to\zeta\in \E(\theta)$, then $\xi\in \E(\theta)$.


Similarly if $\eta_T\in F_T(\theta):= d\psi_T\left(
V\left(\psi_{-T}(\theta)\right)\right)$ is such that
$d\pi(\theta)\cdot \eta_T=d\pi(\theta)\cdot\xi$. Then
$\lim_{T\to +\infty}\eta_T\in \F(\theta)$ and
$\lim_{T\to +\infty}(\xi - \eta_T)=0$.

\qed


\bigskip

\begin{Corollary}\label{C:B.2}
If the orbit of $\theta$ has no conjugate points then
\begin{itemize}
\item[(a)] $\langle X(\theta)\rangle\subseteq \E(\theta)\cap \F(\theta)$.
\item[(b)] $\E(\theta)\cup\F(\theta)\subseteq T_\theta\Sigma$.
\end{itemize}
\end{Corollary}

\medskip

\noindent{\bf Proof:}


The property $\langle X(\theta)\rangle \subseteq \E(\theta)\cap \F(\theta)$ is
a direct consequence of proposition~\ref{P:B.1}. It remains to show that
$\E(\theta)\cup \F(\theta)\subset T_\theta \Sigma$. Observe that since
$-dH = \om(X,\cdot)$, then
$$
T_\theta \Sigma = 
\{\,\xi\in T_\theta T^*M\,\vert\, \om(X(\theta),\xi)=0\,\}\,.
$$
Since $\F(\theta)$ is Lagrangian and $X(\theta)\in\F(\theta)$, then
$\om(X(\theta),\xi)=0$ for all $\xi\in\F(\theta)$. 
Hence $\F(\theta)\subseteq T_\theta \Sigma$. 
Similarly $\E(\theta)\subseteq T_\theta \Sigma$.

\qed

\medskip

\begin{Corollary}\label{C:B.3}
If $M$ is connected and $\Sigma$ is a regular energy level without
conjugate points, then $\pi(\Sigma)=M$.
\end{Corollary}

\medskip

\noindent{\bf Proof:}
For $\theta\in\Sigma$, $d\pi(\theta):\F(\theta)\to T_{\pi(\theta)}M$ is an
isomorphism. By corollary \ref{C:B.2}, $\F(\theta)\subset T_\theta\Sigma$.
Hence $\pi:\Sigma\to M$ is an open map. Since $M$ is connected then
$\pi(\Sigma)=M$.
\qed






\bigskip

\noindent{\bf The metric entropy.}

\bigskip

Let $\La$ be the set of total measure having the Oseledec's
splitting $T_\theta T^*M = E^s(\theta)\oplus E^c(\theta)\oplus E^u(\theta)$.

\bigskip

\begin{Proposition} \label{P:G.5}

For all $\theta\in\La$
\begin{itemize}
\begin{itemize}
\begin{itemize}
\item[(a)] \qquad\qquad $E^s(\theta)\subseteq \E(\theta) 
                \subseteq E^s(\theta) \oplus E^c(\theta)$.
\item[(b)] \qquad\qquad $E^u(\theta)\subseteq \F(\theta) 
                \subseteq E^u(\theta) \oplus E^c(\theta)$.
\item[(c)] \qquad\qquad $\displaystyle{\lim_{T\to +\infty} \frac 1T
            \int_0^T \text{\rm tr }\bigl[ H_{pq} + H_{pp}\,\U\bigr]
            \bigl(\psi_t(\theta)\bigr)\; dt}$.

\end{itemize}
\end{itemize}
\end{itemize}
\end{Proposition}

\medskip
\noindent{\bf Proof:}

We only proof
\begin{equation}\label{E:G.2d-i}
E^u(\theta)\subseteq \F(\theta)\subseteq E^u(\theta) \oplus E^c(\theta)\,,
\end{equation}
which shall be used for theorem E, the other inclusions
are similar.

We first prove the second inclusion.
Observe that for $\ov{v}=(0,v)\in V(\theta)$ we have that
$d\psi_t(\theta)\, \ov{v} = (Y_\theta(t)\, v, *)$. Then, by
proposition~\ref{P:G.4}, we have that $V(\theta)\cap E^s(\theta)=\{0\}$.
For every $\theta\in\La$ there exists a subspace
$R(\theta)\subset E^u(\theta)\oplus E^c(\theta)$ and a linear map
$L(\theta):R(\theta)\to E^s(\theta)$ such that
$V(\theta)=\text{graph\,}L(\theta)$.

Let $\mu$ be an ergodic invariant measure for $\psi_t$.
Given $\e>0$ let $K\subset \La$ be a compact subset such that
$\mu(K)>1-\e$ and
\begin{itemize}
\item[(a)] $\sup \,\bigl\{\,\lV L(\theta)\rV\,\vert\,
                \theta\in K\,\bigr\}< +\infty$.
\item[(b)] There exists $T>0$ and $0<\eta<1<\la<\eta^{-1}$ such that
           for $t>T$ and $\theta\in K$
           $$
           \lV \psi_t\vert_{E^s(\theta)}\rV < \eta^t
           \quad \text{ and } \quad
           \lV d\psi_t^{-1}\vert_{E^u(\theta)\oplus E^c(\theta)}\rV <\la^t
           $$
\end{itemize}

Let $V_{-s}(\theta):= d\psi_s\bigl(V(\psi_{-s}(\theta))\bigr)$. Then
\begin{align*}
V_{-s}(\theta) &= d\psi_s\bigl(\graph L(\psi_{-s}(\theta))\bigr)
               \\
               &= \graph\left[
                  \bigl((d\psi_s\circ L(\psi_{-s}(\theta)\bigr)
                  \circ
                  \left(d\psi_s\vert_{R(\psi_{-s}(\theta)}\right)^{-1}
                  \right]
\end{align*}
For $\theta\in K$ take a sequence $s_n\to +\infty$ such that
$\psi_{-s_n}(\theta)\in K$. Since
$d\psi_{s_n} \bigl(R(\psi_{-s_n}(\theta))\bigr)
\subseteq E^u(\theta)\oplus E^c(\theta)$,
by properties (a) and (b), we have that
$$
\lV d\psi_{s_n}\circ L(\psi_{-s_n}(\theta))\circ
\left(d\psi_{-s_n}\vert_{d\psi_{s_n} R(\psi_{-s_n}(\theta))}\right)\rV
\le \mu^{s_n}\, \la^{s_n}\, \sup_{\theta\in K} \lV L(\theta)\rV.
$$
Since $\mu^{s_n} \la^{s_n}\to 0$ when $n\to +\infty$, we have that
$$
\F(\theta)=\lim_{n\to +\infty}  V_{-s_n} (\theta)
          =\lim_{n\to +\infty} d\psi_{s_n} \bigl(R(\psi_{-s_n}(\theta))\bigr)
          \subseteq E^u(\theta)\oplus E^c(\theta)\,.
$$
Therefore the second inclusion in~\eqref{E:G.2d-i} is satisfied
for $\theta\in K$. Since $\mu(K)\ge 1-\e$ and $\e$ and $\mu$ are
arbitrary, this inclusion holds for a set of total measure in $\La$.


We prove now the first inclusion.
There exist (cf. \cite{A-M} theorem 3.1.19) a continuous riemannian
metric $\langle\,,\,\rangle$ on $T^*M$ and a 
continuous family of linear isomorphisms
$J(\theta):T_\theta T^*M \hookleftarrow$ such that $J(\theta)^2=-I$ and the
symplectic forms is written as $\om(x,y) = \langle x ,J(\theta)y\rangle$
for all $x,y\in T_\theta T^*M$ and $\theta\in T^*M$. From now we use 
this metric.  Let $\{e_1(\theta),\ldots,e_n(\theta)\}$
be an orthonormal basis of $\F(\theta)$ and define
$e_{n+i}(\theta) =Je_i(\theta)$
for $i=1,\ldots,n$. Since the subspace $\F(\theta)$ 
is  Lagrangian, the subspace 
$J\F(\theta)= \text{span\,}\{e_{n+1}(\theta),\ldots,e_{2n}(\theta)\}$ 
is the orthogonal complement of $\F(\theta)$ 
with respect to $\langle\,,\,\rangle$.
The matrices of $\om$ and $d\psi_t(\theta)$ with respect to the family
of basis $\{e_1(\theta),\ldots,e_{2n}(\theta)\}$ are
$$
J = \left[\begin{matrix}
                  &  I    \\
             -I   &
     \end{matrix}\right]
\quad \text{ and } \quad
d\psi_t(\theta) = \left[\begin{matrix}
               A_t(\theta)   &  C_t(\theta)   \\
                             &  B_t(\theta)
     \end{matrix}\right]
$$
Since $d\psi_t$ preserves the symplectic form we have that
$(d\psi_t)^* \, J\, (d\psi_t) = J$. Hence $B^*_t(\theta)= A_t(\theta)^{-1}$
and $B^*_t(\theta)\, C_t(\theta)$ is symmetric.

Since $\F(\theta)\subseteq E^u(\theta)\oplus E^c(\theta)$ then
$$
\lim_{t\to \pm \infty} \frac 1t \log \lV {B^{*}_t}^{-1}(\theta)\, w\rV
= \lim_{t\to \pm \infty} \frac 1t \log \lV A_t(\theta)\, w \rV \ge 0
$$
for all $w\in \re^n$, $\theta\in\La$. This implies that
\begin{equation}\label{E:G.5.1}
\lim_{t\to \pm \infty}\frac 1t \log \lV B^{-1}_t(\theta)\, w \rV \ge 0\,.
\end{equation}

Suppose that $E^u(\theta)\not\subset \F(\theta)$. 
Take $v\in E^u(\theta)\setminus
\F(\theta)$. Then $v = w + z$ with $z\in \F(\theta)$ and $0\ne w\in
J\F(\theta)$. We have that 
$d\psi_{-t}\, v = u_{-t} + \bigl(*,B_t^{-1}(\theta)\, w\bigr)$ with 
$u_{-t}\in\F(\psi_{-t}(\theta))\subseteq E^u(\psi_{-t}(\theta))\oplus
E^c(\psi_{-t}(\theta))$. 
Since $B^{-1}_t(\theta)\, w\in J\F(\psi_{-t}(\theta))$
and $\F$ and $J\F$ are orthogonal, we have that
$\lV d\psi_{-t}(v)\rV \ge \lV B_t^{-1}(\theta)\, w\rV$. Hence from
\eqref{E:G.5.1} we obtain
$$
\lim_{t\to +\infty} \frac 1t \log \lV d\psi_{-t}(\theta)\, v\rV \ge 0\,.
$$
This contradicts the choice $v\in E^u(\theta)$.

\qed


\bigskip

Finaly we prove (c).
Let $\pi_\theta:\F(\theta)\to H(\theta)$ be the restriction of $d\pi$.
Then $\pi_\theta^{-1} (v) = (v,\, \U(\theta)\, v)$ and by corollary
\ref{C:G.3}
\begin{equation}\label{E:G.E.1}
\sup_{\theta\in\Sigma} \lV \pi_\theta^{-1} \rV < +\infty\,.
\end{equation}
Fix $\theta\in\La$, let $Z_\theta(t):V(\theta)\to H(\psi_t(\theta))$
be defined by
$Z_\theta(t)\, v = \pi_{\psi_t(\theta)}\; d\psi_t(\theta)\left(v,
\U(\theta) v\right)$. Then
$$
d\psi_t(\theta)\vert_{\F(\theta)} = 
\pi_{\psi_t(\theta)}^{-1} \circ Z_\theta(t)\circ \pi_\theta
$$
By~\eqref{E:G.2d-i} and~\eqref{E:G.E.1}, we have that
\begin{align*}
\chi (\theta) &= \lim_{T\to +\infty}\frac 1T \log
                      \lv\det d\psi_t(\theta)\vert_{\F(\theta)} \rv
\\
              &= \lim_{T\to +\infty}\frac 1T \log \lv\det Z_\theta(t)\rv\,.
\end{align*}
Since the isomorphisms $H(t)=Z_\theta(t)$, 
$V(t)=\U(\psi_t(\theta))\, Z_\theta(t)$
satisfy the Jacobi equation~\eqref{E:G.Jacobi}, we have that
$$
\dot{Z}_\theta = (H_{pq} + H_{pp}\, \U)\, Z_\theta\,.
$$
Since $\det Z_\theta(t) \ne 0$ for all $t\in \re$, 
then $\det Z_\theta(t)$ has constant sign and
\begin{align*}
\frac{d\,}{dt}\lv \det Z_\theta (t)\rv
            = \frac{d\,}{dt} \det Z_\theta (t)
           &= \text{tr}\left(H_{pq} + H_{pp}\,\U\right)\,
              \lv\det Z_\theta(t)\rv\,,
\\
\frac{d\,}{dt}\log\lv\det Z_\theta(t)\rv
           &=\text{tr}\left[ H_{pq} + H_{pp}\, \U\right]\, ,
\\
\lim_{T\to +\infty}\frac 1T\log\lv\det d\psi_t(\theta)\vert_{\F(\theta)}\rv
           &=\lim_{T\to +\infty} \frac 1T \int_0^T \text{tr}
             \left[ H_{pq} + H_{pp}\,\U\right]\, dt\,.
\end{align*}
\qed



\bigskip
\noindent{\bf The exponential map.}
\bigskip                     


The following proposition proves item (c) in theorem~\ref{T:Pat}.
We present a modern proof that shall be needed in the sequel.

\bigskip

\begin{Proposition}\label{P:G.6}{\bf [Hartman]}\cite{H}
Let $H:T^*M\to\re$ be a convex Hamiltonian. Then
\begin{itemize}
\item[(a)] A half open segment $\{\,\psi_t(\theta)\,\vert\, t\in [0,a[\,\}$,
           $a\in ]0,+\infty[\cup\{+\infty\}$ has no conjugate points if and
           only if there exists a Lagrangian subspace $E\subset T_\theta T^*M$
           such that $d\psi_t(E)\cap V(\psi_t(\theta))=\{0\}$ for all $t\in
           ]0,a[$.
\item[(b)] A closed segment $\{\, \psi_t(\theta)\,\vert\, t\in [0,a]\,\}$ has
           no conjugate points if and only if there exists a Lagrangian
           subspace $E\subset T_\theta T^*M$ such that
           $d\psi_t(E)\cap V(\psi_t(\theta))=\{0\}$ for all $t\in [0,a]$.
\item[(c)] If a closed segment $\{\, \psi_t(\theta)\, \vert\, t\in [0,a]\,\}$
           has no conjugate points, then there exists $\de>0$ such that the
           segment $\{\,\psi_t(\theta)\,\vert\, t\in [-\de,a+\de]\,\}$ has no
           cojugate points.
\end{itemize}
\end{Proposition}

\noindent{\bf Proof:}

{\bf (a)} If the half open segment is disconjugate, then the Lagrangian
          subspace $E=V(\theta)$ satisfies
          $d\psi_t(V(\theta))\cap V(\psi_t(\theta))=\{0\}$.

    Conversely, suppose that such Lagrangian subspace $E$ is given.
Let $E(t):= d\psi_t(E)$. Then the projection 
$d\pi : E(t)\to H(\psi_t (\theta))$ is an isomorphism. 
Let $S(t):H(\psi_t(\theta))\to V(\psi_t(\theta))$ be
the linear isomorphism such that $E(t)=\text{graph }(S(t))$ for $t\in ]0,a[$.
Since $E(t)$ is Lagrangian, then $S(t)$ is symmetric and it satisfies
the Ricatti equation~\eqref{E:Ricatti}. Let $Z(t) : E(t)\to H(\psi_t(\theta))$
be given by $Z(t)\, w = d\pi\, d\psi_t(\theta)\, w$.
Then
\begin{equation}\label{E:G.6.1}
\dot{Z} = (H_{pq} + H_{pp} \, S)\, Z
\quad , \quad
Z(0)=d\pi\vert_E .
\end{equation}
Let $w\in E$ and $(h_1(t),v_1(t)) = d\psi_t(w)\in E(t)$,
then $h_1(t)=Z(t)\,w$, $v_1(t) = S(t)\, Z(t)\, w$.
Let $\xi\in T_\theta T^*M$ and $(h(t),v(t))=d\psi_t(\xi)$. Suppose that
$h(c)=0$, $c\in ]0,a[$. Using these $h_1(t)$, $v_1(t)$,
the same arguments as in~\eqref{E:G.5}, \eqref{E:G.6}, \eqref{E:ache}
give that
\begin{equation}\label{E:G.6.2}
h(t)= Z(t)\, Z(c)^{-1}\, h(c)
+ Z(t)\, \int_c^t Z^{-1}(s) \, H_{pp} \, (Z^*(s))^{-1}
\bigl( Z(c)^* v(c)\bigr)\, ds\, .
\end{equation}
Since $h(c)=0$, then
\begin{equation}\label{E:G.6.3}
\big\langle Z(t)^{-1}\, h(t)\, ,\, Z(c)^* v(c)\big\rangle
= \int_c^t
\big\langle H_{pp}\, (Z(s)^*)^{-1}\, Z(c)^* v(c)
\, ,\,
(Z(s)^*)^{-1}\, Z(c)^* v(c)\big\rangle\, ds \, .
\end{equation}
In particular $h(t)\ne 0$ for all $t\in ]0,a[\setminus\{c\}$.
Hence the segment $]0,a[$ does not have conjugate points.

It remains to prove the case $c=0$. Let $0<b<a$. We prove that $\theta$ is not
conjugate to $\psi_b(\theta)$. We already know that $\psi_b(\theta)$ has no
conjugate points in $\{\,\psi_t(\theta)\,\vert\, 0<t<a\,\}$. For $0<d<b$ let
$E_d=d\psi_{b-d}(V(\psi_d (\theta)))=\graph(K_d)$, where
$K_d:H(\psi_b(\theta))\to V(\psi_b(\theta))$ is the corresponding solution
of the Ricatti equation~\eqref{E:Ricatti}. Let $\e>0$ be such that $b<b+\e<a$
and let $E_{b+\e}=\psi_{-\e}(V(\psi_{b+\e}(\theta)))=\graph(K_{b+\e})$.
By proposition~\ref{P:G.1.5}(b) and remark~\ref{R:G.1.a}, the family $K_d$ is
monotone decreasing on $d$ and has the (lower) bound $K_{b+\e}$. Hence there
exists $\E := \lim_{d\to 0^+}E_d = d\psi_b(V(\theta))$ and $\E\cap
V(\psi_b(\theta))=\{0\}$.

{\bf (c)} If the closed segment $[0,a]$ has no conjugate
points, then $d\psi_t(V(\theta))\cap V(\psi_t(\theta))=\{0\}$ for $0<t\le a$.
Then $d\psi_t(V(\theta))\cap V(\psi_t(\theta))=\{0\}$ for $0<t\le a+\de$ for
some $\de>0$. By item (a), using $E(t)=d\psi_t(V(\theta))$, the segment
$[0,a+\de]$ has no conjugate points. The same argument using
$F(t)=d\psi_{a+\de -t}(V(\psi_{a+\de}(\theta)))$ shows that there exists
$\de_1>0$ such that the segment $[-\de_1,a+\de]$ has no conjugate points.

{\bf (b)} Suppose that the segment $[0,a]$ is disconjugate.
By item (c) there exists $\de>0$ such that the segment $[-\de, a+\de]$ is
disconjugate. Now apply item (a) to the segment $[-\frac{\de}2,a+\de[$.
Conversely, if there is a Lagrangian subspace $E\subset T_\theta T^*M$ with
$d\psi_t(E)\cap V(\psi_t(\theta))=\{0\}$ for all $0\le t \le a$, then the
argument of equation~\eqref{E:G.6.3} applies for all $c\in [0,a]$.

\qed
                     

It may be impossible to find such  Lagrangian subspace $E$ of
theorem~\ref{P:G.6} (a),(b) satisfying $E\subset T_\theta \Sigma$
as explained in remark~\ref{R:G.9}. Compare this with  theorem~\ref{T:Pat}.
{\it If} there exists a {\it continuous}  
invariant Lagrangian bundle $E(\theta)$,
one can always suppose that $E(\theta)\subset T_\theta \Sigma$
by taking
$(E(\theta)\cap T_\theta\Sigma)\oplus \langle X(\theta)\rangle$. 
{\it Then} it must satisfy the transversality condition~\ref{T:Pat}(a).



\begin{Proposition}\label{P:G.7}

For $\theta\in\Sigma$ define $W(\theta):=
(V(\theta)\cap T_\theta \Sigma)\oplus \langle X(\theta)\rangle$.
Suppose that the orbit of $\theta$ has no conjugate points.
Then for all $t\ne 0$,
$d\psi_t\bigl(W(\theta)\bigr)\cap V\bigl(\psi_t(\theta)\bigr)=\{ 0 \}$.
\end{Proposition}

\noindent{\bf Proof:}

Suppose that there exists $b>0$ such that $(0,w_b)\in d\psi_b(W(\theta))\cap
V(\psi_b(\theta))\ne \{ 0\}$. 
Let $(k_0,w_0):= d\psi_{-b}(0,w_b)\in W(\theta)$.
Since the segment $\{\, \psi(\theta)\,\vert\, t\in [0,+\infty[\,\}$ has no
conjugate points, then $k_0\ne 0$. Write $H_p(t):=H_p(\psi_t(\theta))$,
$H_q(t):= H_q(\psi_t(\theta))$. Since $W(\theta) = (V(\theta)\cap T_\theta
\Sigma)\oplus \langle X(\theta)\rangle$ and $X(\theta) = (H_p(0),-H_q(0))$,
then $k_0=\a\, H_p(0)$ for some $\a\ne 0$. Multiplying $w_b$ by $\frac 1\a$,
we can assume that $\a=1$. Let
$$
(h(t),v(t)) := d\psi_t(X(\theta)) - (k_0,v_0))\in d\psi_t(W(\theta))\subseteq
T_{\psi_t(\theta)} \Sigma\,.
$$
Then
\begin{align}
h(0) &= H_p(0)-k_0 = 0 \quad , \quad v(0)\ne 0. \notag \\
h(b) &= d\pi \, [X(\psi_t(\theta)) - (0,w_b)] = H_p(b)\,. \label{E:G.8.1}
\end{align}
Since $(h(0),v(0))\in T_\theta\Sigma$, then
\begin{equation}\label{E:G.8.2}
0 = dH(h(0),v(0)) = H_q \, h(0) + H_p\, v(0) = H_p(0)\cdot v(0)\,.
\end{equation}

Let $\E(t) = \E(\psi_t(\theta))$ be the stable Green bundle of
$\psi_{\re}(\theta)$, let $\SS(t)$ be from~\eqref{E:G.U(t)} and
let $Z(t) := \h(t) : H(\psi_t(\theta))\to V(\psi_t(\theta))$ be from
claim~\ref{LimitSolution}. So that $Z(t)\, w = d\pi\, d\psi_t (w,\SS(t)\, w)$
and $Z(t)$ satisfies~\eqref{E:G.6.1}. Using formula~\eqref{E:G.6.2} for $c=0$,
(with $Z(t)=\h(t)$, $Z(0)=I$ and $h(0)=0$) we obtain that
$$
h(t) = Z(t) \, \int_0^t Z^{-1}(s)\, H_{pp}(s)\, (Z^*(s))^{-1} v(0)\; ds\,.
$$
Hence, since $v(0)\ne 0$, we have that
\begin{equation}\label{E:G.8.3}
\langle Z(b)^{-1} h(b),\, v(0)\rangle
= \int_0^b \langle H_{pp} \, (Z^*(s)^{-1} v(0)), (Z^*(s)^{-1} v(0))\rangle \,
ds > 0 \,.
\end{equation}
Since $X(\theta)\in \E(\theta)$, then
\begin{align*}
(H_p(b), - H_q(b)) &= X(\psi_b(\theta)) = d\psi_b(X(\theta)) =
                      d\psi_b(H_p(0),\SS(0)\, H_p(0)) \\
                   &= (Z(b)\, H_p(0), \SS(b)\cdot Z(b)\, H_p(0))\,.
\end{align*}
\, From the first component we get that $Z(b)^{-1} H_p(b) = H_p(0)$. Using
\eqref{E:G.8.1} we have that $Z(b)^{-1} h(b) = H_p(0)$. 
Replacing this equation on~\eqref{E:G.8.3}, we obtain that
$$
\langle H_p(0),  v(0)\rangle > 0 \,.
$$
This contradicts equation~\eqref{E:G.8.2}.

\qed


\medskip

\begin{Remark}\label{R:G.9}
\quad

The same proof applies for the following statement:
If $H:T^*M\to\re$ is convex and there is a Lagrangian subspace
$E\subset T_\theta \Sigma$ such that $d\psi_t(E)\cap V(\psi_t(\theta))=\{ 0\}$
for $t\in [0,a]$, then
$d\psi_t\bigl(W(\theta)\bigr)\cap V\bigl(\psi_t(\theta)\bigr)=\{ 0 \}$
for all $t\in[0,a]$.

In particular this hypothesis holds if $\{\,\psi_t(\theta)\,\vert\, t\ge
0\,\}$ or $\{\,\psi_t(\theta)\, \vert\, t\le 0\,\}$
have no conjugate points. Just
take the corresponding Green bundles.

The example \EXAMPLES.2 does not satisfy the conclusion of proposition
\ref{P:G.7} on $t\in[0,\pi]$, nevertheless the segment
$\{\,\psi_t(\theta)\,\vert\, t\in [0,\pi]\,\}$ has no conjugate
points. This implies that the Lagrangian subspace given by
proposition~\ref{P:G.6}(b) may not satisfy $E\subset T_\theta \Sigma$.
\end{Remark}


\bigskip

\begin{Corollary}\label{C:G.8}
Let $H:T^*M\to\re$ be convex, $\Sigma = H^{-1}\{e\}$, $q\in\pi(\Sigma)$ and
suppose that the orbits $\{\,\psi_t(\theta)\, \vert\, t\ge 0\,\}$ have
no conjugate points for all $\theta\in\pi^{-1}\{q\}\cap\Sigma$.
Then the exponential map $\exp_q:T^*_q M\to M$ is an inmersion.
\end{Corollary}

\noindent{\bf Proof:}

We have that $\exp_q(t\theta) = \pi\scirc \psi_t(\theta)$ for
$\theta\in\Sigma_q=\pi^{-1}\{q\}\cap \Sigma$. Consider the splitting
$$
T_{t\theta}(T^*_q M)\approx T^*_q M
= T_\theta\bigl(\Sigma_q\bigr) \oplus \langle \theta\rangle
= \bigl(V(\theta)\cap T_\theta\Sigma\bigr) \oplus \langle \theta\rangle\,.
$$
The derivative $d\exp_q(t\theta):T^*_q M\to T_{\pi(\psi_t\,\theta)} M$ is given
by
\begin{align}
d\exp_q(t\theta)\vert_{V(\theta)\cap T_\theta \Sigma}
      &= d\pi\circ d\psi_t(\theta)\vert_{V(\theta)\cap T_\theta\Sigma} \, ,
\label{E:G.C8.0} \\
d\exp_q(t\theta)(\theta)
      &= d\pi\bigl(X(\psi_t(\theta))\bigr)
       = d\pi\circ d\psi_t\bigl(X(\theta)\bigr)\,.
\label{E:G.C8.1}
\end{align}
Therefore
\begin{align}
d\exp_q(t\theta)\bigl(T^*_q M\bigr)
      &= d\pi\circ d\psi_t \bigl(V(\theta)\cap T_\theta \Sigma
          \oplus \langle X(\theta)\rangle \bigr)
\notag\\
      &= d\pi\circ d\psi_t\bigl( W(\theta)\bigr)\,.
\label{E:G.C8.2}
\end{align}
Since $\dim W(\theta)=n$, then $\dim d\psi_t(W(\theta))=n$.
By proposition~\ref{P:G.7} $d\psi_t(W(\theta))\cap V(\psi_t(\theta))=\{0\}$.
Hence $d\pi:d\psi_t(W(\theta))\to T_{\pi(\psi_t\,\theta)}M$ is a linear
isomorphism. From~\eqref{E:G.C8.2} $d\exp_q(t\theta)$ is a linear
isomorphism for all $t>0$, $\theta\in\Sigma_q$.

\qed



\stepcounter{section}

\bigskip
\bigskip

\centerline{\large \bf \thesection. The transversal behaviour.}

\bigskip
\medskip

\def\N{{\boldsymbol{N}}}
\def\cN{{\cal N}}

Given $\theta\in \Sigma$, consider $H_p(\theta)=d\pi\, X(\theta)\ne 0$.
Let $\N(\theta)\subseteq T_\theta\Sigma$ be defined by
\begin{equation}\label{E:T.0}
\N(\theta):= \{\; \xi\in T_\theta\Sigma\,\vert\; \langle \,d\pi\;\xi,\,
d\pi\, X(\theta)\,\rangle_{\pi(\theta)}=0\;\}\,.
\end{equation}
Then $\N(\theta)\oplus\langle X(\theta)\rangle = T_\theta \Sigma$.
Fix $\theta_0\in\Sigma$ and a smooth coordinate system
$(q_1,\ldots,q_n,t)$ of $M\times\re$ along the projection
$(\pi(\psi_t(\theta_0)),t)$ of the orbit of $\theta_0$ such that
\begin{itemize}
\item[(\thesection.a)] $\left.\frac{\partial\,}{\partial q_1}
           \right\vert_{(\pi(\psi_t\theta_0),t)}
           = d\pi\, X(\psi_t(\theta_0))$.
\item[(\thesection.b)] $\frac{\partial\,}{\partial q_2},
           \ldots,\frac{\partial\,}{\partial q_n}$
           is an orthonormal basis for $d\pi\,\N(\psi_t(\theta_0))=
           \langle \tfrac{\partial\;}{\partial q_1}\rangle^\perp$
           along $\pi(\psi_t(\theta_0),t)$.
\end{itemize}

Write $p_i =dq_i\vert_{T_qM}$, $i=1,2,\ldots,n$. From 
(\thesection.a) we have that
$$
1 = \left.\tfrac{d\,}{dt}\, q_1\right\vert_{(\pi(\psi_t\theta_0),\, t)}
  = H_{p_1}(\psi_t (\theta_0)) \ne 0 \,.
$$
Then the equation $H(q_1,q_2,\ldots,q_n;p_1,p_2,\ldots,p_n)=h$ can be solved
locally for $p_1$:
$$
p_1 = -K(Q,P,T;h)\,,
$$
where $P=(p_2,\ldots,p_n)$, $Q=(q_2,\ldots,q_n)$, $T=q_1$. 
This solution can be
extended to a simply connected neighbourhood $W$ of the orbit $\{\,
(\psi_t(\theta_0),\, t)\,\vert\, t\in\re\,\}\subseteq T^*M\times \re$.

\, From now on we omit the $\re$-coordinate in $W\subseteq T^*M\times \re$.
Let $\phi_T$ be the reparametrization of the flow $\psi_t$ on $W$ 
such that it preserves the foliation $q_1=T=\text{constant}$. 
In particular $d\phi_T(\theta_0)$ preserves the transversal 
bundle $\N(\phi_T(\theta_0))$ along
the orbit of $\theta_0$. Define $\tau(T,\theta)$ by
$\phi_T(\theta)=\psi_{\tau(T,\theta)}(\theta)$.
Then $\tau(T,\theta_0)=T$ for all $T\in\re$.
The following reduction appears in Arnold \cite{Ar1}.

\begin{Lemma}\label{L:T.1}
The orbits
$\phi_T(\theta) = (T=q_1, Q(T), p_1(T), P(T); \tau(T,\theta))\in W$
satisfy the equations
$$
\frac{d Q}{dT} = \frac{\partial K}{\partial P}
\qquad , \qquad
\frac{d P}{dT} = -\frac{\partial K}{\partial Q}
\;,
$$
where $Q=(q_2,\ldots,q_n)$, $P=(p_2,\ldots,p_n)$, $T=q_1$
and $K(Q,P;T)$ is defined by
$H(T,q_2,\ldots,q_n;-K,p_2,\ldots,p_n)=h$.
\end{Lemma}


\noindent{\bf Proof:}
Consider the canonical 1-form $\Theta = p\, dq$ on $TM$ and 
the symplectic form $\om = d\Theta= dp\wedge dq$. On $T\Sigma$ we have that
$\om(X(\theta),\cdot)=-dH=0$. Let
$\frac{d\,}{dT} \phi_T(\theta) =
\left( Y(\phi_T(\theta)) ,\, \frac{d\,}{dT} \,\tau(T,\theta)\right)$
be the vectorfield of $\phi_T$. Then $Y(\phi_T(\theta))$ is a multiple
of the vectorfield $X(\phi_T(\theta))$ of $\psi_t$, and then
$\om\bigl(Y(\phi_T(\theta)),\,\cdot\, \bigr) = 0$ on
$TW$. We have
\begin{gather*}
\Theta = p\, dq = P\, dQ - K\, dT , \\
\om\vert_{T\Sigma} = d\Theta\vert_{T\Sigma} = \sum_{i=2}^n
     \left[ dP_i\wedge dQ_i - K_{Q_i}\, dQ_i\wedge dT
      - K_{P_i}\,dP_i\wedge dT\right]
.
\end{gather*}
The matrix of $\om\vert_{T\Sigma}$ is given by
$$
\left[ \begin{matrix}
 0  &  -I  &  -K_Q^*  \\
 I  &   0  &  -K_P^*  \\
 K_Q  &  K_P &  0
\end{matrix} \right]
\;\;
\begin{matrix} \}\scriptstyle{\; Q}\hfill  \\
               \}\scriptstyle{\; P}\hfill  \\
               \}\scriptstyle{\; T=q_1}
\end{matrix}
$$
Then the vector $\left(K_P,-K_Q,1=\frac{dq_1}{dT}\right)=: Z(\phi_T(\theta))$
satisfies $\om(Z(\phi_T(\theta)),\cdot)=0$ on
$\Sigma \approx \{\,(T,Q,p_1,P)\;\vert\; p_1=-K(Q,P,T)\,\}$.
Since the form $\om$ has maximal rank on $\Sigma$, we have that
$Z(\phi_T(\theta))=Y(\phi_T(\theta))$ for $\theta\in W$.
This proves the lemma.

\qed

\bigskip

Since for $\vartheta\in W$,
$\Phi_T(\vartheta)=\psi_{\tau(T,\vartheta)}(\vartheta)$, then
\begin{align}
d\phi_T(\vartheta)\cdot\xi
   &= d\psi_{\tau(T,\vartheta)}\cdot\xi
 +\Big(\left.\frac{\partial\tau}{\partial\vartheta}
   \right\vert_{(T,\vartheta)}\cdot\xi\Big)
   \left.\frac{d\,}{dt}\psi_t(\vartheta)\right\vert_{\tau(T,\vartheta)}
\notag\\
   &= d\psi_{\tau(T,\vartheta)}\cdot\xi + \a(T,\xi)\, X(\phi_T(\vartheta))\;,
\label{E:T.1}
\end{align}
where $\a(T,\xi) = \left.\frac{\partial \tau}{\partial\vartheta}
                   \right\vert_{(T,\vartheta)}\cdot\xi$.

Let $\La(\psi_t(\theta_0)):T_{\psi_t(\theta_0)}\Sigma 
\to \N(\psi_t(\theta_0))$
be the projection along the direction of the vectorfield, i.e.
$$
\La \, \xi = \xi + \be(\xi)\; X(\psi_t(\theta_0))
\text{ with $\be(\xi)\in\re$ such that } \La\, \xi\in \N(\psi_t(\theta_0))\,.
$$

Through the proof of lemma~\ref{L:T.2} all the quantities will be understood
in local coordinates $T=q_1$, $Q=(q_2,\ldots,q_n)$, $P=(p_2,\ldots,p_n)$ of
$W\cap\Sigma$. In particular $T_{\phi_T(\theta_0)}\Sigma =
\re\times\re^{n-1}\times\re^{n-1}$ for all $T\in\re$, with coordinates
$\left(\frac{\partial\,}{\partial T},
\frac{\partial\,}{\partial Q},\frac{\partial\,}{\partial P}\right)$. Since
along the orbit of $\theta_0$ we have that
$$
\dot{q} = (\dot{T},\dot{Q}) = (1,\0_{n-1}) = (H_{p_1},H_P)
\qquad\text{ along } \phi_T(\theta_0)=(T,\0_{n-1},\0_{n-1}),
$$
then the subspace $\N(\phi_T(\theta_0))$ in~\eqref{E:T.0} is
written as $\cN:=\{0\}\times\re^{n-1}\times\re^{n-1}$ in coordinates.
Define $\P:T_\vartheta\Sigma = \re\times\re^{n-1}\times\re^{n-1}\to
\cN=\{0\}\times\re^{n-1}\times\re^{n-1}$ as the projection
$$
\P(h_1,H,V) := (0,H,V).
$$
Using $\beta=-h_1$ on~\eqref{E:T.1} and $H_p=(1,\0)$, 
the projection $\La$ is written as
$$
\La(h_1,H,V) = (0, H, V + h_1\cdot H_Q)\,,
$$
where $H_Q:=(H_{q_2},\ldots,H_{q_n})$. Define the matrizes
\begin{align*}
\H(T) &= \left[\begin{matrix}
H_{pq}  &  H_{pP}  \\
-H_{Qq} & -H_{QP}
\end{matrix}\right]_{(2n-1)\times (2n-1)}
\\
\K(T) &= \left[\begin{matrix}
 \0_{1\times n} & \0_{1\times (n-1)} \\
 K_{Pq}  &   K_{PP}  \\
-K_{Qq}  &  -K_{QP}
\end{matrix}\right]_{(2n-1)\times (2n-1)}
\end{align*}
along the orbit of $\theta_0$.



\bigskip


\begin{Lemma}\label{L:T.2}
Along the orbit $\theta_T:=\phi_T(\theta_0)$ of $\theta_0$, 
we have that
\begin{itemize}
\item[(a)] The matrix $K_{PP}(\theta_T)$ is positive definite.
\item[(b)] For $\xi\in\N(\theta_T)$,
           $\K(\theta_T)\,\xi = \La(\theta_T)\,\H(\theta_T)\,\xi$.
\item[(c)] The operator $\K(\theta_T)\vert_{\N(\theta_T)}$ is
           uniformly bounded on $T\in\re$.
\end{itemize}
\end{Lemma}

\noindent{\bf Proof:}



{\bf (b)} Write $\theta_T = \phi_T(\theta_0)=\psi_T(\theta_0)=(T,\0,\0)$.
          \, From lemma \ref{L:T.1},
          the Jacobi equation for $\phi_T$ is written as
$$
\tfrac{d\,}{dT} \left[ \P\, d\phi_T\,\xi\right]
     = \K\left[\P\, d\phi_T\, \xi \right],
$$
where $d\phi_T\xi =(h_1(T),H(T),V(T))\in\re\times\re^{n-1}\times\re^{n-1}$.
for any $\xi\in T_{\theta_0}\Sigma=\re\times\re^{n-1}\times\re^{n-1}$.
Also, from~\eqref{E:G.Jacobi},
$$
\tfrac{d\,}{dt}\left[ d\psi_t\, \xi \right]
     = \H\left[ d\psi_t\, \xi \right].
$$
Since $d\phi_T(\theta_0)$ preserves tha foliation $q_1=\text{constant}$, then
it preserves the subspace $\cN$. Therefore
$$
\P\,d\phi_T\, \xi = d\phi_T\, \xi = \La\, d\phi_T\,\xi
\quad \text{ for } \xi \in \cN.
$$
In particular, the first coordinate of $d\phi_T\,\xi$ is
$h_1(T)\equiv 0$ and also $\tfrac{d\,}{dT}h_1\equiv 0$. 
Hence if $\xi\in \cN$, then
\begin{align*}
\tfrac{d\,}{dT} \left[ \P\, d\phi_T\,\xi\right]
  &= \tfrac{d\,}{dT} \left[ d\phi_T\,\xi\right]
\\
  &= \tfrac{d\,}{dT} \left[\, d \psi_T\,\xi + \a(T)\, X(\theta_T)\,\right]
\\
  &= \H(d\psi_T\,\xi) + \dot{\a}(T)\, X(\theta_T) + \a(T)\,
             \tfrac{d\,}{dT}\,X(\theta_T)
\\
  &= \H\big[ d\phi_T\,\xi - \a(T)\, X(\theta_T)\big]
      + \dot{\a}(T)\, X(\theta_T) + \a(T)\, \tfrac{d\,}{dT}\, X(\theta_T)
\,,
\end{align*}
where $\a(T)=\a(T,\xi)$ is from \eqref{E:T.1}.

Since $\tfrac{d\,}{dT}X(\theta_T)=\H\, X(\theta_T)$, then
$$
\tfrac{d\,}{dT}
 \left[ \P\, d\phi_T\, \xi\right]
= \H \left[ d\phi_T\, \xi\right]
+ \dot{\a}(T)\, X(\theta_T) \, .
$$
Hence
\begin{equation}\label{E:T.2}
\K\, d\phi_T\, \xi
     = \H\,\left[ d\phi_T\, \xi\right]
          + \dot{\a}(T)\, X(\theta_T)
\quad\text{ for } \xi\in\cN.
\end{equation}
Observe that this equation must hold in all the $2n-1$ coordinates
of $T_\vartheta\Sigma$. 
Since $\text{Image}(\K)\subseteq\cN$, then $\La\,\K=\K$.
So
\begin{align*}
\K(d\phi_T\, \zeta)
     &= \La\,\K(d\phi_T\,\zeta)
        = \La \,[\,\H\, d\phi_T\, \zeta + \dot{\a}(T,\zeta)\, X(\theta_T)\, ]
\\
     &= \La \, \H(d\phi_T\, \zeta) \qquad\text{ for }\zeta\in\cN.
\end{align*}
Since $d\phi_T:\N(\theta_0)=\cN\to \N(\theta_T)=\cN$
is surjective, this proves item (b).


{\bf (a)} Let $V\in\re^{n-1}$, then
$$
V^*\,\K_{PP}\, V = \bigl[(0,V^*),\0_{n-1}\bigr]\,\K\,
\left[\begin{matrix} \0_n \\ V \end{matrix}\right].
$$
Since $\xi=(\0_n,V)\in\cN$, we can use equation~\eqref{E:T.2} to obtain
\begin{align*}
V^*\, K_{pp} \, V
  &= \bigl[(0,V^*),\0_{n-1}\bigr]\,
     \left[\H\, \left[\begin{matrix} \0_n \\ V\end{matrix}\right]
      + \dot{\a}(T,d\phi_{-T}\, \xi)\,
        \left[\begin{matrix} H_p^* \\ -H_Q^* \end{matrix}\right]
     \right]
\\
  &= 0 + V^*\, H_{PP}\, V +\dot{\a} \,\bigl( (0,V^*)\cdot H^*_p\bigr)
\\
  &= V^*\, H_{PP}\, V + \dot{\a}\, \langle (1,\0_{n-1}),(0,V^*)\rangle
\\
  &= V^*\, H_{pp} \, V > 0\,.
\end{align*}


We now prove (c). Since the operator $\H(\vartheta)$ is uniformly bounded on
$\vartheta\in\Sigma$, from (b) it is enough to prove that the projection
$\La$ is bounded on the orbit of $\theta_0$. For, it is enough to see that
the angle $\varangle(X(\theta_T),\N(\theta_T))$ is uniformly bounded
away from 0. Given a vector
$$
\xi = (h,v) = \sum_{i=1}^n
\left( h_i \, \tfrac{\partial\,}{\partial q_i}
     + v_i \, \tfrac{\partial\,}{\partial p_i}
\right) \in \N(\theta_T) \,,
$$
written in our coordinates (\thesection.a), (\thesection.b), 
define the norm $\lv \xi\rv :=\left(\sum_{i=1}^n 
h_i^2 + v_i^2 \right)^{\frac 12}$. Suppose that $\lv\xi\rv=1$.
Since $\xi\in\N(\theta_T)$, then $H_p\cdot h=0$. Moreover,
$H_p=\frac{\partial\,}{\partial q_1}$ and hence
\begin{align*}
\frac{\lv X(\theta_T)\cdot \xi\rv}{\lv X(\theta_T)\rv}
&= \frac{\lv H_p\cdot h - H_q\cdot v\rv}
           {(\lv H_p\rv^2 + \lv H_q\rv^2)^{\frac 12}}
   = \frac{\lv H_q\rv\,\lv v\rv}{\sqrt{1+\lv H_q\rv^2}}
\\
&\le 1\cdot \max_{0\le x\le A}\frac x{\sqrt{1+x^2}}
   = \frac A{\sqrt{1+A^2}}<1 \,,
\end{align*}
where $A:= \sup_{T\in \re} \lv H_q(\theta_T)\rv < +\infty$
with the norm in our coordinates $\tfrac{\partial \,}{\partial p_i}$.

\qed


\bigskip


\begin{Corollary}\label{C:T.3}

Along the orbit $\theta_T=\phi_T(\theta_0)$, we have
\begin{itemize}
\item[(a)] The orbit of $\theta_0$ under $\phi_T$ has no conjugate points.
\item[(b)] Existence of the Green bundles for $d\phi_s\vert_{\N(\theta_T)}$:
\begin{align*}
\E^\top(\theta_T)
    &= \lim_{s\to +\infty}
      d\phi_{-s}\left(V(\theta_{T+s})\cap \N(\theta_{T+s})\right)
\\
\F^\top(\theta_T)
    &= \lim_{s\to -\infty}
      d\phi_{-s}\left(V(\theta_{T+s})\cap \N(\theta_{T+s})\right)
\end{align*}
Moreover, $\E^\top(\theta)=\E(\theta)\cap\N(\theta)$
 and $\F^\top(\theta) = \F(\theta)\cap \N(\theta)$ for all
 $\theta\in \Sigma$.

\item[(c)] Horizontal growth of iterates of vertical vectors:
           For all $R>0$ there exists $S=S(R,\theta_T)>0$ such
           that for all $\lv s \rv > S(R,\theta_T)$ and all
           $\xi\in\N(\theta_T)\cap V(\theta_T)$ we have that
           $\lv d\pi (d\phi_s(\theta_T)\cdot \xi)\rv > R\, \lv \xi \rv$.

\item[(d)] Define
           $$
           B^\top (\theta_T) = \bigl\{ \, \xi\in\N(\theta_T) \,\vert\,
           \sup_{s\in\re} \lv d\pi\cdot d\psi_s\cdot \xi \rv<+\infty
           \, \bigr\}\,.
           $$
           Then $B^\top(\theta_T)\subseteq
           \E^\top(\theta_T)\cap \F^\top(\theta_T)$.

\item[(e)] Let $\Psi_s:= d\phi_s\vert_\N=\La\circ d\psi_s\vert_\N$, then
           \begin{align*}
           d\phi_s\vert_{\N(\theta_T)}
           &= \La(\theta_{T+s})\circ d\Psi_s \circ \La(\theta_T)
           = \Psi_s \,,
           \\
           \Psi_{s+t} &= \Psi_s \circ \Psi_t \, .
           \end{align*}
\end{itemize}
\end{Corollary}

\noindent{\bf Proof:}

Let $U(\theta_T):= V(\theta_T)\cap \N(\theta_T)
                 = V(\theta_T)\cap  T_{\theta_T}\Sigma$. 
\, From~\eqref{E:T.1} we have that                 
$$
d\phi_T(\theta_0)\cdot \xi
      = d\psi_T(\theta_0) \cdot \big[\xi
            + (\a(T,\theta_0)\cdot \xi)\, X(\theta_0) \big] \, .
$$
Hence $d\psi_T(U(\theta_0))\subseteq d\phi_T(W(\theta_0))$, where
$W(\theta):= \bigl(V(\theta)\cap T_\theta\Sigma\bigr) 
\oplus \langle X(\theta)\rangle$.
\, From proposition~\ref{P:G.7},
$$
U(\theta_T)\cap d\phi_T(U(\theta_0))\subseteq V(\theta_T)\cap
d\psi_T(W(\theta_0))=\{0\}
\qquad\text{ for all } T\ne 0\,.
$$
This implies item (a).

\, From lemma~\ref{L:T.2}, the first part of item~(b) and 
items~(c) and~(d) have the same proofs as propositions~A,
\ref{P:G.4} and proposition~\ref{P:B.1} respectively.

Since $d\phi_s$ and $d\psi_s$ satisfy \eqref{E:T.1} and
$d\phi_s$ preserves the bundle $\N(\theta_T)$, we have that
$$
d\phi_s\vert_{\N(\theta_T)}
     = \La(\theta_{T+s})\circ d\psi_s\vert_{\N(\theta_T)}
     = \La(\theta_{T+s})\circ d\psi_s \circ \La(\theta_T)
     = \Psi_s\, .
$$
Moreover,
$$
\Psi_{s+t} = d\phi_{s+t}\vert_{\N(\theta_T)}
           = d\phi_s\vert_{\N(\theta_{T+t})}
                \circ d\phi_t\vert_{\N(\theta_{T})}
           = \Psi_s \circ \Psi_t\,.
$$
This proves item (e).

Finally, we have
\begin{align*}
\E^\top(\theta_0)
     &= \lim_{T\to +\infty}
        \left[ d\phi_{-T}\bigl(V(\theta_T)\cap \N(\theta_T)\bigr)\right]
\\
     &= \La(\theta_0) \, \lim_{T\to + \infty}
     \left[ d\psi_{-T}\bigl(V(\theta_T)\cap T_{\theta_T}\Sigma\bigr)\right]
\\
     &\subseteq \La(\theta_0)\cdot \E(\theta_0)
            \subseteq \E(\theta_0)\, .
\end{align*}
Because $X(\theta_0)\in \E(\theta_0)$.
Thus $\E^\top(\theta_0)\subseteq \E(\theta_0)\cap \N(\theta_0)$.
Since $\dim \E^\top(\theta_0) = \dim U = n-1
= \dim(\E(\theta_0)\cap \N(\theta_0))$, we have that
$\E^\top(\theta_0)= \E(\theta_0)\cap \N(\theta_0)$. The equation
$\F^\top(\theta_0) = \F(\theta_0)\cap \N(\theta_0)$ is proven similarly.
This completes the proof of item (b).

\qed



\bigskip
\bigskip

\noindent{\bf Proof of proposition B.}

\medskip

Let $\theta_0\in\Ga$ and let $\tau>0$ be the period of $\theta_0$.
Suppose that $\Ga$ is hyperbolic, then by proposition~\ref{P:G.5},
$\E(\theta_0) \subseteq E^s(\theta_0) \oplus \langle X(\theta_0)\rangle$
and $\F(\theta_0) \subseteq E^u(\theta_0) \oplus \langle X(\theta_0)\rangle$.
Hence $\E(\theta_0)\cap \F(\theta_0) = \langle X(\theta_0)\rangle$.

Now observe that if we construct the coordinates $(q_1,\ldots,q_n)$ in
(\thesection.a), (\thesection.b) so that they are periodic with
period $\tau$, then $d\phi_\tau(\theta_0):\N(\theta_0)\hookleftarrow$
is the derivative of the Poincar\'e map of $\psi_t$ defined on the
cross-section $\{\, \vartheta \;\vert\; q_1(\vartheta)=0\,\}$.

Suppose that $\E(\theta_0)\cap \F(\theta_0)=\langle X(\theta_0)\rangle$.
By corollary~\ref{C:T.3} (a) and (b) we have that $B^\top(\theta_0)\subseteq
\E^\top(\theta_0)\cap \F^\top(\theta_0) = \langle X(\theta_0)\rangle
\cap \N(\theta_0) = \{ 0 \}$. This implies that $\Ga$ is hyperbolic.

\qed








\bigskip
\bigskip

\stepcounter{section}

\centerline{\large{\bf \thesection. Quasi-hyperbolic actions.}}

\bigskip
\medskip

Let $B$ be a compact metric space and $\pi:\EE\to B$ a vector bundle provided
with a continuous norm $\lv \quad \rv_p$ on each fibre $\pi^{-1}\{p\}$. Let
$\Psi$ be  an $\re$-action $\Psi:\re\to \text{Isom}(\EE)$, i.e.
\begin{itemize}
\item[(i)] There exists a continuous flow $\psi_t$ on $B$
           such that $\pi\circ \Psi_t = \psi_t \circ \pi$.
\item[(ii)] $\Psi_t:\EE(p)\to \EE(\psi_t(p))$ is a linear isomorphism,
            where $\EE(p):=  (\pi\vert_{\EE})^{-1}\{p\}$, $p\in B$.
\item[(iii)] $\Psi_{s+t} = \Psi_s\circ \Psi_t$.
\end{itemize}

We say that the action $\Psi_t$ is {\it quasi-hyperbolic} if
$$
\sup_{t\in\re} \lv \Psi_t(\xi)\rv = +\infty
\qquad \text{ for all } \qquad
\xi\in\EE,\; \xi\ne 0.
$$
We say that $\Psi$ is {\it hyperbolic} if there exists an
invariant continuous splitting
$\EE(p)=E^u(p)\oplus E^s(p)$, $p\in B$ and $C>0$, $\la>0$ such that
\begin{itemize}
\item[(\thesection.a)] $\Psi_t\bigl(E^u(p)\bigr)= E^u\bigl(\psi_t(p)\bigr)$,
                       $\Psi_t\bigl(E^s(p)\bigr)=E^s\bigl(\psi_t(p)\bigr)$,
                       for all $x\in B$, $t\in\re$.
\item[(\thesection.b)] $\lv \Psi_t(\xi)\rv\le C\,\ee^{-\la t}\,\lv\xi\rv$
                       for all $t>0$, $\xi\in E^u(p)$, $p\in B$.
\item[(\thesection.c)] $\lv\Psi_{-t}(\xi)\rv \le C\,\ee^{-\la t}\, \lv\xi\rv$
                       for all $t>0$, $\xi\in E^u(p)$, $p\in B$.
\end{itemize}

The aim of this section is to prove the following

\bigskip

\noindent{\bf \ref{T:Q-A}. Theorem.}
{\it
If the action $\Psi$ is quasi-hyperbolic and the non-wandering set
$\Om(\psi\vert_B)=B$, $\psi\circ\pi=\pi\circ\Psi$, 
then $\Psi\vert_\EE$ is hyperbolic.}

\bigskip

Applying this theorem to the case in which $B=M$ is a compact manifold,
$\psi_t$ is a differentiable flow on $M$, $\Psi_t=d\psi_t$ and $\EE$ is a
$d\psi_t$-invariant continuous bundle such that
$\EE\oplus\left\langle \frac{d\,}{dt}\psi_t\right\rangle = TM$, we get

\bigskip

\begin{Theorem}\label{T:QA} {\bf [Freire] \cite{Freire}}

If $M$ is a closed manifold and $\psi$ is a quasi-Anosov flow on $M$ such that
$\Om(\psi)=M$, then $\psi$ is Anosov.
\end{Theorem}

\medskip

The proof of theorem~\ref{T:Q-A} is similar to that of
theorem~\ref{T:QA}. We include it here for completeness.

Suppose that the action of $\Psi$ is quasi-hyperbolic.
For $p\in B$ define
\begin{align*}
E^s(p) &:= \Big\{\, v\in \EE(p)\, \Big\vert \,
          \sup_{t>0}\lv \Psi_t(p)(v) \rv < +\infty \, \Big\}\, ,
\\
E^u(p) &:= \Big\{ \, v\in \EE(p) \, \Big\vert \,
          \sup_{t<0}\lv \Psi_t(p)(v) \rv < +\infty \, \Big\}\,.
\end{align*}

Observe that by the quasi-hyperbolicity we have that
$$
E^s(p)\cap E^u(p) = \{ 0 \} \qquad \text{for all } p\in B.
$$

\begin{Lemma}\label{L:QA.1}
There exists $\tau>0$ such that for all $p\in B$,
$$
\lV \Psi_\tau \vert_{E^s(p)} \rV < \tfrac 12
\quad , \quad
\lV \Psi_{-\tau}\vert_{E^u(p)} \rV < \tfrac 12
$$
for all $p\in M$.
\end{Lemma}


\noindent{\bf Proof:}

Suppose that the lemma is false for $E^s$. The proof for $E^u$ is similar.
Then there exist sequences $x_n\in B$,
$v_n\in E^s(x_n)$, $\lv v_n\rv=1$ such that
$\lv \Psi_n(v_n)\rv \ge \frac 12$ for all $n\in\na$.
We claim that there exists $C$ such that
$$
\sup_{t\ge 0} \lV \Psi_t (p) \vert_{E^s(p)}\rV < C < +\infty\, .
$$
for all $p\in B$.

Suppose the claim is true. Let $w_n:= \Psi_n(v_n)$ and
$y_n:=\psi_n(x_n)$. We have that $\frac 12 \le \lv w_n\rv \le C$
for all $n$ and
$$
\lv \Psi_t(w_n)\rv = \lv \Psi_{t+n}(v_n)\rv
\le C \qquad \text{for all } t\ge -n\, .
$$
Since $B$ is compact there exists a convergent subsequence
of $(y_n,w_n)$. If $y_n\underset{n}{\to}y$ and $w_n\underset{n}{\to}w$,
we have that $y\in B$, $w\in \EE(y)$, $\lv w \rv \ge \frac 12$ and
$$
\lv \Psi_t(w)\rv \le C
\qquad \text{for all } t\in {\Bbb \re}\, .
$$
This is a contradiction.

Now we prove the claim. Suppose it is false.
Then there exists $x_n\in B$, $t_n\ge 0$, $v_n\in E^s(x_n)$,
$\lv v_n\rv = 1$, such that
\begin{equation}\label{E:QA.1}
\sup_n \lv \Psi_{t_n} (v_n) \rv = +\infty\, .
\end{equation}
Let $s_n>0$ be such that
$$
\lv \Psi_{s_n}(v_n) \rv
> \tfrac 12 \, \sup_{s\ge 0}\lv \Psi_s(v_n) \rv
\ge \tfrac 12 \, \lv \Psi_{t_n}(v_n) \rv\, .
$$
By \eqref{E:QA.1} we have that $s_n\underset{n}{\to} +\infty$. Let
$y_n:=\psi_{s_n}(x_n)$ and
$$
w_n := \frac{\Psi_{s_n}(v_n)}
            {\lv \Psi_{s_n}(v_n)\rv}
\, .
$$
Then $\lv w_n\rv =1$ and if $t>-s_n$ we have
$$
\lv \Psi_t(w_n)\rv
= \frac{\lv \Psi_{t+s_n}(v_n) \rv}
       {\lv \Psi_{s_n}(v_n) \rv}
\le \frac{2\, \lv \Psi_{t+s_n}(v_n)\rv}
         {\sup_{s\ge 0}\lv \Psi_s(v_n)\rv}
\le 2\, .
$$
Since $\lv w_n\rv=1$ and $y_n\in M$, there exists a convergent
subsequence $(y_n,w_n)\to (y,w)$. We would have that $y\in B$,
$w\in \EE(y)$, $\lv w\rv=1$ and
$$
\lv \Psi_t(w)\rv \le 2
\qquad \text{for all } t\in\re \, .
$$
This is a contradiction.

\qed


\bigskip



\begin{Lemma}\label{L:QA.2}
There exists $K>0$ such that for all $x\in B$, $v\in \EE(x)$
$$
\lv \Psi_t(v)\rv \le K \, \bigl( \,\lv v \rv + \lv \Psi_s(v)\rv \,\bigr)
\qquad \text{for all }  \quad 0 \le t \le s .
$$
\end{Lemma}

\medskip

\noindent{\bf Proof:}

Suppose it is false. Then there exist $x_n\in B$, $0\ne v_n\in \EE(x)$ and
$0\le t_n\le s_n$, such that
$$
\lv \Psi_{t_n}(v_n)\rv
         \ge n \bigl(\, \lv v_n \rv + \lv \Psi_{s_n}(v_n) \rv \, \bigr) \, .
$$
Then $t_n\to +\infty$ and $s_n - t_n\to + \infty$ when $n\to +\infty$.
We can assume that
\linebreak
$\lv \Psi_{t_n}(v_n)\rv = \sup\limits_{0\le t \le s_n} \lv \Psi_t(v_n)\rv$.
Let
$$
w_n := \frac{\Psi_{t_n}(v_n)}{\lv \Psi_{t_n}(v_n)\rv}\, \cdot
$$
For $-t_n<t< s_n - t_n$, we have that
$$
\lv \Psi_t(w_n) \rv
     = \frac{\lv \Psi_{t+t_n}(v_n)\rv}{\lv \Psi_{t_n}(v_n)\rv}
     \le \frac 1{\lv \Psi_{t_n}(v_n)\rv} \,
         \sup_{0\le t \le s_n} \lv \Psi_t(v_n)\rv
     = 1.
$$
Taking a subsequence, if necessary, we can assume that $x_n\to x$ and $w_n \to
w\in\EE(x)$. Then $\lv \Psi_t(w)\rv \le 1$ for all $t\in\re$, with $\lv
w\rv=1$. This is a contradiction.

\qed


\bigskip

\noindent
{\bf Proof of theorem~\ref{T:Q-A}: }

Standard methods (cf. Hirsch-Pugh-Shub \cite{HPS}) show that
the continuity of the strong stable and unstable bundles
is redundant in the definition of a hyperbolic action. So it
remains to prove that for all $x\in B$, we have that
$E^s(x) \oplus E^u(x) = \EE(x)$.
Given $x\in B$, since $x\in \Om(\psi\vert_B)$, there exist
$x_n\to x$, $s_n\to +\infty$ such that
$\psi_{s_n} x_n\to x$. Let $E_n$ be a subspace of
$\EE(x_n)$ such that
\begin{gather*}
\dim E_n = \dim \EE(x) - \dim E^s(x) \, , \\
E^s(x) \oplus \lim_n E_n = \EE(x) \, .
\end{gather*}

We claim that there exists $C>0$ such that
$$
\lV \Psi_{-t} \big\vert\, {\Psi_{s_n}(E_n)}\rV \le C
\qquad \text{for all } 0\le t\le s_n .
$$

Suppose the claim is true. Taking a subsequence if necessary, we can assume
that the limit $\lim_n \Psi_{s_n}(E_n)$ exists. Then
$$
\bigl\Vert \Psi_{-t} \big\vert\, {\lim_n \Psi_{s_n}(E_n)} \bigr\Vert \le C
\qquad \text{for all } t>0.
$$
Then $\lim_n \Psi_{s_n}(E_n) \subset E^u(x)$ and
$\dim E^u(x) + \dim E^s(x) \ge \dim E_n + \dim E^s(x) = \dim\EE(x)$.
Since $E^s(x) \cap E^u(x) = \{ 0 \}$, this completes the proof.

Now we proof the claim. Suppose it is false, then there exist
$v_n\in \Psi_{s_n}(E_n)$, $\lv v_n\rv=1$ and $0<t_n<s_n$ such that
$\lv  \Psi_{-t_n}(v_n)\rv \ge n$. By lemma~\ref{L:QA.2}, we have that
$$
n\le \lv \Psi_{- t_n}(v_n) \rv
\le K \, \bigl( \lv \Psi_{-s_n}(v_n)\rv + \lv v_n\rv \bigr)\, .
$$
Hence $\lv \Psi_{-s_n}(v_n)\rv \ge \frac{n-K}{K}$. From lemma
\ref{L:QA.2} we also have that
$$
\frac{\lv \Psi_t ( \Psi_{-s_n}(v_n))\rv}{\lv \Psi_{-s_n}(v_n)\rv}
\le K + \frac K{\lv \Psi_{-s_n}(v_n)\rv}
\qquad \text{for } 0< t< s_n \, .
$$
Let $w_n := \dfrac{\Psi_{-s_n}(v_n)}{\lv \Psi_{-s_n}(v_n)\rv}$.
The estimates above give that
$$
\lv \Psi_t(w_n) \rv \le K + \frac {K^2}{n-K}
\qquad \text{for } 0 < t < s_n\, .
$$
If $w_n\to w$, then $\lv w\rv=1$ and $w\in\lim E_n$, thus $w\notin E^s(x)$.
But $\lv \Psi_t(w)\rv \le K$ for all $t\ge 0$. This is a contradiction.

\qed


\bigskip
\bigskip

{\noindent}{\bf Proof of theorem C.}

\bigskip

For $\theta\in\Sigma$, let
$$
\N(\theta) := \bigl\{\, \xi\in T_\theta\Sigma\,\vert\,
              \langle \, d\pi\cdot \xi\,,\, d\pi\,
               X(\theta)\,\rangle_{\pi(\theta)} = 0 \, \bigr\},
$$
and let $\La(\theta):T_\theta\Sigma\to\N(\theta)$ be the projection along the
direction of $X(\theta)$:
$$
\La(\theta)\cdot\xi = \xi + \beta(\xi)\, X(\theta)
\qquad \text{ such that } \La(\theta)\cdot \xi \in \N(\theta)\,.
$$
Let $\Psi_s(\theta)= \La(\psi_s(\theta))\circ d\psi_s\vert_{\N(\theta)}$.
By corollary~\ref{C:T.3} (e), $\Psi_s$ defines an $\re$-action on the
vector bundle $\pi:\N\to\Sigma$.

Suppose that the Green bundles satisfy $\E\cap\F=\langle X\rangle$ on 
$\Sigma$. Then by corollary~\ref{C:T.3}(b) and~\ref{C:T.3}(d),
$$
B^\top(\theta)\subseteq \E^\top(\theta)\cap \F^\top(\theta)
               = \E(\theta)\cap \F(\theta)\cap \N(\theta)
               =\{0\}\,.
$$
Therefore the action $\Psi_s$ is quasi-hyperbolic. Since $\psi_s$
preserves the Liouville measure, which is positive on open sets, then
$\Om(\psi\vert_\Sigma)=\Sigma$. 
Hence $\Psi_s$ is a hyperbolic action.

This implies that $\phi_t$ is Anosov. By the methods in Hirsch-Pugh-Shub
\cite{HPS}, we get that $\psi_t$ is Anosov. We sketch the proof below.

For $\theta\in\Sigma$ let $\cE^u(\theta)\subseteq\N(\theta)$ be the subspace
given by item (\thesection.b) in the definition of hyperbolic action. 
Then $\cE^u$ is a continuous subbundle of $T\Sigma$. Define
$$
{\cal F}:= \bigl\{ \, L:\cE^u\to\re \text{ continuous } \big\vert \;
           L(\theta):\cE^u(\theta)\to\re \text{ is linear }
           \forall \theta\in \Sigma \, \bigr\},
$$
with $\lV L \rV =
\sup_{\xi\in\cE^u\setminus\{0\}}\frac{\lv L(\xi)\rv}{\lv\xi\rv}$.

To each functional in ${\cal F}$ associate a subbundle $W_L$ of $T\Sigma$
by
$$
W_L(\theta):= \text{Graph}(L(\theta))=
\bigl\{\, \xi+(L(\theta)\cdot\xi)\, X(\theta)\; \big\vert\;
\xi\in\cE^u(\theta)\,\bigr\}.
$$
Let $\tau>0$ be such that $C\, \ee^{-\la \tau}<\ee^{-\la} < 1$,
where $C$ and $\la$ are from (\thesection.b). Consider the following
``graph transformation'' $T:{\cal F}\to {\cal F}$, corresponding
to $W_{TL}=d\psi_\tau(W_L)$ and defined by
$$
d\psi_\tau\bigl(\xi + L(\xi)\, X(\theta)\bigr)
= \Psi_\tau(\xi) + \bigl[ T(L)\, (\Psi_\tau(\xi))\bigr]\, X(\psi_\tau(\theta)),
\; \theta\in\Sigma, \; \xi\in\cE^u(\theta).
$$
We claim that $T$ is a contraction. Indeed
\begin{align*}
\lv\frac{T(L_1-L_2)\, (\Psi_\tau \xi)}{\lV \Psi_\tau\,\xi\rV}\rv
\; \lV X(\psi_\tau \theta)\rV
     &\le \frac{\lV \, d\psi_\tau [(L_1(\xi)-L_2(\xi))\,
          X(\theta)]\rV}{\text{e}^{\la}\; \lV\xi\,\rV}
\\
     &\le \frac{\lv (L_1-L_2)(\xi)\rv}{\text{e}^\la\;\lV\xi\rV}\;
      \lV X(\psi_\tau\theta)\rV\,.
\end{align*}
Hence $\lV T(L_1)-T(L_2)\rV \le \ee^{-\la}\, \lV L_1-L_2\rV$.

The fixed point $L^*$ of $T$ gives the (continuous) strong unstable
bundle of $\psi_t$. Indeed, if $E^u=W_{L^*}$, then $E^u$ is clearly
$d\psi_t$-invariant. Moreover, if $\zeta\in E^u(\theta)$, then
$\zeta=\xi+(L^*\xi)\, X(\theta)$ with $\xi=\La\zeta\in\cE^u(\theta)$.
Since $L^*$ is continuous then there exists $Q_1>0$ such that
$\lv\zeta\rv\le Q\, \lv\xi\rv = Q\,\lv\La\zeta\rv$ for all
$\zeta\in\cE^u$. Since $\N(\theta)$ is transversal to $\langle
X(\theta)\rangle$ and both are continuous subbundles of $T\Sigma$, ($\Sigma$
compact), then the angle $\varangle(X(\theta),\N(\theta))$ is bounded
below and hence there exists $Q_2>0$ such that $\frac 1{Q_2}
\lv\La\zeta\rv = \frac 1{Q_2}\lv \xi\rv < \lv \zeta\rv$.
Then

\begin{align*}
\lv d \psi_t (\zeta)\rv
  &= \lv\, \Psi_t(\xi) + (T(L^*)\cdot\xi)\, X(\psi_t \theta)\,\rv
\\
  &\ge \frac 1{Q_2}\, \lv \Psi_t(\xi)\rv
   \ge \frac{C^{-1}\,\ee^{\la\,t}}{Q_2}\,\lv\xi\rv
   \ge \frac{\ee^{\la t}}{C\, Q_1\, Q_2}\, \lv\zeta\rv\,.
\end{align*}

Finally,
$$
\dim E^u = \dim \text{Graph}(L^*) = \dim \cE^u = n-1\,.
$$
The existence of a continuous strong stable subbundle of dimension $n-1$ is
proven similarly.

\qed





\stepcounter{section}

\bigskip
\bigskip


\centerline{\large \bf \thesection. The Index Form.}

\bigskip
\medskip



Let $L=p\, H_p - H$ be the Lagrangian associated to $H$. 
Consider $\frac{\partial L}{\partial v}(x,v):T_{(x,v)}
(T_xM)\approx T_x M \to\re$ as an element $L_v\in T^*M$. 
The Lagrange transform $\cF_L(x,v)=(x,L_v)$ identifies $q=x$, $p=L_v$.
Similarly, using $\cF_L^{-1}=\cF_H$, we get that $v=H_q$ under the same
identification. Also
$$
H= v\, L_v - L = p\, H_p - L(q,H_p)\,.
$$
Hence $H_q=-L_q=-L_x$. With these identifications the Hamiltonian equations
\eqref{Ham} become
\begin{alignat}{3}
\tfrac{d\,}{dt}\, x &= v
                   \qquad &&\longleftrightarrow \qquad
                               \tfrac{d\,}{dt} \, q &&= H_p\,,
\notag \\
\tfrac{d\,}{dt} \, L_v &= L_x
                   \qquad &&\longleftrightarrow \qquad
                               \tfrac{d\,}{dt} \, p &&= - H_q\,,
\label{E:I.ELH}
\end{alignat}
Hence, identifying $L_x=H_q\in T_{(p,q)}(T^*M)$, the Euler-Lagrange
equation~\eqref{E:I.ELH} is understood as a first order differential
equation on $T^*M$, where $\frac{d\,}{dt}L_v$ is the
tangent vector to the path $t\mapsto L_v(\ga(t),\dot{\ga}(t))$ in $T^*M$.

We derive the Jacobi equation in this Lagrangian setting.
Let $\ga(t)$ be a solution of the Euler-Lagrange equation~\eqref{E-L}.
 Considering a variation $f(s,t)$ of
$\ga(t)=f(0,t)$ made of solutions $t\mapsto f(s,t)$ of the
Euler-Lagrange equation~\eqref{E-L} and taking the covariant
derivative $\frac{D\,}{ds}$, we obtain the
{\it Jacobi equation\/}
\begin{equation}\label{E:I.1}
\tfrac{D\,}{dt} \left( L_{vx}\, k + L_{vv}\, k'\right)
= L_{xx}\, k + L_{xv}\, k'\, ,
\end{equation}
where $k=\frac{\partial f}{\partial s}(0,t)$,
$\dot{k} = \frac{D\,}{dt}\frac{\partial f}{\partial s}$ and the
derivatives of $L$ are evaluated on $\ga(t)=f(0,t)$.
Here we have used that
$\tfrac{D\,}{ds}\tfrac{\partial F}{\partial t}
=\tfrac{D\,}{dt}\tfrac{\partial F}{\partial s}$
for the variation map $F(s,t)=L_v\bigl(f(s,t),\tfrac{\partial f}{\partial
s}(s,t)\bigr)\in T^*M$,
where $\tfrac{D\,}{ds}$ and $\tfrac{D\,}{dt}$ are the covariant 
derivatives on the riemannian manifold $T^*M$. 
The linear operators $L_{xx}$, $L_{xv}$, $L_{vv}$ coincide with 
the corresponding matrices of partial derivatives in local coordinates.
The solutions of \eqref{E:I.1} satisfy
$$
D\vr_t\bigl(k(0),\dot{k}(0)\bigr) =
\bigl(k(t),\dot{k}(t)\bigr)\in T_{\ga(t)}(TM)\, ,
$$
where $\vr_t$ is the Lagrangian flow on $TM$.

Let $\Om_T$ be the set of continuous piecewise $C^2$ vectorfields
$\xi$ along $\ga_{[0,T]}$.
Define the {\it index form\/} on $\Om$ by
\begin{equation}\label{E:I.2}
I(\xi,\eta)=\int_0^T \left(\dot{\xi}\, L_{vv}\, \dot{\eta}
                    + \dot{\xi}\, L_{vx}\, \eta
                    + \xi\, L_{xv}\, \dot{\eta}
                    +\xi\, L_{xx}\, \eta \right)\, dt\, ,
\end{equation}
which is the second variation of the action functional for
variations $f(s,t)$ with $\frac{\partial f}{\partial s}\in\Om_T$.
For general results on this form see Duistermaat \cite{Diust}.

\bigskip

 The following transformation of the index form
is  taken from Hartman \cite{H} and originally due to Clebsch \cite{Clebsch}.
Let $\theta\in TM$ and suppose that the orbit of $\theta$, $\vr_t(\theta)$,
$0\le t\le T$ does not have conjugate points.
Let $E\subset T_\theta T^*M$ be a Lagrangian subspace such that
$d\psi_t(E)\cap V(\psi_t(\theta))=\{0\}$ for all $0\le t \le T$. Such subspace
$E$ always exists by proposition~\ref{P:G.6}(b). Let $E(t):=d\psi_t(E)$ and
let $H(t),\; V(t)$ be a
 matrix solution of the Hamiltonian Jacobi equation~\eqref{E:G.Jacobi}
such that $\det H(t)\ne 0$ and
$E(t)=\text{Image}\, (H(t),V(t))\subset T_{\psi_t(\theta)}(T^*M)$
is a Lagrangian subspace. In particular $H(t)$ satisfies the
Lagrangian Jacobi equation \eqref{E:I.1}.
We have that
\begin{equation}\label{E:I.3}
\left[\begin{matrix}
H(t)  \\
V(t)
\end{matrix}\right]
=
D\FL\,(\vr_t(\theta)) \cdot
\left[\begin{matrix}
H(t)  \\
H'(t)
\end{matrix}\right]
=
\left[\begin{matrix}
I         &   0          \\
L_{vx}    &   L_{vv}
\end{matrix}\right]
\;
\left[\begin{matrix}
H  \\
H'
\end{matrix}\right]
\end{equation}
\, From \eqref{E:I.3} and \eqref{E:I.1}, we have that
\begin{equation}\label{E:I.4}
\begin{aligned}
V  &= L_{vv} \, H' + L_{vx}\, H \\
V' &= L_{xv} \, H' + L_{xx}\, H
\end{aligned}
\end{equation}
Moreover, since $E(t)$ is a Lagrangian subspace, then the
Hamiltonian solution of the Ricatti equation \eqref{E:Ricatti}
$S(t)=V(t)\, H(t)^{-1}$ is symmetric, i.e.
\begin{equation}\label{E:I.5}
H(t)^* \, V(t) = V(t)^* \, H(t).
\end{equation}

Let $\eta\in\Om_T$ and define $\zeta\in\Om_T$ by $\eta(t)=H(t)\, \zeta(t)$.
Then the integrand of $I(\eta,\eta)$ in \eqref{E:I.2} is
$$
\left( L_{vv}\, H\, \zeta' + V\, \zeta\right)\,
\left( H'\, \zeta + H \, \zeta'\right)
+
\left(V'\, \zeta + L_{xv}\, H\, \zeta'\right)\, H \zeta\, .
$$
Since
$\left(L_{vv} \, H \zeta'\right)\cdot(H'\zeta)
= (H\zeta')\cdot (L_{vv}\, H'\zeta)$
and
$(L_{xv}\, H\zeta')\cdot (H\zeta) =
(H\zeta')\cdot (L_{vx}\, H\zeta)$,
the integrand can be written as
$$
L_{vv}\, H \zeta'\cdot H\zeta'
+
H\zeta' \cdot V\zeta
+
V\zeta\cdot H'\zeta
+
V\zeta \cdot H\zeta'
+
V'\zeta\cdot H\zeta \, .
$$
Using \eqref{E:I.5}, we have that
$V^*  H\zeta'\cdot\zeta = H^*V\zeta'\cdot \zeta = V\zeta'\cdot H\zeta$.
hence the integrand in \eqref{E:I.2} is
$L_{vv}\, H\zeta'\cdot H\zeta' + (V\zeta\cdot H\zeta)'$.
Since $\eta$ and $\zeta$ are continuous, we have that
\begin{equation}\label{E:I.6}
I(\eta,\eta)=\int_0^T \big(L_{vv}\; H\zeta' \cdot H\zeta'  \big)\, dt
+ H\zeta\cdot V\zeta\big\vert_0^T
\qquad
\text{if }\quad
\eta=H\zeta\in\Om_T\, .
\end{equation}



Let $\FH(q,p)=(q,H_p)$ be the Legendre transform of the Hamiltonian.
Then $\FL$ and $\FH$ are inverse maps, in particular
$$
(D\FH)^{-1}
  = \left[\begin{matrix} I & 0 \\ H_{qp} & H_{pp} \end{matrix}\right]^{-1}
  = \left[\begin{matrix} I & 0 \\ L_{vx} & L_{vv} \end{matrix}\right]
  = D\FL \, ,
$$
and hence $L_{vv}=(H_{pp})^{-1}$. We obtain that
\begin{equation}\label{E:I.7}
I(\xi,\eta)=\int_0^T (H\zeta')^* \,(H_{pp})^{-1}\,(H\rho')\, dt
           + (H\zeta)^* (V\rho)\big\vert_0^T\, .
\end{equation}
for $\xi=H\zeta\in\Om_T$, $\eta=H\rho\in\Om_T$. This formula can also be
written as
\begin{equation}\label{E:I.7.5}
I(\xi,\eta)=\int_0^T (H\zeta')^* \,(H_{pp})^{-1}\,(H\rho')\, dt
           + \xi^*\,S\,\eta \big\vert_0^T\!,
\end{equation}
where $S(t)=V(t)\,H(t)^{-1}$ is the corresponding solution to the  Ricatti
equation \eqref{E:Ricatti}.

\bigskip

\begin{Corollary}\label{C:I.1}
If $\theta\in T^*M$ and the segment 
$\{\,\psi_t(\theta)\,\vert\, t\in [0,T]\,\}$
has no conjugate points then the index form is positive definite on
$$
\Ga_T = \bigl\{\, \xi:[0,T]\to TM\, 
\big\vert\, \xi(t)\in T_{\pi\psi_t\theta} M,\,
\xi \text{ is piecewise }C^2, \, \xi(0)=0,\, \xi(T)=0\,\bigr\}\, .
$$
\end{Corollary}

\bigskip
\noindent{\bf Proof:}

Let $\xi\in\Ga_T$, $\xi\ne 0$. Write $\xi(t)=H(t)\zeta(t)$.
Since $\det H(t)\ne 0$, $\zeta(0)=0$, $\zeta(T)=0$ and $\zeta(t)\nequiv 0$,
then $\zeta' \nequiv 0$. Now use formula \eqref{E:I.7}.

 \qed
 \bigskip

We now extend formula \eqref{E:I.7} to the case in which $H(t)$ may be singular
(or $\zeta'(t)$, $\eta'(t)$ may be discontinuous)
at a finite set of points. Let $\{t_1,\ldots, t_N\}$ be the points in  $[0,T]$
such that $\det H(t_i)=0$. Then
\begin{equation}\label{E:I.8}
I(\xi,\eta)=\int_0^T (h\zeta')^* \, (H_{pp})^{-1} \, (H\rho')\, dt
+ (H\zeta)^* \! (V\rho)\big\vert_0^T
- \sum_{i=1}^N (H\zeta)^* \! (V\rho)\big\vert_{t_i^-}^{t_i^+}\; .
\end{equation}
where $\xi=H\zeta$, $\eta = H\rho$. But now $\zeta$, $\rho$ are piecewise $C^2$
but may be discontinuous at $t_1,\ldots,t_N$. Alternativelly, we can use a sum
of formulas \eqref{E:I.7} or \eqref{E:I.8} using different Lagrangian subspaces
$d\psi_t(E_i)$ and corresponding solutions $(H_i,V_i)$ on disjoint subintervals
$]t_i,t_{i+1}[\subset [0,T]$.
\bigskip

Observe that for any convex Hamiltonian on $T^*M$ and any $\theta\in M$
there exists $\e>0$ such that the segment $\{\psi_t(\theta)\,\vert\, \lv
t\rv<\e\}$ has no conjugate points. Indeed, let $Y_\theta(t)\, v = d\pi\,
d\psi_t(\theta)\, (0,v)$, where $(0,v)\in V(\theta)$. Then from the Jacobi
equation \eqref{E:G.Jacobi} we have that
$$
\left.\frac{D\,}{dt} Y_\theta(t)\right\vert_{t=0} = H_{pp}(\theta).
$$
Since $Y_\theta(0)=0$ and $H_{pp}(\theta)$ is nonsingular, then
$Y_\theta(T)$ is an isomorphism for $\lv t\rv<\e$, some $\e>0$.
In particular $d\psi_t(\theta)\, V(\theta)\cap V(\theta)=\{0\}$
for $\lv t\rv<\e$.

We say that a curve $\ga(t)\in M$, $t\in[0,T]$ is {\it minimizing}
if it minimizes the action functional
$$
\int_0^T L(\de(t),\de'(t))\, dt
$$
over all absolutely continuous  curves $\de(t)\in M$, $0\le t\le T$, such that
$\de(0)=\ga(0)$ and $\de(T)=\ga(t)$.




\medskip

\begin{Corollary}\label{C:I.NoConjugatePoints}
If $\theta\in T^*M$ and the segment  
$\{\,\pi\psi_t(\theta)\,\vert\, t\in [0,S[\,\}$
is minimizing, then it has no conjugate points.
\end{Corollary}

\noindent{\bf Proof:}

Suppose it is false. Let $\psi_T(\theta)$, $T<S$ be the first conjugate point
in $\{\,\pi\psi_t(\theta)\,\vert\, t\in [0,S[\,\}$. Then there exists
$\xi=(0,v)\in V(\theta)$ such that $d\psi_T(\theta)\,\xi\in V(\psi_T(\theta))$.
Let $\eta(t)=d\pi\, d\psi_t(\theta)\,\xi$.
Using the limit of \eqref{E:I.7} on the  interval $[0,T[$
we have that $I(\eta,\eta)\vert_0^T = 0$.

Let $\e>0$ be such that (cf. prop.~\ref{P:G.6}) the segment
$\{\psi_t(\theta)\,\vert\, T-\e\le t\le T+\e\}$ has no conjugate points. Let
$\zeta(t) = Y_{\psi_{T+\e}(\theta)}(t-T-\e)\, w= d\pi\,
d\psi_{t-T-\e}(\psi_{T-\e}(\theta))\, (0,w)$, $T-\e\le t \le T+\e$,
where $w$ is such that $\zeta(T-\e)=\eta(T-\e)$. Let $\widehat{\eta}=\eta(t)$
if $T-\e\le t\le T$ and $\widehat{\eta}(t)=0$ if $T\le t \le T+\e$. Then using
$H(t)= Y_{\psi_{T+\e}(\theta)}(t-T-\e)$  on \eqref{E:I.7}, we have that
$$
I(\widehat{\eta},\widehat{\eta})\big\vert_{T-\e}^{T+\e}
         > I(\zeta,\zeta)\big\vert_{T-\e}^{T+\e}.
$$

Extend $\zeta$ and $\widehat{\eta}$ by $\zeta(t)=\widehat{\eta}(t)=\eta(t)$
for $0\le t \le T-\e$. Then
$$
I(\zeta,\zeta)\big\vert_0^{T+\e} 
< I(\widehat{\eta},\widehat{\eta})\big\vert_0^{T+\e} = 0.
$$
Now consdider the variation $f(s,t) = \exp_{\pi \psi_t(\theta)} s\,
\widehat{\eta}$. We have that $f(s,0)=\pi(\theta)$,
$f(s,T+\e)=\pi(\psi_{T+\e}(\theta))$, $\frac{\partial f}{\partial s}(0,t)
=\widehat{\eta}(t)$ and if
$$
A(s) = \int_0^{T+\e} L\left( f(s,t),\tfrac{\partial f}{\partial
s}(s,t)\right)\, dt
$$
is the action of $f(s,t)$, then 
$A'(0)=0$ because $f(0,t)=\pi\, \psi_t(\theta)$
satisfies the Euler-Lagrange equation and
$$
A''(0) =
I(\widehat{\eta},\widehat{\eta})<0\,.
$$
Therefore the segment $\{\, \pi\, \psi(\theta)\,\vert\, 0\le t \le T+\e\,\}$
is not minimizing.

\qed











\stepcounter{section}

\bigskip
\bigskip

\centerline{\large\bf \thesection. Proof of theorem D.}

\bigskip

We begin by quoting the following theorem by R. Ma\~n\'e \cite{Ma2}. Given a
(periodic) Lagrangian $L:TM\times S^1\to\re$ and $\om\in H^1(M,\re)$, let
${\cal M}^\om (L)$ be the set of minimizing measures of $L+\ov{\om}$, where
$\ov{\om}$ is any 1-form in the class $\om$.

\begin{Theorem}{\bf (Ma\~n\'e \cite{Ma2}) }
\begin{itemize}
\item[(a)]  For every $\om\in H^1(M,\re)$ there exists 
            a residual subset ${\cal O}(\om) 
            \subset C^\infty(M\times S^1)$ such that $\psi\in {\cal O}(\om)$
            implies $\#{\cal M}^\om (L+\psi)=1$.

\item[(b)] There exist residual subsets ${\cal O}\subset 
           C^\infty (M\times S^1)$ and ${\cal H}\subset H^1(M,\re)$ 
           such that $\psi\in {\cal O}$ and $\om\in {\cal H}$ 
           imply $\# {\cal M}^\om (L+\psi)=1$.
\end{itemize}
\end{Theorem}

Now take an autonomous Lagrangian $L$ and $\om=0\in H^1(M,\re)$. Then the item
(a) implies the first part of theorem D. Let ${\cal O}$ be the residual
subset given by this theorem. Let $\AA$ be the subset of ${\cal O}$ of
potentials $\psi$ for which the measure on ${\cal M}(L+\psi)$ 
is supported on a periodic orbit. Let ${\BB}:= {\cal O}\setminus {\AA}$ 
and let ${\AA}_1$ be the subset of $\AA$ on which the minimizing 
periodic orbit is hyperbolic. We prove that $\AA_1$ is relatively 
open on $\AA$. For, let $\psi\in \AA_1$ and
$$
{\cal M}(L+\psi) = \{ \mu_\ga\}
$$
where $\mu_\ga$ is the invariant probability measure supported on the
hyperbolic periodic orbit $\ga$ for the flow of $L+\psi$. We claim that if
$\phi_n\in\AA$, $\phi_n \to \psi$ and ${\cal M}(L+\phi_k)=\{\mu_{\eta_k}\}$,
then $\eta_n\to\ga$. Indeed, since $L$ is superlinear, the velocities in the
support of the minimizing measures $\mu_k:=\mu_{\eta_k}$ are bounded
(cf. \cite{Mather}, \cite{Ma1}), and
hence there exists a subsequence $\mu_k\to\nu$ converging weakly* to a
some invariant measure $\nu$ for $L+\psi$. Then if $\nu\ne\mu_\ga$,
\begin{equation}\label{E:D.0}
\lim_k S_{L+\phi_k}(\mu_k)= S_{L+\psi}(\nu) > S_{L+\psi}(\mu_\ga)\, .
\end{equation}
Thus if $\de_k$ is the analytic continuation of the hyperbolic periodic orbit
$\ga$ to the flow of $L+\phi_k$ in the original energy level $c(L+\psi)$,
since $\lim_k S_{L+\psi_k}(\mu_{\delta_k})=S_{L+\psi}(\mu_\ga)$,
for $k$ large we have that,
$$
S_{L+\phi_k}(\mu_{\de_k})< S_{L+\phi_k}(\mu_{\eta_k})\, ,
$$
which contradicts the choice of $\eta_k$. Therefore $\nu=\mu_\ga$. For
energy levels $h$ near to $c(L+\psi)$ and potentials $\phi$ near to $\psi$,
there exist hyperbolic periodic orbits $\ga_{\phi,h}$ which are the
continuation of $\ga$. Now, on a small neighbourhood of a hyperbolic orbit
there exists a unique invariant measure supported on it, 
and it is in fact supported in the periodic orbit. 
Thus, since $\eta_n\to \ga$, then  $\eta_n$ is hyperbolic. 
Hence $\phi_n\in\AA_1$ and $\AA_1$ contains a neighbourhood of $\phi$
in $\AA$.

Let ${\cal U}$ be an open subset of $C^\infty (M,\re)$ such that $\AA_1={\cal
U}\cap\AA$. We shall prove below that $\AA_1$ is dense in $\AA$. This implies 
 that $\AA_1\cup{\BB}$ is generic. For, let
${\cal V}:=\text{int }( C^\infty(M,\re)\setminus{\cal U})$, then ${\cal
U}\cup{\cal V}$ is open and dense in $C^\infty(M,\re)$. Moreover,
${\cal V}\cap\AA=\o$ because
$\AA\subseteq \ov{\AA_1}\subseteq \ov{\cal U}$ and
${\cal V}\cap \AA\subseteq \AA\setminus\ov{\cal U}=\o$.
Since ${\cal O}=\AA\cup{\BB}$ is generic and
\begin{align*}
({\cal U}\cup{\cal V})\cap(\AA\cup{\BB})
        &= ({\cal U}\cap\AA) \cup\bigl(({\cal U}\cup{\cal
                 V})\cap{\BB}\bigr) \\
        &\subseteq \AA_1\cup{\BB}\,,
\end{align*}
then $\AA_1\cup\BB$ is generic.                         

\medskip

\noindent{\bf Note: }
The perturbation to achieve hyperbolicity in the case of a periodic orbit
and a singularity follow the same spirit. However to prove it in the
case of a singular point is much easier
because the Jacobi equation (the linear part of the flow) is autonomous. 
We suggest to read it first.

\medskip

We have to prove that $\AA_1$ is dense in $\AA$, i.e.  given that a
Lagrangian $L$ has a unique minimizing measure supported on a
periodic orbit $\ga$, then there exists a perturbation by an
arbitrarily small $C^\infty$-potential $\phi$, such that
the new Lagrangian $L+\phi$ has a unique minimizing measure supported
on a hyperbolic periodic orbit.



Fix $\rho\in\AA$ and let $\Ga$ be the periodic
orbit in ${\cal M}(L+\rho)$, we can assume that $\rho=0$.
By the graph property (cf. \cite{Mather}, \cite{Ma1}),
the projection $\pi\vert_\Ga:\Ga\to M$, $\pi(x,v)=x$ is injective.
In particular $\pi(\Ga)\subseteq M$ is a simple closed curve.
Choose  coordinates $\ov{x}=(x_1,\ldots,x_n):U\to S^1\times \re^{n-1}$
on  a tubular neighbourhood of $\pi(\Ga)$ such that
$\ov{x}(\Ga)=S^1\times\{{\boldsymbol{0}}\}$ and
$\left\{ \,\frac{\partial\,}{\partial x_1},\, \frac{\partial\,}{\partial x_2},
\ldots , \frac{\partial\,}{\partial x_n}\,\right\}$ is an orthonormal
frame over the points of $\pi(\Ga)$. In particular 
$\frac{\partial\;}{\partial x_1}$
 is paralel to $\pi (\Ga)$. Define $\tL = L +\phi$
with $\phi(x_1,\ldots,x_n)=\frac 12\,f(\ov{x})\, \e\,
(x_2^2 + x_3^2 + \cdots + x_n^2)$, where $f(\ov{x})$ is a $C^\infty$
nonnegative bump function with support in $U$ which is $1$ on a small
neighbourhood of $\pi(\Ga)$. Clearly $\phi$ can be made
$C^\infty$ arbitrarily small. On $\pi(\Ga)$ we have that
$\tL_{vv} = L_{vv}$, $\tL_{vx} = L_{vx}$ and $\tL_{xx} = L_{xx} + \e\,
\left[\begin{smallmatrix} 0 & \boldsymbol{0}
        \\ \boldsymbol{0} & I \end{smallmatrix}\right]$.
Observe that since $\tL\ge L$ then ${\cal M}(L+\phi)=\{\mu_\Ga\}$.

Let $H$ and $\tH = H -\phi$ be the Hamiltonians associated 
to $L$ and $\tL$ respectively. Let $\Ga$ be the corresponding 
periodic orbit for $H$ and $\tH$ on $T^*M$.
We have to prove that the periodic
orbit $\Ga$ is hyperbolic for the flow of $\tH$. Since it is minimizing, by
corollary~\ref{C:I.NoConjugatePoints} the
orbit $\Ga$ has no conjugate points. By proposition B it is enough to prove
that the Green bundles $\widetilde{\E}$, $\widetilde{\F}$ satisfy
$\widetilde{\E}(\theta)\cap \widetilde{\F}(\theta)=\langle
\widetilde{X}(\theta)\rangle$ on a point $\theta\in\Ga$, where
$\widetilde{X}$ is the Hamiltonian vectorfiled of $\tH$.

Fix $\theta\in\Ga$ and let
$\tZ_T(t)= d\pi\circ d\tpsi_t \circ \bigl[
d\pi\vert_{d\tpsi_{-T}(V(\tpsi_T(\theta)))}\bigr]^{-1}$.
Let $(\tZ_T, \tV_T)$ and $\tK_T(t) = \tV_T(t)\, (\tZ_T(t))^{-1}$,
$0\le t < T$,
be the corresponding solutions of the Jacobi equation
\eqref{E:G.Jacobi} and the Ricatti equation \eqref{E:Ricatti}
respectively: $\text{graph}(\tK_T(t)) =
d\tpsi_t(d\tpsi_{-T}(V(\tpsi_T\,\theta)))$, $\tZ_T(T)=0$, $\tZ_T(0)=I$.




Define $\N(\theta) := \{\, w\in T_{\pi(\theta)} M \,\vert \,
\langle w,\dot{\ga}\rangle=0\,\}$. Then $\N(\theta)$ is the subspace of
$T_{\pi(\theta)}M$ generated by the vectors
$\frac{\partial \,}{\partial x_2}$,$\ldots$,
$\frac{\partial \,}{\partial x_n}$.
Let $v_o\in \N(\theta)$, $\lv v_o\rv=1$ and let $\xi^T(t) := \tZ_T(t)\,v_o$.
Denote by $\tI_T$ and $I_T$ be the index forms on $[0,T]$ for $\tL$ and $L$
respectively. Using the solution $(\tZ_T,\, \tV_T)$ on formula \eqref{E:I.7},
we obtain that
\begin{equation}\label{E:D.1}
\tI_T(\xi^T,\, \xi^T) = - (\tZ_T(0)\, v_o)^*\, (\tV_T(0)\, v_o)
                     = - {v_o}^*\, \tK_T(0)\, v_o\, .
\end{equation}
Moreover, in the coordinates 
$(x_1,\ldots,x_n;\frac{\partial\;}{\partial x_1},
\ldots ,  \frac{\partial\;}{\partial x_n})$
on $TU$ we have that
\begin{align}
\tI_T(\xi^T,\, \xi^T) &= \int_0^T \big(\dot{\xi}^T\, \tL_{vv} \, \dot{\xi}^T
                                + 2\, \dot{\xi}^T\, \tL_{xv}\, \xi^T
                                + \xi^T\, \tL_{xx}\, \xi^T\big) \, dt
\notag\\
                     &= \int_0^T \big(\dot{\xi}^T\, L_{vv} \, \dot{\xi}^T
                                + 2\, \dot{\xi}^T\, L_{xv}\, \xi^T
                                + \xi^T\, L_{xx}\, \xi^T\big) \, dt
                         + \int_0^T \e\, \sum_{i=2}^n \lv\xi_i^T\rv^2\, dt\, .
\label{E:D.1.5}
\end{align}

We have that $\tZ_T(0)=I$ and for all $t>0$,
$\lim_{t\to\infty} \tZ_T(t)=\widetilde{\h}(t)$
with $\widetilde{\h}(t)$ given on claim \ref{LimitSolution} 
for $\tH$. Writing $\pi_N(\xi)=(\xi_2,\xi_3,\ldots \xi_n)$ then 
$\lv\pi_N \widetilde{\h}(0)\,v_0\rv=\lv v_0\rv=1$ because 
$v_0 \in {\bf N}(\theta)$.
Hence there exists $\lambda>0$ and $T_0>0$ such that 
$\lv\pi_N \xi^T(t)\rv=\lv\pi_N\tZ_T(t)\,v_0\rv>\frac{1}{2}$
for all $0\le t\le\lambda$ and $T>T_0$. Therefore
\begin{equation}\label{E:D.2}
\tI_T(\xi^T,\,\xi^T)\ge I_T(\xi^T,\, \xi^T) +\frac{\e\la}{4}\, .
\end{equation}

Let $(\boldsymbol{h}(t),\, \boldsymbol{v}(t))=
d\psi_t\circ\left(d\pi\vert_{\E(\theta)}\right)^{-1}$ be the solution of the
Jacobi equation for $H$ given by claim \ref{LimitSolution} and let
$\SS (\psi_t(\theta)) = \boldsymbol{v}(t)\, \boldsymbol{h}(t)^{-1}$ be the
corresponding solution of the Ricatti equation, with
$\text{graph}[\SS (\psi_t(\theta)]=\E(\psi_t(\theta))$. 
Using formula \eqref{E:I.7}, and writing 
$\xi^T(t)= \boldsymbol{h}(t)\, \zeta(t)$, we have that
\begin{align}
I_T(\xi^T,\, \xi^T)
        &= \int_0^T (\boldsymbol{h}\, \dot{\zeta})^*\,
                 H_{pp}^{-1}\, (\boldsymbol{h}\, \dot{\zeta})\, dt
        + 0
        - \bigl(\boldsymbol{h}(0)\, \zeta(0)\bigr)^{@!@,*}
                  \bigl(\boldsymbol{v}(0)\,\zeta(0)\bigr)
\notag\\
I_T(\xi^T,\, \xi^T)
        &\ge - v_o^* \, \SS (\theta)\, v_o\, .
\label{E:D.3}
\end{align}

\, From \eqref{E:D.1}, \eqref{E:D.2} and \eqref{E:D.3}, we get that
$$
v_o^* \, \SS (\theta)\, v_o \ge v_o^*\, \tK_T \, v_o + \frac{\e\la}4\, .
$$
\, From proposition~\ref{P:G.1.5}, we have that
$\lim_{T\to +\infty} \tK_T(0) = {\widetilde{\SS }}(\theta)$, where
$\text{graph}(\widetilde{\SS }(\theta))=\widetilde{\E}(\theta)$,
the stable Green bundle for $\tH$. Therefore
\begin{equation}\label{E:D.4}
v_o^* \, \SS (\theta)\, v_o
   \ge v_0^*\, \widetilde{\SS }(\theta)\, v_o +\frac{\e\la}4\, .
\end{equation}

Similarly, for the unstable Green bundles we obtain that
\begin{equation}\label{E:D.5}
v_o^*\, \U(\theta)\, v_o + \la_2
\le v_o^* \, \widetilde{\U}(\theta) \, v_o
\quad \text{ for } v_o \in\N(\theta), \lv v_o\rv=1.
\end{equation}
for some $\la_2>0$ independent of $v_o$.

\, From proposition~\ref{P:G.1.5} we have that
$\U(\theta)\succcurlyeq \SS (\theta)$. From \eqref{E:D.4} and
\eqref{E:D.5} we get that
$\widetilde{\U}\vert_\N \succ \U\vert_\N \succcurlyeq \SS \vert_\N
\succ \widetilde{\SS }\vert_\N$. Since
$\widetilde{\E}(\theta) = \text{graph}(\widetilde{\SS }(\theta))$
and $\widetilde{\F}(\theta)=\text{graph}(\widetilde{\U}(\theta))$, we
get that $\widetilde{\E}(\theta)\cap \widetilde{\F}(\theta) \subseteq
\langle \widetilde{X}(\theta)\rangle$. Then proposition B shows that
$\Ga$ is a hyperbolic periodic orbit for $L+\phi$.



This proves that $\AA_1$ is dense in $\AA$. Let $\AA_2$ be the subset of
$\AA_1$ of potentials $\psi$ for which the minimizing hyperbolic periodic
orbit $\Ga$ has transversal intersections $W^s(\Ga)\pitchfork W^u(\Ga)$.
Then $\AA_2$ is relatively open on $\AA_1$. Indeed, if $\phi_n\in\AA_1$ and
$\phi_n\to\psi\in\AA_2$ and ${\cal M}(L+\phi_n)=\{\mu_{\ga_n}\}$,
${\cal M}(L+\psi)=\{\mu_\Ga\}$, we have seen that the $\ga_n$'s are
continuations of the hyperbolic orbit $\Ga$ for nearby flows (on nearby
energy levels): $\ga_n=\Ga_{\phi_n,h_n}$. Since compact subsets (fundamental
domains) of the invariant manifolds $W^s(\Ga_{\phi,h})$ 
depend continuously in the $C^1$ topology on $(\phi,h)$, 
then the transversality property 
$W^s(\Ga_{\phi,h})\pitchfork W^u(\Ga_{\phi,h})$ persists on a neighbourhood
of $(\psi,h_0)$. In particular it holds for $\ga_n=\Ga_{\phi_n,h_n}$ for
$(\phi_n,h_n)$ sufficiently close to $(\psi,h_0)$.


By the same arguments as above, it is enough to prove  that $\AA_2$
is dense in $\AA$. For it we need the following lemma,

\bigskip

\begin{Lemma}\label{L:D.2}

Let $\Ga$ be a hyperbolic periodic orbit without conjugate points of
a convex Hamiltonian. Then for all $\theta\in W^s(\Ga)$ there exists
$S=S(\theta)>0$ such that the segment 
$\{\,\psi_t(\theta)\,\vert\, t\ge S\,\}$
has no conjugate points.
\end{Lemma}

\medskip


\noindent{\bf Proof:}

Let $\E$ be the stable Green subspace of $\Ga$. Then $\E(\vartheta)\cap
V(\vartheta)=\{0\}$ for all $\vartheta\in\Ga$. Since the weak stable
manifold $W^s(\Ga)$ is tangent to $\E$; then for all $\theta\in W^s(\Ga)$
there exists $S=S(\theta)>0$ such that $E(t):= T_{\psi_t(\theta)} W^s(\Ga)$
satisfies $E(t)\cap V\bigl(\psi_t(\theta)\bigr)=\{0\}$.
Moreover, $\dim E(t) = n$ and
for all $u\in E(t)$, $\lim_{s\to +\infty} d\psi_s(u) 
\in \langle X(\theta)\rangle$. Since $\psi_t$
preserves the symplectic form $\om$, we have that
$\om(u,v)=\lim_{s\to +\infty} \om(d\psi_s(u),\, d\psi_s(v))=0$ for all $u,\,
v\in E(t)$. Hence $E(t)$ is Lagrangian and
$E(t)\cap V\bigl(\psi_t(\theta)\bigr)=\{0\}$
for all $t>S$. Then the lemma follows from proposition~\ref{P:G.6}.

\qed

\medskip

Now we prove that $\AA_2$ is dense in $\AA$. Let
$\rho\in \AA\setminus\AA_2$ and make the perturbation $\tL=(L+\rho)+\phi$
explained above so that $\rho+\phi\in\AA_1$. We can assume that $\rho+\phi=0$
and also that $\rho+\phi\notin\AA_2$, otherwise there is nothing to prove.
Write ${\cal M}(L)=\{\mu_\Ga\}$ and let $\theta\in W^s(\Ga)\cap W^u(\Ga)$ be
such that $\dim\bigl(T_\theta W^s(\Ga)\cap T_\theta W^u(\Ga)\bigr)>1$. 
Observe that the $\a$ and $\om$ limit sets of $\theta$ are $\Ga$. 
In particular the orbit of $\theta$ has no autoacumulation points. 
Let $S=S(\theta)>0$ be from lemma \ref{L:D.2} and $\epsilon>0$ small. 
Then there exists $\tau>S(\theta)$ and a neighbourhood $W \subset M$ of
$\pi(\psi_t(\theta))$ such that
$\{\,t\in\re\,\vert\, \pi\psi_t(\theta)\in W\,\}=]\tau,\tau+3\epsilon[$ 
and $W\cap\pi(\Ga)=\o$.
Let $U\subset M\times\re$ be a tubular neighbourhood of
$\{\,(\pi\psi_t(\theta);t)\,\vert\, t>S(\theta)\,\}$. Choose coordinates
$\ov{y}=(y_1,\ldots,y_n;t):U\to\re^n\times\re$ such that
$\ov{y}\bigl(\pi\psi_t(\theta),t\bigr)=\bigl(\a(t),\0;t\bigr)\in 
\re\times \re^{n-1}\times \re$, and
$\left\{\frac{\partial\,}{\partial y_1},\frac{\partial\,}{\partial y_2},
\cdots , \frac{\partial\,}{\partial y_n}\right\}$ is an orthonormal
frame over the points $\pi\psi_t(\theta)$, $t>S(\theta)$. Let
$\varphi(y_1,\ldots,y_n)=
\frac 12\, f(\ov{y})\, \de\, (y_2^2+y_3^2\cdots+y_n^2)$,
where $f(\ov{y})$ is a $C^\infty$ bump function with support in $W$ which is
$1$ on a small neighbourhood of $\bigl\{\,\pi\psi_t(\theta)\,\vert\,
t\in[\tau+\epsilon,\tau+2\epsilon]\,\bigr\}$. By choosing a small $\de>0$ the
function $\varphi$ can be made $C^\infty$ arbitrarily small. Choose $\de$ 
small enough such that the orbit $\Ga$ remains hyperbolic.
 Then we still have that
${\cal M}(L+\phi)=\{\mu_\Ga\}$ and the orbit of $\theta$ is the same
for both flows. Write $\tL = L + \rho+\phi$, $\hL=\tL+\psi$ and
$\tI_T$, $\hI_T$ the corresponding index froms on
$\{\, \psi_t(\theta)\,\vert\, t\in[\tau,T]\,\}$. Since
$\{\, \psi_t(\theta)\,\vert\, t>S(\theta)\,\}$ has no conjugate points, the
stable Green subspace $\widehat{\E}(\psi_t(\theta))$, $t>S(\theta)$ and the
stable solution of the Ricatti equation $\widehat{\SS}(\psi_t(\theta))$ exist.
The same arguments as in equations \eqref{E:D.1.5}--\eqref{E:D.4} give for
$\xi^T=\widehat{Z}_T(t)\, v_o$, $v_o\in\N(\psi_\tau(\theta))$, $\lv v_o\rv=1$, 
$\widehat{Z}_T(t)=d\pi\circ d\widehat{\psi}_{t-T}\circ\bigl[d\pi
\vert_{d\psi_{-T}(V(\widehat{\psi}_{T+\tau}(\theta)))}\bigr]^{-1}$, that
$$
\hI_T(\xi^T,\, \xi^T)
        =\tI_T(\xi^T,\, \xi^T)
        + \int_{\tau+\epsilon}^{\tau + 2\epsilon}
                       \de\, \sum_{i=2}^n \lv\xi_i^T\rv^2\, dt\, ,
$$
$$
v_0^*\, \SS\bigl(\psi_\tau (\theta)\bigr)\, v_o
         \ge v_o\, \widehat{\SS}\bigl(\psi_\tau (\theta)\bigr)\, v_o
          + \frac{\de\,\widehat{\la}}4\,,
$$
for some $\widehat{\la}>0$. In particular
\begin{equation}\label{E:D.8}
\widehat{\E}\bigl(\psi_\tau(\theta)\bigr)
\cap\widetilde{\E}\bigl(\psi_\tau(\theta)\bigr)=\bigl\langle
X(\psi_\tau\theta)\bigr\rangle\,.
\end{equation}

By the same arguments as in proposition~\ref{P:G.5}, we have that
$\widehat{\E}\bigl(\psi_\tau(\theta)\bigr)=$
\linebreak
$= T_{\psi_\tau(\theta)}\bigl(W^u_{\tL}(\psi_\tau\theta)\bigr)$.
Observe that
$T_{\psi_\tau(\theta)}\bigl(W^u_{\tL}(\psi_\tau\theta)\bigr)
= T_{\psi_\tau(\theta)}\bigl(W^u_\tL(\psi_\tau \theta)\bigr)$
didn't change. Write
$T_{\psi_\tau(\theta)}\bigl(W^u_\tL(\psi_\tau \theta)\bigr)
= {\Bbb V} \oplus {\Bbb W}$, with
${\Bbb V}\subseteq
\widetilde{\E}(\psi_\tau\theta)$
and
${\Bbb W}\cap \widetilde{\E}(\psi_\tau\theta)=\{0\}$.
By diminishing $\de$ if necessary we can get \eqref{E:D.8}
and $\widehat{\E}(\psi_\tau\theta)\cap{\Bbb W}=\{0\}$.
Thus
$$
T_{\widehat{\psi}_\tau(\theta)}W^s_{\hL}(\Ga)
\cap
T_{\widehat{\psi}_\tau(\theta)}W^u_\hL(\Ga)
= \widehat{\E}(\psi_\tau\theta)
  \cap T_{\psi_\tau(\theta)}W^u(\Ga)
=\bigl\langle\widehat{X}(\psi_\tau \theta)\bigr\rangle\,.
$$

Finally, by a finite number of these perturbations
on a fundamental domain of $W^s(\Ga)$ one can remove all tangencies of
$W^s(\Ga)$ and $W^u(\Ga)$ and obtain a potential in $\AA_2$ arbitrarily
$C^\infty$-near $\rho\in\AA$.

\qed




\bigskip

\noindent{\bf The case of a singularity.}

\medskip

Suppose that the minimizing measure $\mu$ is supported on a singularity
$(x_0,0)$ of the Lagrangian flow. From the Euler-Lagrange equation~\eqref{E-L}
we get that
\linebreak
$L_x(x_0,0)=0$. Differentiating the energy function
\eqref{E:Energy} we see that $(x_0,0)$ is a singularity of the energy level
$c(L)$. Moreover, the minimizing property of $\mu$ implies that $x_0$ is a
minimum of the function $x\mapsto L_{xx}(x,0)$. In particular,
$L_{xx}(x_0,0)$ is positive semidefinite in linear coordinates in $T_{x_0}M$.

Choose coordinates $\ov{y}=\{y_1,\ldots ,y_n\}$ on a neighborhood of $x_0$ 
such that $\ov{y}(x_0)=\0$ and $L_{vv}(x_0,0)$ is the identity on the basis
$\bigl\{\,\frac{\partial\,}{\partial y_1}\big\vert_{x_0},
\cdots,\frac{\partial\,}{\partial y_n}\big\vert_{x_0}\,\bigr\}$.
Use the coordinates $\bigl(\,\{y_1,\ldots,y_n\}\,,\,
\bigl\{\, \frac{\partial\,}{\partial y_1},\cdots,
\frac{\partial\,}{\partial y_n}\,\bigr\}\bigr)$ on a neighbourhood of $x_0$. 
On the orbit $\vr_t(x_0,0)\equiv (x_0,0)$ the matrices $L_{xx}$, $L_{xv}$,
$L_{vv}$ are constant with respect to the time $t$. The Jacobi
equation~\eqref{E:I.1} at $(x_0,0)$ becomes
$$
\ddot{k} = L_{xx}(x_0,0)\; k\; ,
$$
where $L_{vv}(x_0,0)=I$. Let $f$ be a nonnegative $C^\infty$ function
on $\re^n$ with support on $\lv y\rv<\e<1$ and $f\equiv 1$ on a neighbourhood
of $\0$. Let $\phi(y)=\frac 12\,\delta\, f(y)\, (y_1^2+\cdots+y^2_n)$.
Adjusting $\de>0$ we can make $\phi$ arbitrarily $C^\infty$ small.
Let $\tL:=L+\phi$. Then the atomic measure $\mu$ supported on $(x_0,0)$ is
still the unique minimizing measure for $\tL$ and the Jacobi equation
for the orbit $\widetilde{\vr}_t(x_0,0)\equiv (x_0,0)$ is
\begin{equation}\label{E:D.S1}
\ddot{k} = \bigl( L_{xx}(x_0,0) + \de\, I\bigr)\; \dot{k}\,.
\end{equation}
Since $L_{xx}(x_0,0)$ is positive semidefinite, then
$A:=\bigl(L_{xx}(x_0,0)+\de\, I\bigr)$ is positive definite.
Equation~\eqref{E:D.S1} is linear
\begin{equation}\label{E:D.S2}
\left[\begin{matrix} k' \\ k'' \end{matrix}\right]
= \left[\begin{matrix}
         0  &  I  \\
         A  &  0
   \end{matrix}\right]
\; \left[\begin{matrix} k \\ k' \end{matrix}\right]
\end{equation}
with constant coefficients where $k\equiv 0$ is a hyperbolic singularity.

To obtain the transversality property $W^s\pitchfork W^u$ at homoclinic orbits
we can use the same arguments as above if we have a replacemnet for
lemma~\ref{L:D.2}. The argumenmts of lemma~\ref{L:D.2} apply if we show that
the stable subspace of $(x_0,0)$ is transversal to the vertical subspace.
For, it is enough to prove that the eigenvectors $(x,y)$ of the (non-singular)
matrix in \eqref{E:D.S2} have coordinate $x\ne 0$. These eigenvectors satisfy
$y=\la x$, $Ax=\la^2 x$, $(\la\in\re)$. Hence if $x=0$, then $(x,y)=(0,0)$ is
not an eigenvector.

\qed


\stepcounter{section}

\bigskip
\bigskip

\centerline{\large \bf \thesection. The Exponential map.}


\bigskip
\bigskip



The following lemma was proven for geodesic flows
by Freire and Ma\~n\'e \cite{Fre-Ma} and has intrinsic interest
(compare with proposition \ref{P:G.4}).

\smallskip

\begin{Lemma}\label{L:AI.1}

Let $H$ be a convex Hamiltonian. Choose a riemannian metric 
$\lV\quad\rV$ on $M$. If $\psi_t$ has no conjugate points 
on $\Sigma = H^{-1}\{e\}$, then there exists $B>0$ such that
$$
\lV Y_\theta (t) \cdot v\rV > B\, \lV v \rV
$$
for all $\theta\in\Sigma$, $\lv t\rv >1$ and $v\in T_\theta M$.
Where $Y_\theta (t)\cdot v = d\pi\circ d\psi_t(\theta)\cdot (0,v)$,
\linebreak
$(0,v)\in H(\theta)\oplus V(\theta)$.


\end{Lemma}


\bigskip



We sketch the proof of theorem F.
In the case in which
$\Theta(X)>0$ on $\Sigma$ one can define the vectorfield
$Y=\frac{1}{\lv\Theta(X)\rv}\, X$. This vectorfield preserves
the 1-form $\Theta$ on $T\Sigma$.  Extending quadratically
the Hamiltonian to $T^*M$ by $\H(t\,\theta)= t^2$, where
$\theta\in\Sigma$, we obtain a new Hamiltonian $\H$, which
is convex and has no conjugate points in $\Sigma$.
The Hamiltonian vectorfield $\YY$ of
$\H$ is called the (convex) {\it simplectification}
of the contact vectorfield $Y$. The transversal bundle
$\N(\theta)=\ker \Theta\vert_{T_\theta\Sigma}$ is invariant
under the flow of $\YY$. Let $E_q:T_q\tM\to\tM$ be the
exponential map for $(\YY,\Sigma)$.
The derivative $dE_q(t\theta)$ is given in
formulas~\eqref{E:G.C8.0} and~\eqref{E:G.C8.1}.
The norm of the directional derivative in formula~\eqref{E:G.C8.1}
is clearly bounded below. By lemma~\ref{L:AI.1} the norm
of the derivative $dE_q\vert_{V(\theta)\cap T_\theta\Sigma}$ in
formula~\eqref{E:G.C8.0} is uniformly bounded away from zero.
Since
$V(\theta)\cap T_\theta\Sigma\subset \N(\theta)$,
and the angle
$\varangle\bigl(\N(\theta),X(\theta)\bigr)$
is bounded below, we obtain that $\lV dE_q\rV$ is bounded away from zero.
This implies that
$E_q:T_q\tM\to \tM$ is a covering map and hence a diffeomorphism.
In particular, the universal cover $\tM$ of $M$ is homeomorphic
to $\re^n$. By corollary~\ref{C:G.8}, the exponential map $\exp_q$ for $X$
is a local diffeomorphism. Once we know that $E_q$ is a diffeomorphism,
it is easy to show that $\exp_q$ is bijective and hence a diffeomorphism.







\bigskip



\noindent
{\bf Proof of Lema \ref{L:AI.1}:}

Since $Y_\theta (t)$ satisfies \eqref{E:G.Y} (with $H_{qp}$ and $H_{pp}$
uniformly bounded on $\Sigma$)  and by corolary~\ref{C:G.3},
$S_\theta(t)$ is uniformly bounded for $\lv t \rv >1$, then there exists
$D_1 >1$ such that
$$
\lV \dot{Y}_\theta(t)\cdot w\rV \le D_1 \, \lV Y_\theta (t) \cdot w \rV
\text{ for all } \lv t \rv >1, \; w\in V(\theta), \;
\theta\in \Sigma.
$$
Let
$$
A:= (1+D_1)^2 \, \sup_{\theta\in\Sigma}\lV H_{pp}(\theta)^{-1}\rV .
$$

Let $H_\theta(t):= Y_{\psi_{-2}(\theta)}(t+2)\circ
\left(Y_{\psi_{-2}(\theta)}(2)\right)^{-1}$,
then $\det H_\theta(t)\ne 0$ for $t>-2$, $H_\theta(0)=I$ and there exists
a linear homomorphism $V_\theta(t):H(\theta)\to V(\psi_t(\theta))$ such
that $(H_\theta,V_\theta)$ is a matrix solution of the 
Jacobi equation~\eqref{E:G.Jacobi} and 
Image$(H_\theta(t),V_\theta(t))=d\psi_{t+2}(V(\psi_{-2}(\theta)))$ is a
Lagrangian subspace. Let
$$
D_2 := \max_{
\begin{smallmatrix}{-2\le t\le 2}\\{\theta\in \Sigma}\end{smallmatrix}}
\lV H_\theta(t)^*\, (H_{pp})^{-1}\, H_\theta(t)\rV  < +\infty .
$$

Let $\rho:\re\to [0,1]$ be a $C^\infty$ function such that
$\rho(t)=0$ for $\lv t\rv\ge 1$ and $\rho(0)=1$. Let
$$
C:= D_2\, \int_{-1}^1 \lv\rho'(t)\rv^2\, dt\,.
$$
Let
\begin{equation}\label{E:AI.9}
0<B<(A\, C)^{\frac 12}.
\end{equation}

Suppose that $\lV Y_\theta(T)\, v_o\rV< B\, \lV v_o\rV$ for some
$\theta\in\Sigma$, $v_o\in T_{\pi(\theta)}M$, $\lV v_o\rV=1$ and $\lv T\rv>1$.
Assume that $T>1$, the case $T<-1$ is proven similarly. Let
$J(t)\in H(\psi_t(\theta))=T_{\pi \psi_t (\theta)}M$ be defined by
$J(t)=0$ for $-1\le t \le 0$, $J(t)=Y_\theta(t)\, v_o$ for $0\le t \le T$ and
$J(t)=(T+1-t)\, \tau_t\, J(T)$ for $T\le t\le T+1$, where 
$\tau_t:T_{\pi \psi_T(\theta)}M\to T_{\pi\psi_t(\theta)}M$ 
is the parallel transport along the path
$t\mapsto \pi\psi_t(\theta)$. Then $J(t)$ is continuous and
piecewise $C^2$.


Let $v(t)\in V(\theta)\approx T_{\pi(\theta)}M$ be defined by
$J(t)=Y_\theta(t)\, v(t)$. Then $v(t)=0$ for $-1\le t\le 0$ and
$v(t)=v_o$ for $0\le t \le T$. Let $\{e_1,\ldots,e_n\}$ be a
basis of $T_{\pi(\theta)}M$ and write $v(t)=\sum_{i=1}^n v_i(t)\, e_i$,
$Y_i(t)=Y(t)\, e_i$. Then the covariant derivative along $\pi \psi_t(\theta)$
of $J(t)=Y(t)v(t)$ is
$$
\frac{D\,}{dt}(Yv)
= \frac{D\,}{dt}\left(\sum_{i=1}^n v_i(t)\, Y_i(t)\right)
= \sum_i \left(\frac {DY_i}{dt}\right)v_i
+ \sum_i \left(\frac{dv_i}{dt}\right) Y_i
\, ,
$$
$$
Y\, v' = \frac{D\,}{dt}(Y v) - \frac{DY}{dt} v \,.
$$
For $T\le t\le T+1$, since $J(t)=Y_\theta(t)\, v(t)=(T+1-t)\, \tau_t\, J(T)$,
we have that
\begin{align*}
\lV Y v'\rV &\le \lV \tfrac{D\,}{dt}(Yv)\rV + \lV \tfrac{DY}{dt}\, v\rV \\
            &\le \lV \tfrac{D\,}{dt} J(t)\rV + D_1\,\lV J(t) \rV \\
            &\le \lV J(T)\rV + D_1\, \lV J(T)\rV \\
            &\le (1+D_1)\, B\, .
\end{align*}

Using $Y_\theta(t)$ on formula \eqref{E:I.8} we have that
\begin{align*}
I(J,J) &= \int_T^{T+1} (Y v')^* \, (H_{pp})^{-1}\, (Yv')\, dt \\
       &\le (1+D_1)^2\, B^2\, \lV H_{pp}^{-1}\rV = A\, B^2.
\end{align*}

Let $Z(t)\in H(\psi_t(\theta))=T_{\pi\psi_t(\theta)}M$ be defined by 
$Z(t)=\rho(t)\, H_{\theta}(t)\, v_o$.
Then $Z(t)= Y_\theta(t)\,\xi(t)$ for $t\ne 0$, with $\xi(t)=0$
for $\lv t\rv>1$. Since $d\psi_t(\theta)\cdot V(\theta)=
\text{ Image }(Y_\theta(t)\,,\, S_\theta(t)\, Y_\theta(t))$, we have that
$S_\theta(t)\, Y_\theta(t)\to I$ when $t\to 0$. Using $Y_\theta(t)$
on formula \eqref{E:I.8}, we have that the only non zero term is
$$
I(Z,J) = - Z(0^+)\cdot v_o + Z(0^-)\cdot 0 = -\lV v_o\rV^2 = -1 .
$$

Write $Z(t)=H(t)\, \zeta(t)$, $\zeta(t) =\rho(t)\, v_o$. Using
$H(t)$ and $V(t)$ on fromula \eqref{E:I.7}, we have that
\begin{align*}
I(Z,Z) &= \int_{-1}^1 (H\zeta')^*\,(H_{pp})^{-1}\, (H\zeta')\, dt \\
       &=\int_{-1}^1 (\rho'(t))^2\,(H(t)\, v_o)^*\,
         (H_{pp}^{-1})\, ( H(t)\, v_o)\, dt \\
       &\le D_2\, \int_{-1}^1 \vert \rho'(t)\vert^2\, dt = C .
\end{align*}

Since by \eqref{E:I.2} the index form is symmetric, for $\la\in\re$
we have that
$$
I(Z-\la\, J\, , \, Z-\la\, J)\le C+2\, \la +\la^2\, A\, B^2.
$$
Since by \eqref{E:AI.9} $4 - 4\, A\, B^2\, C>0$, this polynomial
is $\la$ has two real roots. Therefore $I(Z-\la J\, ,\, Z-\la J)<0$
for some value of $\la$. Hence there must be conjugate points
in the orbit segment $\{\,\psi_t(\theta)\,\vert\, t\in[-1,T+1]\,\}$.
\qed





\bigskip
\bigskip



\begin{Proposition}\label{P:AI.2}

Let $e$ be a regular value of $H$.
If $\theta(X)>0$ on the energy level $\Sigma=H^{-1}\{e\}$ 
and the flow of $X$ has no
conjugate points, then there exists a Hamiltonian $\H$ on $T^*M$ with
vectorfield $Z$ such that
\begin{itemize}
\item[(i)] $\H$ is convex and $\H^{-1}\{e\}=\Sigma$.
\item[(ii)] $Z=\frac 1{\theta(X)}\cdot X$ on $\Sigma$.
\item[(iii)] The flow $\zeta_t$ of $Z$ has no conjugate points on  $\Sigma$.
\item[(iv)] $\zeta_t$ preserves the 1-form $\Theta$ on $T\Sigma$.
\item[(v)]  If $N(\theta):= \frac{d\,}{ds}s\theta\vert_{s=1}$, then
            for $\theta\in\Sigma$ we have
            $$
            d\zeta_t(\theta)\cdot N(\theta) = N(\zeta_t(\theta))
                             + t\, X(\zeta_t(\theta))\, .
            $$
\end{itemize}
\end{Proposition}


\bigskip

\begin{Remark}\label{R:AI.2.5}
This proposition holds if $\Theta(X)\ne 0$ on a connected compact foward
invariant set $K\subset \Sigma$. In this case we obtain an open connected
neighbourhood $U$ of $K$ such that 
$\inf_{\vartheta\in U}\lv\Theta(X(\vartheta))\rv>0$ 
and a convex Hamiltonian $\H$ without conjugate
points, defined on a neighbourhood of $U$ in $T^*M$. To apply this on
theorem F, we can take
$K=\text{ clousure of
}\psi_{\re^+}\bigl((\pi\vert_\Sigma)^{-1}\{q\}\bigr)\subseteq \Sigma$.
\end{Remark}

\noindent
{\bf Proof:}

\bigskip


Recall that $\om(X,\cdot)= - dH$.
Let $\Theta=p\,dq$ be the canonical 1-form on $T^*M$. Suppose that
$\Theta(X)>0$ on $\Sigma$. Let $Y$ be the vectorfield on $\Sigma$
given by $Y=\frac1{\Theta(X)}\cdot X$. Then $Y$ preserves the 1-form
$\Theta$ on $T\Sigma$ because the Lie derivative
\begin{align*}
L_Y\Theta &= d\, i_Y\, \Theta + i_Y\, d\Theta
               =  d[\Theta(Y)] + \om(Y,\cdot) \\
          &= d(1) - \tfrac 1{\Theta(X)}\, dH
               = 0 \text{ on } T\Sigma.
\end{align*}
Let $\phi_t(q,p)$ be the flow of $Y$ on $\Sigma$. Define
the simplectification $\widetilde{\phi_t}$ of $\phi_t$
(cf. Arnold \cite{Ar2}, \cite{Ar1}) on the cotangent bundle without its
zero section $T^*M\setminus M$ by $\widetilde{\phi_t}(q,\la p)
= \la\, \phi_t(q,p)$, $(q,p)\in\Sigma$. Observe that $\widetilde{\phi_t}$
preserves the canonical 1-form $\Theta$. Indeed, we have that
$\pi\circ\phi_t=\pi\circ\widetilde{\phi_t}$ and if
$\phi_t(q,p)=(q_t,p_t)$,
\begin{align*}
\Theta(q_t,\la\, p_t)\cdot( d\widetilde{\phi_t})
   &= (\la\, p_t)\cdot d\pi\,(d\widetilde{\phi_t}\, w) \\
   &= \la \left[ p_t\cdot(d\pi\, d\phi_t\, w)\right]    \\
   &= \la \, \left[ p\cdot (d\pi\, w)\right]
      = \Theta(q,p)\cdot w\; .
\end{align*}
Let $\widetilde{Y}$ be the vectorfield of $\widetilde{\phi_t}$. We have that
$$
L_{\widetilde{Y}}\Theta = d[\Theta(\widetilde{Y})] + \om(\widetilde{Y},\cdot)
                  = 0\, .
$$
Therefore the function $\widetilde{H}=\Theta(\widetilde{Y})$ is a Hamiltonian
for $\widetilde{Y}$. But $\widetilde{Y}$ has conjugate points on $\Sigma$
and $\widetilde{H}$ is not convex.

Let $\H=\frac 12 \widetilde{H}^2$ and let $Z$ be the Hamiltonian 
vectorfield of~$\H$. We have that $Z=\widetilde{H}\cdot \widetilde{Y}$, 
since $\widetilde{H}\vert_\Sigma\equiv 1$, $Z$ is also 
an extension of $Y$. Let $\zeta_t$ be the flow of $Z$. Observe that
$$
\zeta_t(q,sp) = s\, \widetilde{\phi}_{t\tH(q,sp)}(q,p)
              = s\, \widetilde{\phi}_{t\, \frac 12 s^2}(q,p)\,, 
                  \qquad \text{ for }(q,p)\in \Sigma\,. 
$$
Taking the derivative $\left.\frac{d}{ds}\rv_{s=1}$, we obtain
$$
d\zeta_t(\theta)\cdot N(\theta)
              = N(\zeta_t(\theta)) + t\, X(\zeta_t(\theta))
\quad \text{ for } \theta\in\Sigma.
$$

Write $\zeta(t,\theta)=\psi(s(t,\theta),\theta)$ with $s(t,\theta)>0$,
$\theta\in\Sigma$. Then
\begin{equation}\label{E:AI.rep}
D\,\zeta_t = D\,\psi_s + X(\psi_s(\theta))\;\tfrac{\partial s}{\partial \theta}
\qquad \text{ on } T\Sigma\,.
\end{equation}
Let $\E\subset T\Sigma$ be the stable Green bundle for $X$ on $\Sigma$.
Since $X(\theta)\in\E(\theta)$ and $\E$ is $\psi_t$-invariant, then
equation~\eqref{E:AI.rep} shows that the flow $\zeta_t$ of $Z$ preserves the
Lagrangian bundle $\E\subset T\Sigma$, which satisfies
$\E(\theta)\cap V(\theta)=\{0\}$ for all $\theta\in \Sigma$.
By  proposition~\ref{P:G.6}, $\zeta_t$ has no conjugate
points on $\Sigma$.

The following lemma completes the proof of proposition~\ref{P:AI.2}


\bigskip

\begin{Lemma}\label{L:AI.3}

The Hamiltonian $\H$ is convex, i.e. $\H_{pp}$ is positive definite.
\end{Lemma}


\bigskip

\noindent
{\bf Proof:}

Let $q\in \Sigma$ and $S(q) := T_q^*M \cap \Sigma$.
Since $\H$ is homogeneous of degree 2, then $\H_{pp}$ is homogeneous
of degree 0 and it is enough to prove that $\H_{pp}(q,p)$ is
positive definite for $p\in S(q)$.
Let $p_0\in S(q)$. Let $h:T_{p_0}S(q)\times \re\to T^*_q M$ be
$h(x,\la):= x+ \la \, p_0$. Let $G=\H\circ h$ and 
$K=\Theta(\widetilde{Y}\circ h)$. Then
$G = \frac 12 K^2$. Since $D^2h=0$, then $D^2\H(q_0)$ is positive definite
if $D^2G(\0,1)$ is positive definite. Since $K(\0,\la)=\la$, we have that
$$
G_{\la\la}(\0,1)= \left[ K^2_\la + K\cdot K_{\la\la}\right]
\big\vert_{(0,1)} = 1 \, .
$$
Since in the coordinates $(x,\la)$ the vectors
$\left.\frac{\partial\,}{\partial x_i}\right\vert_{(\0,1)}$,
$i=1,\ldots,n-1$ are tangent at $p_0$ to $S(q)$ and 
$\Theta(\widetilde{Y})\equiv 1$
on $S(q)$, then $K_x(\0,1)=\0$. Hence
\begin{align*}
G_{xx}(\0,1)(v,v) &= \lv K_x\cdot v\rv^2 + K\, K_{xx}(v,v) \\
                  &= K_{xx}(v,v)\, .
\end{align*}
Observe that $K$ is homogeneous of degree 1. Therefore $K_x(\0,1)= K_x(\0,1)$
for all $\la>0$. Hence $K_{x\la}(\0,1)=0$, so that
$$
G_{x\la}(\0,1)\cdot (v,\mu) = \mu \, \left[ K_\la\, (K_x\cdot v)
+ K\, (K_{x\la}\cdot v)\right]=0\, .
$$
And then
\begin{align*}
D^2G(\0,1)(v,\mu)^{(2)} &= \mu^2\, G_{\la\la} + 2\mu(G_{\la x}\cdot v)
                            + G_{xx}(v,v)   \\
                        &= \mu^2 + K_{xx}(\0,1)\cdot (v,v) \, .
\end{align*}
So it is enough to see that $K_{xx}(\0,1)$ is positive
definite.


Let $N$ be the unit normal vector to $T_{p_0}S(q)$ such that
$p_0\cdot N<0$. Such $N$ exists because $H\vert_{S(q)}\equiv e$ and
$p_0\cdot H_p=\theta(X)\ne 0$. Let
$F : T_{p_0}S(q)\times \re\to\re$ be given by $F(x,y)=H(p_0+x+yN)$.
Since $\left.\frac{\partial F}{\partial y}\right\vert_{(0,0)}<0$,
there exists a local $C^2$ function $y:U\subseteq T_{p_0}S(q)\to \re$
such that $F(x,y(x))=e$. Differentiating this equation with respect
to $x$, we obtain  that
\begin{align*}
dy &= -\frac{F_x}{F_y}\\
d^2y &= -\frac 1{F_y}\, D^2 F\cdot(I,dy)^{(2)}.
\end{align*}
Since the vectors 
$\left.\frac{\partial\,}{\partial x_i}\right\vert_{(0,0)}$ are
tangent to the level $F\equiv e$ at $(q,p_0)$, we have that
$F_x(\0,1)=\0$. Hence $dy(\0,1)=\0$ and
$$
d^2 y(\0)\cdot v = - \frac 1{F_y}\, D^2 F\cdot (v,\0)^{(2)}
                = -\frac 1{F_y}\, H_{xx}\cdot(v,v) > 0 \, .
$$

Now let $v\in T_{p_0}S(q)$ be a unit vector and $r(s) =\lv p_0+s\, v\rv$,
$z(s) = y(sv)$, $R(s)=\lv p_0+sv + z(s)\, N\rv$. We have that
$$
r'(s)=\frac{p_0\cdot v + s}{r(s)}
\text{ hence }
\frac 12 \, r(s)^2 = \frac 12 \, \lv p_0\rv^2   + (p_0\cdot v)\, s
                     + \frac 12 \, s^2\, .
$$
Also
$$
R'(s) = \frac{(p_0+sv+ z(s)\, N)\cdot (v + z'\, N)}{R(s)}
      = \frac{p_0\cdot v + s + z'\, (p_0\cdot N) + z\, z'}{R(s)}
$$
therefore, since $\a\, s^2\le z(s)\le  \beta s^2$ for some $0<\a<\be$
and $\lv s\rv$ small.
\begin{align*}
\tfrac  12 R(s)^2 &= \tfrac 12\,  \lv p_0\rv^2 + (p_0\cdot v) s
                   + \tfrac 12\,  s^2 + z\, (p_0\cdot N) + \tfrac 12\,  z^2 \\
                &= \tfrac 12\,  r(s)^2 + z(s) \, 
                    (p_0\cdot N) + \tfrac 12\, z(s)^2 \\
                &\le \tfrac 12\,  r(s)^2 - \tfrac 12\,  \ga s^2 \, ,
\end{align*}
for some $\ga >0$ and $\lv s \rv$ small. Then for some $\de>0$, we have that
\begin{gather*}
\frac{r(s)}{R(s)} \ge 1 + \de \, s^2 \\
K_{xx}\cdot (v,v) = \frac{d^2\,}{ds^2}\, K(sv,1)\big\vert_{s=0}
    = \left. \frac{d^2\,}{ds^2}\left(\frac{r(s)}{R(s)}\right)\right\vert_{s=0}
    \ge \de > 0 \, .  \qed
\end{gather*}

\bigskip
\bigskip

\newpage
\noindent{\bf Proof of Theorem F.}

\bigskip
Trough the proof of theorem $F$ we work with the lifted Hamiltonian to the 
cotangent bundle of the universal cover $\tM$ of $M$.
Let $\zeta_t$ be the flow of the Hamiltonian $\H$ of proposition~\ref{P:AI.2}.
Let $q\in\tM$ and let $E_q:T_q\tM\to\tM$ be the exponential map associated to
the flow $\zeta_t$ and the energy level $\Sigma$. We first prove that $E_q$ is
a diffeomorphism. The derivative $dE_q(t\theta)$ is non-singular by
corollary~\ref{C:G.8} and is given by formulas
\eqref{E:G.C8.0}~and~\eqref{E:G.C8.1}. We
claim that the norm $\lV dE_q(t\theta)\rV$ is uniformly bounded below. 
By lemma~\ref{L:AI.1} the norm $\lV dE_q(t\theta)\vert_{V(\theta)\cap
T_q\Sigma}\rV$ is uniformly bounded below for $t>\epsilon$.
\, From equation \eqref{E:G.C8.1}
$\lV d E_q(t\theta)\vert_{\langle \theta\rangle}\rV$ is also uniformly bounded
below. It remains to see that the angle
$\varangle\bigl[dE_q(t\theta)(V(\theta)\cap T_q\Sigma)\,,\,
d\pi(X(\zeta_t\,\theta))\bigr]$
is uniformly bounded below.

Since $\zeta_t$ preserves the 1-form $\Theta$, the bundle
${\bf K}(\theta)=\ker\,\Theta\vert_{T_\theta\Sigma}$ is invariant
under $d\zeta_t$.
Since $V(\theta)\cap T_q\Sigma\subset {\bf K}(\theta)$, then
\begin{equation}\label{E:AI.4.1}
dE_q(t\theta)\,\bigl( V(\theta)\cap T_q\Sigma\bigr)
\subseteq d\pi\, d\zeta_t(\theta)\,\bigl({\bf K}(\theta)\bigr)
\subseteq d\pi\bigl({\bf K}(\zeta_t(\theta))\bigr)\,.
\end{equation}
For $\theta=(q,p)\in\Sigma$ write $H_p(q,p)=\a\, p+ \, k$,
with $\langle k, p\rangle=0$. Since $\Theta(X)=p\cdot H_p\ne 0$ on $\Sigma$
there exists $0\le \la<1$ such that $|\, k|\le \la\, |H_p|$ on $\Sigma$.
Let $h\in d\pi\bigl({\bf K}(\zeta_t\, \theta)\bigr)$ with $|h|=1$, i.e.
$\langle p,h\rangle=0$, $|h|=1$. Then
\begin{align*}
\langle h,H_p\rangle &= \a\, \langle p,h\rangle
                        +  \, \langle k,h\rangle
                      =  \, \langle k,h\rangle\,.
\\
\lv \langle h,H_p\rangle \rv &\le 1\cdot \la\, \lv H_p\rv\,.
\end{align*}
Hence $\cos \varangle(h,H_p)\le \la< 1$. Therefore the angle
$\varangle\bigl[d\pi({\bf K}(\theta)),d\pi(X(\theta))\bigr]$ is uniformly
bounded below on $\theta\in\Sigma$. By \eqref{E:AI.4.1}
$\varangle\bigl[dE_q(t\theta)\bigl(V(\theta)\cap T_\theta\Sigma\bigr),
d\pi\bigl(X(\zeta_t\, \theta)\bigr)\bigr]
> \varangle\bigl[d\pi\bigl({\bf K}(\zeta_t\,\theta)\bigr), d\pi(X(\zeta_t\,
\theta))\bigr]$
is uniformly bounded below.
This proves that $\lV dE_q(t\theta)\rV$ is uniformly bounded
below on $t\in\re^+$, $\theta\in\Sigma$. Hence the map
$E_q:T_q\tM\to \tM$ is a differentiable covering map.
Since $\tM$ is simply connected, it must be a diffeomorphism.

Now let $\exp_q:T_q\tM\to\tM$ be the exponential map
of $(\psi_t,\Sigma)$.
 Since $\zeta_t$ is a reparametrization of
$\psi_t$ for $t>0$, $\theta\in\Sigma$ there exists $\tau(t,\theta)>0$
such that $\psi_t(\theta)=\zeta_{\tau(t,\theta)}(\theta)$.
By corollary~\ref{C:G.8} $\exp_q$ is a local diffeomorphism. It
remains to prove that it is bijective. Since $\lv\Theta(X)\rv=
\tfrac{\lv X\rv}{\lv Y\rv}$ is bounded then $t\mapsto \tau(t,\theta)$ is a
(bijective) homeomorphism of $\re^+$. This implies that
the map $t\,\theta\mapsto\tau(t,\theta)\,\theta$,
with $\theta\in T^*_q M\cap\Sigma$,
is a homeomorphism of $T^*_q M$.
Since $E_q$ is bijective and $\exp_q(t\theta) =
E_q\bigl(\tau(t,\theta)\,\theta\bigr)$, then $\exp_q$ is
bijective.

\qed







\setcounter{section}{1}
\renewcommand{\thesection}{\Roman{section}}
\newcounter{eg}

%\bigskip
%\bigskip

\newpage

%\bigskip

\centerline{\small \bf APENDIX \thesection.}

\bigskip


\centerline{\large \bf \thesection.  Examples.}

\bigskip
%\bigskip
\medskip

The following formalism for twisted Hamiltonians on surfaces
was derived by G. Paternain and M. Paternain in \cite{Pat2}.
Let $M$ be a closed orientable riemannian surface with riemannian metric
$\langle\;,\;\rangle_x$. Let $K:TTM\to TM$ be the connection map
$K\xi=\nabla_{\dot{x}} v$, where $\xi=\frac{d\,}{dt}\bigl(x(t),v(t)\bigr)$.
Let $\pi:TM\to M$ be the canonical projection. Let $\om_0$ be the symplectic
form in $TM$ obtained by pulling back the canonical symplectic form via the
Legendre transform associated to the riemannian metric, i.e.
$$
\om_0(\xi,\zeta)=
       \langle d\pi\, \xi\,,\, K\zeta\rangle
     - \langle d\pi\,\zeta\,,\, K\xi\rangle\,.
$$
Denote by $\Om$ the area 2-form on $M$. Given a smooth function
$F:M\to \re$, define a new symplectic form $\om_F$ on $TM$ by
$$
\om_F = \om_0 + (F{\scriptstyle{\circ}}\pi) \, (\pi^*\Om)\, .     
$$
This is called a {\it twisted symplectic structure} on $TM$
(cf. \cite{Pat2}). Let $H:TM\to\re$ be the Hamiltonian
$$
H(x,v)=\frac 12 \langle v,v \rangle_x\, .
$$
Consider the Hamiltonian vectorfield $X_F$ corresponding
to $(H,\om_F)$, i.e.
\begin{equation}\label{E:X.4}
\om_F\bigl(X_F(\theta),\cdot\bigr)  = dH\,.
\end{equation}
Define $Y:TM\to TM$ as the bundle map such that
\begin{equation}\label{E:X.5}
\Om_x(u,v) = \langle Y(u), v\rangle_x\,.
\end{equation}
Then $Y(u)= iu$ is the  rotation of angle $+\frac{\pi}2$.
\, From \eqref{E:X.4} and \eqref{E:X.5} we have that
\begin{equation}\label{E:X.5.5}
dH_\theta(\xi)
     = \om_0\bigl( X_F(\theta),\xi\bigr)
     + F\bigl(\pi (\theta)\bigr) \, 
       \bigl\langle Y\bigl(d\pi\, X_F(\theta)\bigr),
d\pi\xi\bigr\rangle_{\pi(\theta)}\,.
\end{equation}
Hence if $\xi=(\xi_1,\xi_2) = (d\pi\,\xi, K\xi)\in H(\theta)\oplus V(\theta)$
and $X_F=(X_1,X_2)\in H(\theta)\oplus V(\theta)$, equation
\eqref{E:X.5.5} becomes
\begin{gather}
\langle \theta,\xi_2\rangle
     = \langle X_1,\xi_2\rangle - \langle X_2,\xi_1\rangle
     + F\; \langle Y(X_1),\xi_1\rangle \,,
\notag  \\
X_F(\theta) = \bigl(\theta,F\, Y(\theta)\bigr)\,.
\label{E:X.6}
\end{gather}
In particular, the orbits of the Hamiltonian flow are of the form
$\bigl(\ga(t),\dot{\ga}(t)\bigr)$ for $\ga:\re\to M$, with
\begin{equation}\label{E:X.7}
\frac{D\,}{dt} \; \dot{\ga} =  F(\ga)\; Y(\dot{\ga}\bigr)\,.
\end{equation}

If $\int_M F\,\Om = 0$ this flow can be seen as a Lagrangian flow as follows.
The condition $\int_M F\,\Om = 0$ implies that the cohomology class of $F\,\Om$ is
zero. Hence there exists a 1-form $\eta$ on $M$ such that $d\eta= F\,\Om$.
Consider the Lagrangian
$$
L(x,v) = \frac 12 \langle v , v\rangle_x  + \eta_x(v) \,.
$$
The corresponding Euler-Lagrange equation is
$$
\frac{D\,}{dt}\; \langle \dot{\ga},\cdot\,\rangle
     = d\eta_x(\dot{\ga})
     =\langle F\, Y(\dot{\ga}),\cdot\, \rangle\, .
$$
which gives the same differential equation \eqref{E:X.7}.
This flow corresponds to a magnetic field with Lorentz force
$F(x)\,Y(v)$.

Observe that the theory developed in the previous sections also
applies to the case in which $\int_M F \,\Om\ne 0$.
Indeed, let $q:U\subseteq M\to \re^2$ be a local chart defined on a simply
connected domain $U\subset M$. Then the $2$-form $F\,\Om$ is exact
on $U$ and there exists a $1$-form $\eta\vert_U$ such that $d\eta=F\,\Om$ on
$U$. The Hamiltonian flow on $TU\subset TM$ is the Lagrangian flow of
$L(x,v)=\frac 12\langle v,v\rangle_x + \eta_x(v)$, $x\in U$.
Let $(q,p)$ be the natural chart on $T^*U\subseteq T^*M$ corresponding
to the chart $q$ and let $\cF^{-1}$ be the inverse of the Legendre transform
corresponding to $L$. Let $(q,w)=\cF^{-1}\circ (q,p)$. Since
$\cF^*(d\Theta) = \om_F$, then $(q,w)$ is a symplectic
chart for $\om_F$ and sends the vertical bundle of $T_{q(U)}\re^n$
to the vertical bundle of $T_U M$.


We derive the Jacobi equation. Let $(c(t),\dot{c}(t))$ be an orbit
of the flow. Consider a variation $\ga_s(t)=f(s,t)$, where $f(0,t)=c(t)$ and
the paths $t\mapsto f(s,t)$ are solutions of \eqref{E:X.7}:
\begin{equation}\label{E:X.9}
\frac{D\,}{dt}\;\dot{\ga}_s = F\; Y(\dot{\ga}_s)\,.
\end{equation}
Denote the variational field of $f$ by
$J(t)=\frac{\partial f}{\partial s}(0,t)$.
Using the formula
\begin{equation*}
\frac{D\,}{ds}\,\frac{D\,}{dt}\,\frac{\partial f}{\partial t}
= \frac{D\,}{dt}\,\frac{D\,}{ds} \,\frac{\partial f}{\partial t}
+ R\left(\frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right)
  \frac{\partial f}{\partial t}\,.
\end{equation*}
and taking the covariant derivative of equation \eqref{E:X.9}
in the direction $J(t)$, we obtain
$$
\ddot{J} + R(\dot{c},J) \, \dot{c}
   =   \frac{D\,}{ds} \bigl[ F\; Y(\dot{\ga}_s)\bigr]\, .
$$
Since $Y(v)$ is linear on the fibers of $TM$ and
$\dot{J}=\frac{D\,}{d s} \frac{\partial f}{\partial t}
= \frac{D\,}{dt} \frac{\partial f}{\partial s}$,
we have
\begin{align*}
\tfrac{D\,}{ds}\, FY(\dot{\ga}_s)
      &= \bigl[ \nabla_J(F\; Y)\bigr](\dot{\ga}_s) + F\; Y(\dot{J})\; ,
\\
      &=(\nabla_J F)\, Y(\dot{c}) + F\; Y(\dot{J}) \,.
\end{align*}
Hence we obtain the Jacobi equation
\begin{equation}\label{E:X.10}
\ddot{J} + R(\dot{c},J)\, \dot{c}
         - (\nabla_J F)\, Y(\dot{c})
         - F\; Y(\dot{J})
         = 0 \,.
\end{equation}


 Consider the orthonormal basis
$\{\,\dot{c} , \, Y(\dot{c})=i\,\dot{c}\,\}$ of $T_{c(t)}M$. We have that
\begin{align*}
\tfrac{D\,}{dt} \; \dot{c}  &= F \, Y(\dot{c}) \,,  \\
\tfrac{D\,}{dt} \, Y(\dot{c})
     &= (\nabla_{\dot{c}} Y)(\dot{c}) + Y\left(\tfrac{D\,}{dt}\, \dot{c}\right)
\\
     &=  - F\, \dot{c}\,.
\end{align*}
Write
$$
J = x\, \dc + y\,  Y(\dc)\,.
$$
Then
\begin{align*}
\dJ &=(\dx -F\, y)\, \dc + (\dy + F \, x) \, Y(\dc) \,,\\
\ddJ &= (\ddx - 2\,F \,\dy -\dot{F}\, y - F^2\, x)\, \dc
        + (\ddy + 2\,F\, \dx + \dot{F}\, x - F ^2 \, y) \, Y(\dc) \,,\\
F\, Y(\dJ) &= - F \, (\dy + F \, x) \, \dc
                + F \, (\dx - F \,  y) \, Y(\dc) \, .
\end{align*}
Replacing these formulas on equation \eqref{E:X.10} we obtain
\begin{gather}
\ddx - (F \, y)' = 0 \,,
\label{E:X.11} \\
\ddy +  K\, y + F\dot{x} =   y\, (\nabla_{Y(\dc)}F) \,,
\label{E:X.12}
\end{gather}
where $K=\bigl\langle R\bigl(\dc,Y(\dc)\bigr)\,\dc \,,\, Y(\dc)\bigr\rangle$
is the sectional curvature along $c(t)$. Fix the energy level $\Sigma =
H^{-1}\{\frac 12\} = SM$. If $(J,\dJ)\in T_{\dc(t)}\Sigma
\subset H(\theta)\oplus V(\theta)$, we have that
$dH(J,\dJ)=\langle \dc , \dJ\rangle = 0$. Hence
\begin{equation}\label{E:X.13}
\dx - F \, y = 0 \,.
\end{equation}
This implies equation \eqref{E:X.11}, and from \eqref{E:X.12} and
\eqref{E:X.13}, we get
\begin{equation}\label{E:X.14}
\ddy + \bigl(K+F^2 - \nabla_{Y(\dot{c})}F\bigr)\, y = 0 \, .
\end{equation}
Hence the Jacobi equations on $T_{\dot{c}(t)}\Sigma$ are given by
\eqref{E:X.13} and \eqref{E:X.14}.


\bigskip
\bigskip
\bigskip

\noindent{\bf Example 1.} {\it A convex Hamiltonian without conjugate points
and no continuous invariant transversal bundle.}

\bigskip

Take $K\equiv -1$, $F\equiv 1$. Then equations \eqref{E:X.13} and
\eqref{E:X.14} become
\begin{align}
\dx &= y \,,      \notag\\
\ddy &= 0\,.      \label{E:S1.J}
\end{align}
The solutions have the form
\begin{align}
x(t) &= \tfrac 12 \,a\,t^2 + b\, t + c \,,
\notag\\
y(t) &= a\, t + b \, ,
\label{E:X.15} \\
\dy(t) &= a \, .
\notag
\end{align}
The derivative of the Hamiltonian flow $\psi_t$ on $\bigl( c(t),\dc(t)\bigr)$
is given by
\begin{align}
d\psi_t\bigl( J(0),\dJ(0)\bigr)
        &= \bigl( J(t),\dJ(t)\bigr) \in H(\psi_t\theta)\oplus V(\psi_t \theta)
\notag\\
        &= \bigl(x\,\dc + y\, Y(\dc)\,;\,0\,\dc + (\dy + x)\, Y(\dc)\bigr)
\notag\\
        &= \bigl(x(t)\,\dc+y(t)\,Y(\dc)\,;\, z(t)\, Y(\dc)\bigr) \,,
\label{E:S1.Coor}
\end{align}
where $z(t)=\dy(t)+x(t)$ and $\theta=\bigl(c(0),\dc(0)\bigr)$.

Every orbit $(c,\dc)$ has no conjugate points because the only
solution of \eqref{E:X.15} with $x(0)=y(0)=0$ and $y(T)=0=x(T)$ , $T\ne 0$ is
$x(t)=y(t)\equiv 0$. The Green bundles coincide and are generated
by $X(\theta)=(\dc; - Y(\dc))$ and $(Y(\dc);\0)$. The last vector can be found
by solving $x(0)=0$, $y(0)=1$, $x(T)=y(T)=0$ and letting $T\to \pm\infty$.

Now suppose that there exists an invariant continuous subbundle
$\N(\theta)\subseteq T_\theta\Sigma$ which is transversal to the vectorfield
$X(\theta)$. Let $\theta\in\Sigma$ be a recurrent point and let
$t_n\underset{n}\to +\infty$ be such that $\psi_{t_n}(\theta)\underset{n}\to
\theta$. Let $\xi\in\N(\theta)\setminus\{0\}$.  If $d\psi_t (\xi) =
\bigl(x(t),y(t);z(t)\bigr)=\bigl(J(t);\dJ(t)\bigr)$, then
\begin{align}\label{E:X.16}
\lim_{n\to \pm \infty} \frac 1{\lv x(t_n)\rv}\; d\psi_{t_n}(\xi)
  &= \lim_n \frac 1{x(t_n)} \; \bigl(x(t_n),y(t_n);z(t_n)\bigr)
\notag \\
  &=(1,0;1) = X(\theta)\,.
\end{align}
By the continuity of $\N$ we have
$$
X(\theta) = \lim_n \frac 1{x(t_n)} d\psi_t(\xi) \in \N(\theta)\,.
$$
This contradicts the transversality hypothesis.

The same argument as in equation \eqref{E:X.16} shows that for every
$\theta\in\Sigma$, the angle
$\varangle\bigl(d\psi_t\bigl(V(\theta)\cap
T_\theta\Sigma\bigr),X(\psi_t \theta)\bigr) \underset{t\to
+\infty}{\longrightarrow} 0$.



In this case the trajectories of the Hamiltonian flow are 
the unit tangent vectors
of the horospheres of the riemannian metric, i.e.
\begin{equation}\label{E:S1.1}
\psi_t(\theta) =- Y\bigl(h_t(Y(\theta))\bigr)\,,
\end{equation}
where $h_t$ is the horocycle flow. In particular, the exponential map
$\exp_q: T_q\tM \to \tM$ is a diffeomorphism for all $q\in M$.
This flow has no periodic orbits because the horocycles 
$W^{uu}\subset \Sigma = T^1\tM/_{\pi_1(M)}$ are homeomorphic to $\re$.

$$
\begin{matrix}
\text{\epsfxsize=4cm\epsfbox{ex1.eps}}  \\
\text{\small \sc example \thesection.1}
\end{matrix}
$$

We prove \eqref{E:S1.1}. Let $\theta\in T_q\tM$, $\lV\theta\rV=1$.
Let $\phi_t:T\tM\hookleftarrow$ be the geodesic flow of~$\tM$.
Consider a sphere
$$
S_T(\theta) = \bigl\{\, q\in\tM\,\vert\,
d_{\tM}\bigl(q,\pi\phi_T(\theta)\bigr)=T\, \bigr\}
$$
and a parametrization by arc length $\ga_T(s)$ of $S_T(\theta)$ with
$\ga_T(0)=\pi(\theta)$. Then $\frac{D\,}{ds}\dot{\ga}_T(s)$ is
orthogonal to $\dot{\ga}_T(s)$. Moreover, it has constant norm, because it is
the image of $\frac{D\,}{ds}\dot{\ga}_T(0)$ by the isometry given by
a rotation about the center $\pi\phi_T(\theta)$ of $S_T(\theta)$. Therefore
$\ga_T$ satisfies
$$
\frac{D\,}{ds}\, \dot{\ga}_T = \la_T\; Y(\dot{\ga}_T)
$$
for some constant $\la_T>0$. Passing to the limit when $T\to +\infty$ we obtain
that the reparamerizations by arc length $\ga$ of the horospheres satisfy
\begin{equation}\label{E:S1.2}
\tfrac{D\,}{ds} \, \dot{\ga} = \mu\; Y(\dot{\ga})\,,
\end{equation}
with $\mu = \lim_{T\to +\infty} \la_T$, (in fact $T\to \la_T$ is strictly
decreasing). We claim that $\mu=1$. Indeed, the solutions of \eqref{E:S1.2} of
are closed circles if $\mu$ is big enough. The infimum $\mu_0$
of such $\mu$'s corresponds to the horospheres . For $\mu>\mu_0$ the flow has
conjugate points corresponding to the period (length) of these closed circles
(see example 3). From the Jacobi equation \eqref{E:X.14} with $F=\mu$,
  we see that $\Sigma=H^{-1}\{\frac 12\}$ has
conjugate points for $\mu>1$ and the flow is Anosov for $\mu<1$. Hence
$\mu_0=1$.

Indeed, for $\mu<1$ the solution of~\eqref{E:X.13} and~\eqref{E:X.14} are
$y=A\, \ee^{\la t}+ B\, \ee^{-\la t}$, $x=\mu(\frac{A}\la \, \ee^{\la t} -
\frac{B}\la\, \ee^{-\la t} + C)$, with $\la=\sqrt{1-\mu^2}$. Writing
$\xi=\bigl(x(0),y(0);\dx(0),\dy(0)\bigr)$ we have that
$d\psi_t(\xi)=\bigl(x(t),y(t);\dx(t),\dy(t)\bigr)$. For $A\ne 0$ the
function $t\mapsto\lV d\psi_t(\xi)\rV$ has exponential growth and for $B\ne 0$,
$t\mapsto\lV d\psi_{-t}(\xi)\rV$ has exponential growth. As in section~\S3,
this implies that the flow is Anosov for $\mu<1$. For $\mu>1$, the Jacobi
equations~\eqref{E:X.13},~\eqref{E:X.14} exhibit conjugate points, by the same
arguments as for equations~\eqref{E:X.18} and~\eqref{E:X.19}.




\bigskip
\bigskip
\bigskip


\openup -.205pt   
\newpage
\noindent{\bf Example 2.} {\it A periodic orbit without conjugate points and
                           no transversal invariant subspace.}

\bigskip

We construct a convex Lagrangian which has the Jacobi equations \eqref{E:S1.J}
on a periodic orbit. Let $\theta\in\Sigma$ be such a periodic point with period
$\tau>0$. Changing the coordinates $(x,y;z)$ on $T_\theta\Sigma$ of
equation~\eqref{E:S1.Coor} to $(x,y;\dy)=(x,y;z-x)$, the
matrix of the derivative $d\psi_\tau(\theta)$ is
$$
d\psi_\tau(\theta) = \left[\begin{matrix}
                   1  &  t  &  \tfrac 12\, t^2  \\
                      &  1  &        t          \\
                      &     &        1
                   \end{matrix}\right]
\; \begin{matrix} \}\, x   \\
                  \}\, y   \\
                  \}\, \dy
       \end{matrix}
\qquad {\Huge\approx} \qquad
\left[ \begin{matrix}
        1  &  1  &      \\
           &  1  &  1   \\
           &     &  1
       \end{matrix} \right]
\quad \begin{matrix} \text{in Jordan} \\ \text{form.} \end{matrix}
$$
Let $\{e_1,e_2,e_3\}$ be the basis of $T_\theta\Sigma$ in which
$d\psi_\tau(\theta)$ is written in canonical form. The transformations
$d\psi_\tau$ and
$$
d\psi_\tau - I = \left[\begin{matrix}
                 0  &  1  &     \\
                    &  0  &  1  \\
                    &     &  0
                 \end{matrix}\right]
$$
have the same invariant subspaces. The vectorfield corresponds to the unique
eigenspace $\langle e_1\rangle$ and the only invariant subspaces are
$\langle e_1\rangle=\langle X(\theta)\rangle$, $\langle e_1,e_2\rangle =
\E(\theta)=\F(\theta)$ and $\langle e_1,e_2,e_3\rangle = T_\theta\Sigma$. None
of them is transversal to $X(\theta)=e_1$.

Now we make the construction. Consider the torus $\T^2 = \re^2/_{5\Z^2}$
with the flat metric and polar coordinates $(r,\theta)$ centered at
$(0,0)\in\re^2/_{5\Z^2}$ defined on $r\le 2$. Let $F:\T^2\to\re$ be
a $C^\infty$ function such that $F(r,\theta)=F(r)$, $F(r=1)=1$,
$\left.\frac{\partial F}{\partial r}\right\vert_{r=1}=-1$,
and $\int_{\T^2}F\,\Om=0$. Let $\eta$ be a 1-form in $\T^2$ such that
$d\eta=F\,\Om$. Define $L(x,v)=\frac 12 \,\langle v,v\rangle_x + \eta_x(v)$.
At $r=1$, the Euler-Lagrange equation~\eqref{E:X.7} is
$$
\tfrac{D\,}{dt}\;\dot{\ga} = F(\ga)\; Y(\dot{\ga})= Y(\dot{\ga})\qquad
\text{ at } r=1\,.
$$
Hence $r=1$, $\dot{\theta}=-1$ is a periodic solution. Moreover, at $r=1$ we
have that $F=1$, $\nabla_{Y(\dc)}F = - \frac{\partial F}{\partial r} = +1$.
Hence the Jacobi equations~\eqref{E:X.13},~\eqref{E:X.14} are the same as
\eqref{E:S1.J}.


\bigskip
\bigskip
\bigskip

\noindent{\bf Example 3.} {\it A {\sl segment} without conjugate points with
crossing solutions.}

\bigskip

Let $W=\bigl\{\,(x,y)\in\re^2\,\big\vert\, x^2+y^2\le 4\,\bigr\}$ and consider
the Lagrangian
$$
L(x,v) = \frac 12 \; \langle v , v\rangle + \eta_x(v) \, ,
$$
where $\langle\;,\;\rangle$ is the euclidean metric and $d\eta=-\Om$.
This Lagrangian corresponds to $F\vert_{W}= -1$ in the formalism above and
can be embedded into a convex Lagrangian on a closed surface.
The Euler-Lagrange equation \eqref{E:X.7} is
\begin{equation}\label{E:X.17}
\dot{v} = - i\, v \, .
\end{equation}
The solutions of \eqref{E:X.17} on the energy level $|v|=1$, 
are all the circles contained in $W$ with
constant angular velocity $\dot{\theta}=-1$. Fix ${\cal C}$: $R(t)=1$,
$\theta(t)=\pi-t$ in polar coordinates in $W$ with center $(0,0)$.
Let $\vartheta=(0,1)_{(-1,0)}$ be the initial vector of this solution. 
Then the circles which are rotations of ${\cal C}$ fixing the 
initial point $p=(-1,0)$ are also solutions of \eqref{E:X.17} 
(see figure \thesection.3) with the same energy
$E=\tfrac 12 \, \lv v \rv = \tfrac 12$. The solution ${\cal C}$ has conjugate
points, but the conjugate points appear at $t=2\pi$. In particular the segment
$\bigl\{\, \psi_t(\vartheta)\,\big\vert\, 0\le t \le \frac{3\pi}2\,\bigr\}$
has no conjugate points. But the intersections with other solutions
starting at $p=(-1,0)$ appear at $t<\frac{3}{2}\pi$ and at 
$t=\pi$ there is a vector $\xi\in V(\vartheta)\cap T_\vartheta\Sigma$ 
such that $d\pi\, (d\psi_t\, \xi) =
d\pi\, X(\psi_t \vartheta) = (0,-1)$. Proposition~\ref{P:G.7}
says that these phenomena can not occur if all the positive 
(negative) orbit of $\vartheta$ has no conjugate points.
The claims stated above can be proved using the Jacobi 
equations \eqref{E:X.13} and \eqref{E:X.14}:
\begin{equation}\label{E:X.18}
\dx + y = 0\;, \qquad   \qquad \ddy + y = 0 \, ,
\end{equation}
whose solutions are
\begin{align}
x(t) &= a \, \cos t + b \, \sin t + c \, , \notag\\
y(t) &= a\, \sin t - b \, \cos t  \,.      \label{E:X.19}
\end{align}
Then if $x(0)=y(0)=0$ and $\dy(0)\ne 0$ then 
$\bigl(x(t), y(t)\bigr)\ne 0$ for all
$0<t<2\pi$. Moreover, if $a=-1$, $b=0$, $c=+1$, then 
$x(0)=y(0)=0$, $y(\pi)=0$, $x(\pi)=+2$.


\bigskip


$$
\begin{matrix}
\text{\epsfxsize=4.5cm\epsfbox{ex3.eps}}  \\
\text{\small \sc example \thesection.3}
\end{matrix}
\qquad\qquad\qquad
\begin{matrix}
\text{\epsfxsize=4cm\epsfbox{ex4.eps}}  \\
\text{\small \sc example \thesection.4}
\end{matrix}
$$



\bigskip
\bigskip

\noindent{\bf Example 4.} {\it A non surjective exponential map without
                           conjugate points.}

\bigskip

Consider the flat torus $\T^2 = \re^2/_{6\Z}=[-3,3]^2/_{6\Z}$ and polar
coordinates $(r,\theta)$ on $\bigl\{\,(x,y)\in\T^2\,\big\vert\,x^2+y^2 <
3\,\bigr\}$ centered at $\0=(0,0)$. Define the Lagrangian
$$
L(x,v) = \tfrac 14 \,\langle v, v\rangle_x + \eta_x(v) + \phi(x)\,,
$$
where $\phi$ is a $C^\infty$ function on $\T^2$ such that
$\phi(r,\theta) = \tfrac 14 r^2 + \cos \pi r$ on $r\le 2$
and $\phi(r,\theta)\ge 1$ outside $r \le 2$;
and $\eta(r,\theta)=-\tfrac 12 \, r^2\, f(r)\, d\theta$ with $f(r)$ a
$C^\infty$ function with support on $r\le \tfrac 52$
such that $f(r)=1$ on $r \le 2$ and $f(r)\le 1$.

Observe that the path ${\cal C}: r(t)\equiv 1$, $\theta(t)=t$ is {\it static}
(cf. Ma\~n\'e \cite{Ma3}). Indeed, the Lagrangian
$$
L(r,\theta;\dr,\dth) = \tfrac 14\,(\dr^2 + r^2 \, \dth^2)
               - \tfrac 12 \, r^2 \, \dth
               + \tfrac 14\, r^2 + \cos \pi r
\qquad \text{ on } r \le 2\, ,
$$
is minimized on $\dr=0$, $\dth = 1$, $r=1$, with $L=-1$;
and on $r \ge 2$, we have that $\phi(r)\ge 1$ and $f\le 1$, so
that
$$
\tfrac 14\, r^2\, \dth^2 + \eta(v)
    \ge r^2\, \left(\tfrac 14 \,\dth^2 - \tfrac 12 \, \dth \right)
     \ge - \tfrac 14 \,{r^2} \ge -1
$$
Hence $L\ge \phi(r) \ge 1$ on $r \ge 2$.


By theorem VII in Ma\~n\'e \cite{Ma3} (see also \cite{CDI}), there exists a
semistatic (hence foward minimizing) solution $\ga(t)$ such that 
$\ga(0)=(0,0)$ and the $\om$-limit of $(\ga,\dot{\ga})$ is 
the (hyperbolic) closed orbit ${\cal C}$. 
By the rotational symmetry of the Lagrangian, for every 1-dimensional
subspace $\cL\subset T_{(0,0)}\T^2$ we obtain a semistatic curve $\ga$, which
is tangent to $\cL$,  with $\ga_\cL(0)=(0,0)$ and
$\om$-limit$(\ga_\cL,\dot{\ga}_\cL)={\cal C}$.
Since the semistatic solutions are minimizing, none of them
$\ga_\cL\vert_{[0,+\infty[}$ have conjugate points. Moreover, by theorem X in
Ma\~n\'e \cite{Ma3} (also \cite{CDI}), ${\cal C}$ and all the
$(\ga_\cL,\dot{\ga}_\cL)$'s are in the same energy level
$E=c(L)$. By the graph property (theorem VII, Ma\~n\'e \cite{Ma3} and
also \cite{CDI}), the $\ga_\cL\vert_{]0,+\infty[}$'s don't intersect
$\pi({\cal C})$ neither intersect each other.








\begin{thebibliography}{99}

\bibitem{A-M} R. Abraham \& J. E. Marsden.
              Foundations of Mechanics. Benjamin: London, 1978.

\bibitem{Ar1} V.I. Arnold.
              Mathematical Methods of Classical Mechanics.
              Graduate Texts in Math., {\bf 60}. Springer, 1989.

\bibitem{Ar2} V. I. Arnold, S.P. Novikov. (Editors)
              Dynamical Systems IV. Symplectic Geometry and its Applications.
              Encicl. of Math. Springer, 1985.

\bibitem{BBB} W. Ballman, M. Brin \& K. Burns,
              {\it On surfaces with no conjugate points.}
              Jour. Differential Geometry, {\bf 25} (1987) 249-273.

\bibitem{Bao&Chern} D. Bao \& S. S. Chern.
              {\it On a notable connection in Finsler geometry.}
              Houston Jour. of Math., vol.~{\bf 19}, No.~{\bf 1},
              (1993), 135-180.

\bibitem{Chern} S. S. Chern. {\it Finsler Geometry is just Riemannian Geometry
                without the Quadratic Restriction}. Notices of the A.M.S.,
                {\bf 43} No.~9, sept. 1996, 959-963.

\bibitem{Clebsch}  A. Clebsch.
                 {\it \"Uber die Reduktion der zweiten Variation
                 auf ihre einfachste Form}.
                 J.~Reine Angew. Math. {\bf 55} (1958) 254-276.



\bibitem{CDI}     G. Contreras, J. Delgado, R. Iturriaga.
                  {\it Lagrangian Flows: The Dynamics of Globally
                  Minimizing Orbits, part II}.

\bibitem{Diust}   J. J. Duistermaat. {\it On the Morse Index in Variational
                  Calculus.} Advances in Math., {\bf 21}, (1976), 173-195.


\bibitem{Eb}      P. Eberlein.
                  {\it When is a geodesic flow of Anosov type? I.}
                  J. Diffferential Geometry, {\bf 8}, (1973), 437-463.

\bibitem{Fra-Ro}  J. Franks \& C. Robinson, {\it A quasi-Anosov diffeomorphism
                  that is not Anosov}, Trans. Amer. Math. Soc., {\bf 223},
                  (1976), 267-278.

\bibitem{Freire}  A. Freire.
                  {\it Fluxos geod\'esicos em variedades
                  riemannianas compactas de curvatura negativa},
                  M.Sc. Thesis, IMPA, 1981.

\bibitem{Fre-Ma} A. Freire \& R. Ma\~n\'e.
                  {\it On the entropy of the geodesic flow in manifolds
                  without conjugate points.}
                  Invent. Math. {\bf 69}, (1982), 375-392.

\bibitem{Foulon1} P. Foulon, {\it G\'eom\'etrie des \'equations
                  differ\'entielles du second order}, Ann. Inst.
                  Henri Poincar\'e, {\bf 45} (1), (1986), 1-28.

\bibitem{Foulon}  P. Foulon.
                  {\it Estimation de l'entropie des syst\`emes lagrangiens
                  sans points conjugu\'es}.
                  Ann. Inst. Henri Poincar\'e {\bf 57}, (1992), 117-146.



\bibitem{Green}   L. W. Green.
                  {\it A theorem of E. Hopf},
                  Michigan Math.~J. {\bf 5} (1958) 31-34.

\bibitem{H}       P. Hartman. Ordinary Differential Equations. John Wiley \&
                  Sons, Inc., 1964.

\bibitem{HPS}     M. Hirsch, C. Pugh, M. Shub.
                  Invariant manifolds. Lecture Notes in Math., {\bf 583}.
                  Springer, 1977.




\bibitem{Innami}  N. Innami.
                  {\it Natural Lagrangian systems without conjugate points}.
                  Ergod. Th. \& Dynam. Sys. {\bf 14}, (1994), 169-180.


\bibitem{Kli}     W. Klingenberg.
                  {\it Riemannian manifolds with
                  geodesic flows of Anosov type},
                  Ann. of Math., {\bf 99}, (1974), 1-73.

%\bibitem{K}       W. Klingenberg.
%                  Lectures on closed geodesics. Springer,
%                  19**

%\bibitem{Manning} A. K. Manning. {\it Topological entropy for geodesic flows.}
%                  Annals of Math., {\bf 110}, (1979), 567-573.

\bibitem{ManheK}  R. Ma\~n\'e.
                  {\it On a theorem of Klingenberg},
                  Dynamical Systems and Bifurcation Theory,
                  M. Camacho, M. Pacifico and F. Takens eds.,
                  Pitman Research Notes in Math. {\bf 160},
                 (1987), 319-345.

\bibitem{Ma1}     R. Ma\~n\'e.
                  {\it On the minimizing measures of Lagrangian dynamical
                  systems}. Nonlinearity, {\bf 5}, (1992), no.~3,
                  623-638.


\bibitem{Ma2}     R. Ma\~n\'e.
                  {\it Generic properties and problems of minimizing
                  measures of lagrangian systems.}
                  Nonlinearity, {\bf 9}, (1996), no. 2, 273-310.

\bibitem{Ma4}     R. Ma\~n\'e.
                  {\it Asymptotic measures of class A solutions of
                  generic Lagrangians}. Unpublished.
\bibitem{Ma3}
                  R. Ma\~n\'e.
                  {\it Lagrangian Flows: The Dynamics of Globally
                  Minimizing Orbits.} To~appear, Proc. Int. Congress in Dyn.
                  Sys., Montevideo 1995, Pitman Research Notes in Math.


%\bibitem{Ma5}     R. Ma\~n\'e. On the topological entropy of geodesic flows.

\bibitem{Mather}  J. Mather.
                  {\it Action minimizing invariant measures for positive
                  definite Lagrangian systems}.
                  Math.~Z. {\bf 207}, (1991), no.~2, 169-207.

\bibitem{Morse}   M. Morse.
                  Calculus of variations in the large.
                  Amer. Math. Soc. Colloquium Publications vol.~XVIII,
                  1934, 1960.

\bibitem{Os}      V.I. Oseledec.
                  {\it A multiplicative ergodic theorem}.
                  Trans. Moscow Math. Soc. {\bf 19}, (1960), 197-231.


\bibitem{Pat1}    G. Paternain \& M. Paternain.
                  {\it On Anosov Energy Levels of Convex Hamiltonian Systems.\/}
                  To~appear. Math. Zeitschrift.

\bibitem{Pat2}    G. Paternain \& M. Paternain.
                  {\it On the topological entropy of autonomous Lagrangian
                  systems}. To~appear.

\bibitem{Pesin}   Ya. Pesin.
                  {\it Lyapounov characteristic exponents and smooth ergodic
                  theory.} Usp. Mat. Nauk, {\bf 32-4}, (1977), 55-111.


%\bibitem{Przy}     F. Przyticki.
%                   {\it An Upper Estimation for Topological Entropy}.
%                   Invent. Math. {\bf 59}, (1980), 205-213.

\bibitem{Rob}      C. Robinson. {\it A quasi-Anosov, flow that is not Anosov}.
                   Indiana Univ. Math. J. {\bf 25}, (1976), no~8,
                   763-767.

\bibitem{Ruelle}   D. Ruelle.
                   {\it An inequality for the entropy of differentiable maps.}
                   Bol. Soc. Bras. Mat. {\bf 9}, (1978), 83-87.


\end{thebibliography}


\end{document}







\centerline{
\parbox{6cm}{Gonzalo Contreras
        \medskip
        \newline\openup -3pt{\footnotesize
               Depto. de Matem\'atica. PUC-Rio.
        \newline R. Marqu\^es de S\~ao Vicente, 225.
        \newline G\'avea 22453-900 Rio de Janeiro.
        \newline Brasil.
        \newline gonzalo@@mat.puc-rio.br}  }
\qquad
\parbox{6cm}{Renato Iturriaga
        \medskip
        \newline\openup -3pt{\footnotesize
                 CIMAT
        \newline La Valenciana.
        \newline Guanajuato.
        \newline M\'exico.
        \newline renato@@fractal.cimat.mx  } \vfill}
}


\date{December, 1996}



