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\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Conjecture}[Theorem]{Conjecture}
\newtheorem{Remark}[Theorem]{Remark}

\def \dim{{\mbox {dim}}\,}
\def\df{\buildrel \rm def \over =}
\def\Vol{\mbox{Vol}}
\def\V{\mbox{Var}}

\def \e{\varepsilon}
\def \vr{\varphi}
\def \re{{\mathbb R}}
\def \lv{\left\vert}
\def \rv{\right\vert}
\def \ov{\overline}
\def \tx{{\widehat{x}}}
\def \0{\lambda_{0}}
\def \la{\lambda}
\def \ga{\gamma}
\def \M{\widetilde{M}}
\def\p{\Phi_{k}}
\def \A{A_{L_{\la}+1/2}}
\def \D{{\mathbb D}}
\def \vr{\varphi}
\def\Z{{\mathbb Z}}

\def\wO{\widetilde\Omega}


\begin{document}
\title[Anosov magnetic flows]{Anosov magnetic flows, critical values
and topological entropy}

\author[K. Burns]{Keith Burns}
\thanks{First author supported by NSF grant DMS-9803346.}
\address{Mathematics Department\\
Northwestern University\\
Evanston IL 60208\\
U.S.A.}
\email {burns@math.northwestern.edu}


\author[G.P. Paternain]{Gabriel P. Paternain}
\address{CIMAT \\
          A.P. 402, 3600 \\
          Guanajuato. Gto. \\
          M\'exico.}
 \email{paternain@cimat.mx}




\curraddr{Department of Pure Mathematics and Mathematical Statistics\\
University of Cambridge\\
Cambridge CB3 0WB\\
 England}
 \email {gpp24@dpmms.cam.ac.uk}

\subjclass{58F15, 58F17, 58F05}



%\date{2000}


\begin{abstract}We study the magnetic flow determined by
a smooth Riemannian metric $g$ and a closed 2-form $\Omega$ 
on a closed manifold $M$.
If the lift of $\Omega$ to the universal cover $\M$ is exact,
we can define a critical value $c(g,\Omega)$
in the sense of Ma\~n\'e \cite{Ma1} for the lift of the flow to $\M$.
We have $c(g,\Omega)<\infty$ if the lift of $\Omega$ 
has a bounded primitive.
This critical value can be expressed in terms of an isoperimetric constant
defined by $(g,\Omega)$, which coincides with Cheeger's isoperimetric constant
when $M$ is an oriented surface and $\Omega$ is the area form of $g$.
When the magnetic flow of $(g,\Omega)$ is Anosov on the unit tangent 
bundle $SM$, we show that
$1/2>c(g,\Omega)$ and any closed bounded form in $\M$ of degree~$\geq 2$ 
has a bounded primitive.

Next we consider the 1-parameter family of magnetic flows on $SM$
associated with the pair $(g,\la\Omega)$ for $\la\geq 0$, where 
$\Omega$ is such that
its lift to $\M$ has a bounded primitive.

We introduce a volume entropy $h_{v}(\la)$ 
defined as the exponential growth rate of the average 
volume of certain balls. We show that
$h_{v}(\la)\leq h_{top}(\la)$, where $h_{top}(\la)$ is
the topological entropy of the magnetic flow of $(g,\la\Omega)$ on $SM$ and
that equality holds if the magnetic flow of $(g,\la\Omega)$ is Anosov on $SM$.
If $\la_1 \leq \la_2$ and the magnetic flows for $(g, \la_1\Omega)$
and $(g, \la_2\Omega)$ are both Anosov on $SM$, then $h_v(\la_1) \geq h_v(\la_2)$.

We construct an example of a Riemannian metric
of negative curvature on a closed oriented surface of higher genus
such that if $\phi^{\la}$ is the magnetic flow associated to
the area form with intensity $\la$, then there are values of the parameter
$0<\la_{1}<\la_{2}$ with the property that $\phi^{\la_{1}}$
has conjugate points and  $\phi^{\la_{2}}$ is Anosov. 
Variations of this example show that it is also possible to exit and
reenter the set of Anosov magnetic flows arbitrarily many times
along the one-parameter family. Moreover, we can start
with a Riemannian metric with conjugate points and end
up with an Anosov magnetic flow for some $\la>0$.
Finally we have a version of the example (in which $\Omega$ is no longer the area form) such that
the topological entropy of $\phi^{\la_{1}}$
 is greater than the topological entropy of the geodesic flow, which in turn is greater than the
topological entropy of $\phi^{\la_{2}}$.




\end{abstract}


\maketitle

\section{Introduction} \label{intro}

Let $M$ be a closed $n$-dimensional manifold endowed
with a $C^{\infty}$ Riemannian metric $g$, 
and let $\pi:TM\rightarrow M$ be the canonical projection.
Let $\omega_{0}$ be the symplectic form on $TM$ obtained 
by pulling back the canonical symplectic form of $T^{*}M$ 
via the Riemannian metric.
Let $\Omega$ be a closed $2$-form on $M$ and consider
the new symplectic form $\omega_{1}$ defined as:
\[\omega_{1}\df \omega_{0}+\pi^{*}\Omega.\]
The 2-form $\omega_{1}$ is a symplectic form and defines what is called
a {\it twisted symplectic structure} \cite{A}.

Let $E:TM\rightarrow {\re}$ be given by
\[E(x,v)=\frac{1}{2}g_{x}(v,v). \]
The magnetic flow of the pair $(g,\Omega)$ is the Hamiltonian 
flow of $E$ with respect to $\omega_{1}$. The magnetic flow models the
 motion of a particle of unit mass and charge under the effect
 of a magnetic field, whose Lorentz force $Y:TM\rightarrow TM$ is the
 bundle map defined by:
\[\Omega_{x}(u,v)=g_{x}(Y_{x}(u),v),\]
for all $x\in M$ and all $u$ and $v$ in $T_{x}M$. In other words, the curve
\[t\mapsto (\ga(t),\dot{\ga}(t))\in TM\]
is an orbit of the Hamiltonian flow if and only if
\begin{equation}
\frac{D\dot{\ga}}{dt}=Y_{\ga}(\dot{\ga}),
\label{1}
\end{equation}
where $D$ stands for the covariant derivative of $g$.
The magnetic flow of the pair $(g,0)$ is
the geodesic flow of the Riemannian metric $g$.
A curve $\ga$ that satisfies (\ref{1}) will be called
a {\it magnetic geodesic}.

Magnetic flows were first considered by V.I. Arnold in \cite{A1/2} and
by D.V. Anosov and Y.G. Sinai in \cite{AS}.
Recent work on these flows has uncovered several remarkable
properties, see \cite{B,Go,G1,G2,Mac,P,P1,PP1/2,PP1,W2,W3}.

\subsection{Critical values}
Let $\wO$ be the lift of $\Omega$ to  the universal 
cover~$\widetilde M$ of~$M$. 
Suppose that $\widetilde\Omega$ is an exact form, i.e., there
exists a smooth 1-form $\Theta$ such that $\wO=d\Theta$.
Let us consider the Lagrangian on $\M$ given by
\[L(x,v)=\frac{1}{2}|v|^{2}_{x}-\Theta_{x}(v).\]
It is well known that the extremals of $L$, i.e., the solutions of
the Euler-Lagrange equations of $L$,
 $$
 \frac{d}{dt}\frac{\partial L}{\partial v}(x,v) = \frac{\partial L}{\partial x}(x,v)
$$
 coincide with the lift to $\M$ of the
magnetic geodesics. The {\it energy function}
 $$
 E(x,v) = \frac{\partial L}{\partial v}(x,v)v - L(x,v) 
 $$
is invariant under the Euler-Lagrange flow. It is easily checked that this
definition of $E$ coincides with the lift to $\M$ of the function $E$ defined above.
Being able to express the magnetic flow
as a Lagrangian flow is an advantage since it allows us to use
variational techniques to derive results for magnetic flows.


The magnetic flow shares with the geodesic flow the property that the level
sets of the function $E$ are invariant. 
There is, however, a significant difference.
The geodesic flow is the same for all energy levels 
up to a uniform change of speed.
For the magnetic flow, on the other hand, 
the behaviour of the flow depends in an essential way on the energy. 
In particular there is a {\it critical value} of the energy at which
there is a decisive change in the behaviour of the flow.
This critical value has been extensively studied, 
notably by A.~Fathi \cite{Fa}, R.~Ma\~n\'e \cite{Ma1} 
and J.~Mather \cite{Mat1, Mat2}.

We now give Ma\~n\'e's definition of the critical value in our context.
The action of the Lagrangian $L$ on
an absolutely continuous curve $\ga:[a,b]\rightarrow \M$ is defined by
\[A_{L}(\ga)=\int_{a}^{b}L(\ga(t),\dot{\ga}(t))\,dt.\]
The critical value is
\[c(L):=\inf\{k\in\re:\;A_{L+k}(\ga)\geq 0\;\mbox{\rm  
for any absolutely continuous closed curve $\ga$}\]
\[\mbox{\rm defined on any closed interval $[a,b]$ }\}.\]

Like any Lagrangian flow, the magnetic flow for $T\M$ 
in the case when $\wO$ is exact can be viewed as the Hamiltonian flow 
defined by the canonical symplectic form on $T^*\M$ and a 
suitable Hamiltonian function $H:T^*\M \to \re$; in this case
\[ H(x,p) = \frac12 |p + \Theta_x|^2. \]
The Legendre transform ${\mathcal L}:T\M\to T^{*}\M$ defined
by
$${\mathcal L}(x,v)=\frac{\partial L}{\partial v}(x,v)$$
 carries orbits of the
Lagrangian flow for $L$ to orbits of the Hamiltonian flow defined
by $H$ and the canonical symplectic form.

The critical value can also be defined in Hamiltonian terms.
We now introduce the {\it critical value of the pair} $(g,\Omega)$
as the real number:
\begin{align*}
c(g,\Omega) &= \inf_{u\in C^{\infty}(\M,\re)}\;\sup_{x\in
  \M}\;H(x,d_{x}u)\\
&= \inf_{u\in C^{\infty}(\M,\re)}\;\sup_{x\in \M}\;
   \frac{1}{2}|d_{x}u+\Theta_{x}|^{2}.
\end{align*}
As $u$ ranges over $C^{\infty}(\M,\re)$ the form $\Theta-du$ ranges over all
primitives of $\wO$, because any two primitives differ by a closed 1-form
which must be exact since $\M$ is simply connected.

We show in Appendix A that $c(L)=c(g,\Omega)$ 
whenever $\wO$ is exact, even if all primitives of $\wO$ are unbounded.
This generalizes Theorem~A in \cite{CIPP}, which gives
$c(L)=c(g,\Omega)$ when $\Omega$ itself is exact.
Our proof closely follows the arguments of Fathi and Maderna in~\cite{FaMa}.
Clearly $c(g,\Omega)\geq 0$ and we prove in Lemma \ref{mqc} 
that if $\Omega$ is non-trivial, then $c(g,\Omega)>0$. 


The Hamiltonian characterization of the critical value can also be
expressed in Lagrangian terms. 
Let us call two Lagrangians $L$ and $\overline L$ {\it equivalent}
if there is a function $u \in C^\infty(\M,\re)$ such that
 $$
 L(x,v) - \overline L(x,v) = d_xu(v)
 $$
 for all $(x,v) \in T\M$. If a Lagrangian is convex and superlinear,
so are all Lagrangians equivalent to it. Equivalent Lagrangians have
the same action on closed curves and define
the same Euler-Lagrange equation and the same energy function on $T\M$.
The Hamiltonian definition of $c(L) = c(g,\Omega)$ means that 
if $k > c(L)$ and we choose $k' \in (c(L),k)$, we can then choose
$u \in C^\infty(\M,\re)$  such that
 $$
 H(x,d_xu) = \max_{v \in T_x\M}\{d_xu(v) - L(x,v)\} \leq k'
 $$
 for all $x \in \M$, and hence
 $$
L(x,v) - d_xu(v) + k' \geq 0
 $$
 for all $(x,v) \in T\M$. The Lagrangian $\overline L$ defined by
$\overline L(x,v) = L(x,v) - d_xu(v)$ satisfies
 $$
\overline L(x,v) + k \geq k - k' > 0
 $$
 for all $(x,v) \in T\M$. Thus if $k > c(L)$ we can find a Lagrangian 
$\overline L$ equivalent to $L$ such that $\overline L + k$ is positive 
and uniformly bounded away from $0$  throughout $T\M$. 


 
The critical value is closely related to an isoperimetric constant.
For a smooth map $\vr$ of the standard unit disk $\D^{2}$ into $\M$, let
$\ell(\partial\vr)$ be the length of $\vr(\partial \D^{2})$ and
\[a(\vr):=\left|\int_{\D^{2}}\vr^{*}(\wO)\right|.\]
We define our isoperimetric constant as

\[\mbox{\rm iso}(g,\Omega)=\inf_{\vr}\frac{\ell(\partial\vr)}{a(\vr)}.\]
This constant is defined even when $\wO$ is not exact, but we show
in Proposition \ref{0iso} that it is always zero in that case.
If $M$ is an oriented surface and $\Omega$ is the area form 
then $\mbox{\rm iso}(g,\Omega)$
coincides with Cheeger's isoperimetric constant introduced in \cite{Ch}.

\vspace{.5cm}

\noindent{\bf Theorem A. }{\it If $\wO$ is exact, then
\[\sqrt{2c(g,\Omega)}=\frac{1}{\mbox{\rm iso}(g,\Omega)}.\]}

\vspace{.5cm}

The proof is given in Section~\ref{proofAB}.

Using Theorem A and A. Katok's methods
from \cite{Ka}, we can give a lower bound for the critical value when
$M$ is an orientable surface with Euler characteristic $\chi<0$. 
Given a Riemmanian metric $g$, let $a_{g}$ be the total $g$-area of $M$. 
By the conformal equivalence theorem
there exists a unique positive scalar $C^{\infty}$ function $\rho$ such that
the metric
$\rho^{2}g$ has constant negative curvature and $a_{\rho^{2} g}=a_{g}$. 
Let $\rho_{g}$ be the {\it conformality coefficient} given by
\[\rho_{g}:=\int_{M}\rho\,d\mu_{g},\]
where $d\mu_{g}$ is the normalized Riemmanian measure.
By the Cauchy-Schwartz inequality, 
$\rho_{g}\leq 1$ and equality holds if and only if
$g$ itself is a metric of constant negative curvature. In Section~\ref{proofAB} we show:

\vspace{.5cm}

\noindent{\bf Theorem B. }
{\it Let $M$ be a closed orientable surface with Euler characteristic
$\chi<0$. For any
pair $(g,\Omega)$ we have:
\[c(g,\Omega)\geq 
\frac{\left(\int_{M}\Omega\right)^{2}}{-4\pi\chi\,\rho_{g}^{2}\,a_{g}}.\]
}

\vspace{.5cm}


\subsection{Anosov magnetic flows}
We are especially interested in the case when the magnetic 
flow on the unit tangent bundle  is Anosov. In the theory of 
Lagrangian systems, an Anosov energy level is a regular level set of the
energy on which the Euler-Lagrange flow is Anosov. In our case 
$S\M = E^{-1}(1/2)$ and  $1/2$ is a  
regular value of the energy function $E$. 

\vspace{.5cm}

\noindent{\bf Theorem C. }
{\it Suppose that the restriction to the unit tangent bundle 
of the magnetic flow of the pair $(g,\Omega)$ is Anosov. Then
\[c(g,\Omega)<1/2.\]
Moreover any closed bounded form of degree~$\geq 2$ 
on $\M$ has a bounded primitive.}


\vspace{.5cm}

The first statement in the theorem can be seen
as a ``twisted version'' of Theorem~B in \cite{CIPP}.
The second statement means that the $L^{\infty}$-cohomology 
of $\M$ vanishes in degree~$\geq 2$.
This extends the observation, made by M. Gromov in~\cite{Gr}
(see \cite[Proposition 7.1]{Pa} for a proof), that if $M$ admits
a metric of negative sectional curvature, then
every closed bounded form of degree $\geq 2$ on $\M$ has a bounded primitive.
We remark that we do not know of any example of a manifold
with an Anosov magnetic flow that does not admit a Riemannian
metric of negative curvature.



Given a real number $\la$, we can consider the restriction to the unit
tangent bundle of the magnetic flow
 associated with the pair $(g,\la\Omega)$ and
we call this flow the {\it $\la$-magnetic flow} 
and denote it by $\phi^{\la}: SM \to SM$.
Similarly, a {\it $\la$-magnetic geodesic} will be a unit speed magnetic
geodesic of the pair $(g,\la\Omega)$. 
The $0$-magnetic flow is the geodesic flow.

The structural stability of Anosov flows means that the set 
of $\la$ for which the $\la$-magnetic flow is Anosov is open. 
We call a component of that set an
{\it Anosov interval}. 
It is obvious that the $\la$-magnetic flow is Anosov 
if and only if the $-\la$-magnetic flow is Anosov.

It follows from Theorem~C that if the $\la$-magnetic flow is Anosov,
then
\[1/2>c(g,\la\Omega)=\la^{2}c(g,\Omega)\geq \la^{2}c(g,\Omega).\]
Unless $\Omega\equiv 0$ we have $c(g,\Omega)>0$ and hence
\[\lambda^{2}<\frac{1}{2\,c(g,\Omega)}\]
if the $\la$-magnetic flow is Anosov. In \cite{PP1/2} G. and M. Paternain
obtained a different bound on the $\la$ for which the $\la$-magnetic flow 
is Anosov. 
At the end of Section~\ref{proofAB} we use Theorem~B  to show that 
their bound is not as sharp as the above estimate.



The fact that the $\la$-magnetic flow must be non Anosov when
$\la$ is large enough naturally raises the question:

\vspace{.5cm}

\noindent{\bf Question.}\;Is it true that if the
$\la_{0}$-magnetic flow is Anosov for some $\la_{0} >0$, then the
$\la$-magnetic flow is Anosov for all $\la\in[0,\la_{0}]$?

\vspace{.5cm}

\begin{figure}

\special{psfile=figure1.eps voffset=-200 hoffset=40 hscale=50 vscale=50 angle=0}

\vspace{7cm}

\caption{Escape from and reentry into the set of Anosov magnetic
flows.}

\label{fig1}

\end{figure}

We answer this question in the negative (see Figure \ref{fig1}).
In Section~\ref{S=egs} we construct a simple and explicit example with
Anosov geodesic flow and more than one Anosov interval. Our
example is a closed oriented surface with negative Gaussian
curvature and $\Omega$ is the area form. We also exhibit a surface
with non Anosov geodesic flow such that the $\la$-magnetic flow is
Anosov for some $\la > 0$.

In \cite{Go,G1,W3}, N. Gouda, S. Grognet and M. Wojtkowski established
geometric conditions on the Riemannian metric and the form
$\Omega$ to ensure that $\phi^{\la}$ is Anosov. For closed oriented surfaces
with negative curvature
and for $\Omega$ the area form, these conditions read:
\[K(x)+\la^{2}<0,\]
for all $x\in M$, where $K$ is the Gaussian curvature of $M$.
If we define the {\it magnetic curvature} as $K_{mag}^\la(x):=K(x)+\la^{2}$,
then their result simply says that if the magnetic curvature
is negative then $\phi^{\la}$ is Anosov.
Our first example shows that their condition is not optimal since
if $K_{mag}^{\la_0} < 0$  implies $K_{mag}^{\la} < 0$  for all
$\la\in[0,\la_{0}]$.

Recently, M. Bialy \cite{Bi} has shown that if we take a Riemannian metric
on the $n$-torus which is conformally flat then for any non-trivial
2-form $\Omega$ the magnetic flow $\phi^{1}$ has conjugate points.
In Bialy's proof it is essential to assume that $M$ is an $n$-torus.
Our second example shows how different the situation is
for surfaces of higher genus. 
The surface in that example has conjugate points 
because there is a closed geodesic along which the curvature is positive. 
But the Anosov $\la$-magnetic flow does not have conjugate points~\cite{pp2}.


\subsection{Topological entropy}
Finally we discuss how topological entropy of the $\la$-magnetic flow 
changes with $\la$. 
Let $h_{top}(\la)$ denote the topological entropy of $\phi^\la_t$ on $SM$. 
Given a point $x \in \M$ and  $T>0$ set
\[B_{mag}(x,\la,T)=\{y\in\M:\;\mbox{there is a $\la$-magnetic geodesic}\qquad\]
\[\qquad\qquad\qquad\mbox{ from $x$ to $y$ with length
$< T$}\}.\]
We call $B_{mag}(x,\la,T)$ a {\it magnetic ball} with center $x$
and radius $T$. Our next theorem shows that we can define
an average volume entropy by considering the exponential growth
rate of the average volume of magnetic balls and that this quantity
enjoys similar properties
to those obtained by A. Manning in \cite{Man1} and
A. Freire and R. Ma\~n\'e in \cite{FM} for geodesic flows.
Let $\mbox{\rm Vol}\,B_{mag}(x,\la,T)$ be 
the Riemannian volume of $B_{mag}(x,\la,T)$;
we shall see at the beginning of Section~\ref{monot} 
that $x\mapsto \mbox{\rm Vol}\,B_{mag}(x,\la,T)$ is invariant under
covering transformations and hence it defines a function on $M$, which
we still denote by $\mbox{\rm Vol}\,B_{mag}(x,\la,T)$.

\vspace{.5cm}

\noindent{\bf Theorem D. }{\it Let $h_v(\la)$ be the 
exponential growth rate of the 
average volume of a magnetic ball, i.e.
\[h_{v}(\la):=\limsup_{T\to\infty}\frac{1}{T}\log\int_{M}\mbox{\rm Vol}\,B_{mag}(x,\la,T)\,dx.\]
Then
\[h_{top}(\la)\geq h_{v}(\la).\]
If the $\la$-magnetic flow is Anosov, then
\[h_{top}(\la)=h_{v}(\la)=
\lim_{T\to\infty}\frac{1}{T}\log\int_{M}\mbox{\rm Vol}\,B_{mag}(x,\la,T)\,dx.\]}

\vspace{.5cm}

A particular case of our theorem was obtained
by S. Grognet in~\cite{G2}.

Theorem D applies to {\it any} magnetic flow; 
it is not even necessary to assume that
the lift $\wO$ of the magnetic field to the universal cover has a primitive.
In the case when $\wO$ does have a primitive $\Theta$, 
the lifts to $\M$ of the 
 $\la$-magnetic geodesics (which we still call $\la$-magnetic geodesics) are  
solutions to the Euler-Lagrange equation for the Lagrangian
 \[L_{\la}(x,v)=|v|_{x}^{2}/2-\la\Theta_x(v).\]
 They are extremals  for the action of $L_\la + 1/2$.
A $\la$-magnetic geodesic $\ga$ between $x$ and $y$ is called 
{\it minimizing} if
 \[ A_{L_\la + 1/2}(\ga) \leq  A_{L_\la + 1/2}(\overline\ga) \]
 for any other curve $\overline\ga$ joining $x$ to $y$ ($\overline\ga$
 is defined on arbitrary time intervals). We show in Lemma~\ref{constspeed}
that the action of $ L_\la + 1/2$ along a curve 
which does not have unit speed will decrease if 
the curve is reparametrized to have unit speed.
Thus a minimizer must have speed one.


A theorem due to Ma\~{n}\'{e} \cite{CDI,Ma1} ensures that any two distinct points in $\M$
are joined by a minimizing $\la$-magnetic
geodesic, provided  
 $$
 1/2 > c(L_\la) = c(g,\la\Omega) = \la^{2}\,c(g,\Omega).
 $$
In this case, i.e.~when $0 \leq \la < 1/\sqrt{2c(g,\Omega)}$, 
we can define the {\it minimal ball} 
\[B_{min}(x,\la,T)=\{y\in\M:\;\mbox{there is a minimizing $\la$-magnetic geodesic}\qquad\]
\[\qquad\qquad\qquad\mbox{ from $x$ to $y$ with length
$< T$}\}.\]
It is obvious that $$B_{min}(x,\la,T) \subset B_{mag}(x,\la,T).$$
 In the case when the $\la$-magnetic flow is Anosov on $SM$, which entails
$0 \leq \la < 1/\sqrt{2c(g,\Omega)}$ by Theorem~C, 
there is only one $\la$-magnetic geodesic from a point of $\M$ to another.
It must perforce be the minimizing $\la$-magnetic geodesic guaranteed by 
Ma\~{n}\'{e}'s theorem, and hence  $$B_{min}(x,\la,T) = B_{mag}(x,\la,T)$$ 
when the  $\la$-magnetic flow is Anosov on $SM$.

We show in Section~\ref{monot} that ${\rm Vol}\,B_{min}(x,\la,T)$ 
is a nonincreasing function on the interval
$0 \leq \la < 1/\sqrt{2c(g,\Omega)}$ for any given $x$ and $T$. It follows from this and
Theorem~D that $h_{top}(\la)$ is nonincreasing on the set of $\la$ such that
the  $\la$-magnetic flow is Anosov on $SM$.


In \cite{PP1/2,PP1} G. and M. Paternain showed that if
we start with an Anosov geodesic flow and $\Omega\not\equiv 0$
then, the function $\la\mapsto h_{top}(\la)$ is {\it strictly}
decreasing in the Anosov interval containing zero. 
But their arguments in \cite{PP1} in fact show more.
They prove that if the $\la$-magnetic flow is Anosov, then
$h_{top}'(\la)\neq 0$ and for this property $0$ does not need to belong
to the Anosov interval that contains~$\la$. 
Combining  this result with the above 
discussion gives

\vspace{.5cm}

\noindent{\bf Theorem E. }{\it  Suppose that  $ 0 \leq \la_{1} < \la_{2}$ and
that the $\la_{i}$-magnetic flow is Anosov for $i=1,2$. Then
\[h_{top}(\la_{1}) > h_{top}(\la_{2}).\]
}

\vspace{.5cm}


Our earlier examples show that there can
be more than one Anosov interval in
$[0,1/\sqrt{2\,c(g,\Omega)})$. This raises the following
natural question.

\vspace{.5cm}

\noindent{\bf Question.}\;Can the topological entropy go up
 in between two Anosov intervals?

\vspace{.5cm}

A further refinement of our examples shows that this is indeed possible.





 \vspace{.5cm}


{\it Acknowledgements:} The first author thanks the Universidad
de la Rep\'ublica in Uruguay for hospitality while this work was started
and the Centro de Investigaci\'on in Matem\'aticas in Mexico while
this work was finished.


\section{Proof of Theorems A and B} \label{proofAB}



We first prove the following result that we mentioned in the introduction.


\begin{Proposition} Suppose that $\wO$ is not exact.
Then $\mbox{\rm iso}(g,\Omega)=0$.
\label{0iso}
\end{Proposition}



\begin{proof}Since $\wO$ is not exact, 
there exists a cycle $c\in H_{2}(\M,\Z)$ such that
\[\int_{c}\wO\neq 0.\]
Since $\M$ is simply connected, the Hurewicz isomorphism theorem ensures that
$\pi_{2}(M)$ is isomorphic to $H_{2}(\M,\Z)$. Hence there exists a smooth map
$f:S^{2}\to\M$ such that
\[\int_{S^{2}}f^{*}(\wO)\neq 0.\]
Endow $S^{2}$ with the canonical metric and fix a point $x\in S^{2}$.
Consider a disk $U_{\e}$ in $S^{2}$ which is given by the complement 
of an open geodesic disk
with center at $x$ and radius $\e$. As $\e\to 0$ we have
\[\int_{U_{\e}}f^{*}(\wO)\to \int_{S^{2}}f^{*}(\wO)\neq 0,\]
\[\ell(f|_{\partial U_{\e}})\to 0,\]
and hence $\mbox{\rm iso}(g,\Omega)=0$.



\end{proof}

In the rest of this section we assume that $\wO$ is exact.


\begin{Lemma}$c(g,\Omega)>0$ if $\Omega$ is non-trivial.
\label{mqc}
\end{Lemma}


\begin{proof}Suppose that $c(g,\Omega)=0$. 
Since the image of $\ga$ is a compact set,
there exists, for any $\varepsilon>0$,  a
 smooth function
$u:\M\to\re$ such that 
$|d_{x}u+\Theta_{x}|<\varepsilon$
for all $x$ in the image of ~$\ga$. 
Hence
 $$
\left|\int_{\ga}\Theta\right| = \left|\int_{\ga}du +\Theta\right|
 \leq \varepsilon\,\ell,
$$
where $\ell$ is the length of $\ga$. Since this holds for all $\varepsilon$,
\[\int_{\ga}\Theta=0.\]
Since $\ga$ was arbitrary, this implies that $\Theta$ is exact
and thus $\Omega\equiv 0$.

\end{proof}







\subsection{Proof of Theorem A}
If $c(g,\Omega) = 0$, then $\Omega$ is trivial by the previous lemma 
and we have $\mbox{\rm iso}(g,\Omega) = \infty$. 
Thus we can assume that $c(g,\Omega) > 0$.

Suppose  $0<k<c(g,\Omega)$. 
Then there exists an absolutely continuous closed curve
$\alpha:[0,T]\to \M$ such that $A_{L+k}(\alpha)<0$. 

Since $\alpha([0,T])$ is a compact set, by Theorem 3.5.2 in \cite{Fa}
there exists a constant $\delta_{0}>0$ such that if $x,y\in \alpha([0,T])$ 
and $\delta\in [0,\delta_{0}]$ satisfy $d(x,y)\leq \delta$,
then there exists a magnetic geodesic defined in $[0,\delta]$ that connects
$x$ to $y$ and that minimizes the action among all absolutely continuous curves
from the fixed interval $[0,\delta]$ into $\M$ that connect $x$ to $y$.
Therefore by dividing the interval $[0,T]$ into sufficiently small subintervals
we can find a curve $\ga:[0,T]\to\M$ which is a piecewise magnetic geodesic and
such that
 \[A_{L+k}(\ga)\leq A_{L+k}(\alpha)<0.\]

%By rounding off the corners if
%necessary, we obtain a curve, which we still denote by $\ga$, with 
%negative $L+k$~action that defines a smooth map of $S^1$ into~$\M$.

%Since local minimzers of the action 
%always exist for the Lagrangian
%$L$ (see \cite[Theorem 3.5.1]{Fa})
%we can find a curve $\ga:[0,T]\to\M$ which is 
%a piecewise magnetic geodesic and
%such that
% \[A_{L+k}(\ga)\leq A_{L+k}(\alpha)<0.\]



\begin{Lemma} Let $\sigma:[0,\ell]  \to \widetilde M$ be an 
absolutely continuous  curve parametrized by arc length. 
The reparametrization of $\sigma$ that minimizes $A_{L+k}$ 
has constant speed $\sqrt{2k}$.
\label{constspeed}
\end{Lemma}

\begin{proof}This is an immediate consequence of Lemma~3-3.2 in \cite{CI2}. 
Alternatively, suppose the reparametrization has speed $v(t)$ at $\sigma(t)$. 
Then the action of $L + k$ along the reparametrization is
$$ \int_0^\ell \frac {v(t)^2}2 + k -
\Theta(v(t)\dot\sigma(t))\,\frac{dt}{v(t)} = \int_0^\ell \frac
{v(t)}2 + \frac k{v(t)}\,dt - \int_\sigma \Theta. $$ Since the
last integral is independent of the reparametrization and the
function $$ v \mapsto \frac v2 + \frac kv $$ has a unique minimum
at $v = \sqrt{2k}$, the action is minimized when $v(t) \equiv
\sqrt{2k}$.
\end{proof}

If necessary, we now reparametrize $\ga$ so that it has constant speed 
$\sqrt{2k}$. 
Since this reparametrization can only decrease the $L+k$ action, 
$\ga$ now has energy $k$ and negative $L+k$ action.
The curve $\ga$ is smooth except for a possible finite number of corners. 
By rounding off these corners if necessary we can obtain
a curve, which we still denote by $\ga$, 
with negative $L+k$~action that defines a smooth map of $S^1$
into~$\widetilde M$ and has energy $k$.
It is clear from the proof of the lemma that the new parametrization gives us
$$
 A_{L+k}(\ga) = \sqrt{2k}\ell - \int_\ga\Theta,
 $$
 where $\ell$ is the length of $\ga$.
Since $A_{L+k}(\ga) < 0$, we obtain
\[\sqrt{2k}<\frac{\int_{\ga}\Theta}{\ell},\]
and since the left hand side is independent of the parametrization we can assume that
$\ga$ is a smooth map from the unit circle in the plane into $\widetilde M$.

On the other hand, since $\M$ is simply connected $\ga$ can be extended 
to a smooth map $\vr: \D^2 \to \widetilde M$ such that $\vr|\partial \D^2 = \ga$. 
For any such extension $\vr$ we have
$$
 \int_{\D^2} \vr^*(\wO) = \int_\ga \Theta.
 $$
We obtain:
\[\sqrt{2k}<\frac{\int_{\ga}\Theta}{\ell}
=\frac{a(\vr)}{\ell(\partial\vr)}\leq \frac{1}{\mbox{\rm iso}(g,\Omega)}.\]
Since $k<c(g,\Omega)$ was arbitrary, this yields:
\[\sqrt{2\,c(g,\Omega)}\leq \frac{1}{\mbox{\rm iso}(g,\Omega)}.\]


To prove that this inequality is in fact an equality we argue by contradiction.
Suppose that there exists $r>0$ such that
\[\sqrt{2\,c(g,\Omega)}<r<\frac{1}{\mbox{\rm iso}(g,\Omega)}.\]
This means that we can find a smooth regular closed curve $\ga:S^{1}\to\M$ 
with length $\ell$ such that
\[r<\frac{\int_{\ga}\Theta}{\ell},\]
which implies:
\[r\,\ell-\int_{\ga}\Theta<0.\]
If we reparametrize $\ga$ to a curve $\tilde{\ga}:[0,T]\to\M$ so that $\tilde{\ga}$
 has energy $k:=r^{2}/2$ we obtain:
\[2kT-\int_{\ga}\Theta<0,\]
which by the Lagrangian definition of the critical value means
that $r^{2}/2=k<c(g,\Omega)$. This contradiction completes the proof
of the theorem.


\qed


\subsection{Proof of Theorem B}
Let $g_{0}=\rho^{2} g$ be the metric of constant negative curvature 
such that $a_{g_{0}}=a_{g}$.
Lift $g$ and $g_{0}$ to metrics $\widetilde{g}$ and $\widetilde{g_{0}}$ in $\M$.
Let $C(R)$ be a geodesic circle of radius $R$ with respect to the metric $\widetilde{g_{0}}$
and let $D(R)$ be the disk bounded by $C(R)$. Let $\Omega_{0}$ be the area form of
$g_{0}$. Write:
\begin{align*}
\Omega\,&=f\,\Omega_{0},\;\mbox{\rm where $f$ is smooth function on $M$},\\
\ell_{0}(R)&:=\widetilde{g_{0}}\mbox{\rm -length of }C(R),\\
\ell(R)&:=\widetilde{g}\mbox{\rm -length of }C(R),\\
a_{0}(R)&:=\widetilde{g_{0}}\mbox{\rm -area of }D(R),\\
a(R)&:=\int_{D(R)}\widetilde{\Omega}=\int_{D(R)}\widetilde{f}\,\widetilde{\Omega}_{0}.
\end{align*}
In the disk $D(R)$ we introduce coordinates $(r,s)$, where $r$ is
the $\widetilde{g_{0}}$-arc length parameter along radial
geodesics and $s$ is the $\widetilde{g_{0}}$-arc length parameter
along concentric circles. Let $k$ be the square root of minus the
curvature of $g_{0}$. We have:
\begin{equation}
\ell_{0}(R)=\frac{2\pi}{k}\sinh\,k R=\frac{da_{0}}{dR}(R),
\label{eq1}
\end{equation}
\begin{equation}
a_{0}(R)=\frac{2\pi}{k^{2}}(\cosh(k R)-1),
\label{eq2}
\end{equation}
\begin{equation}
\ell(R)=\int_{0}^{\ell_{0}(R)}\widetilde{\rho}^{-1}(R,s)\,ds,
\label{eq3}
\end{equation}
\begin{equation}
a(R)=\int_{0}^{R}\int_{0}^{\ell_{0}(r)}\widetilde{f}(r,s)\,dsdr.
\label{eq4}
\end{equation}
The key observation is that the projection to $M$ of a circle in
$\M$ converges to a horocycle when the radius goes to infinity,
and the projection to the unit sphere bundle of $(M,g_{0})$ of the
normalized arc length measure weakly converges to an invariant
measure for the horocycle flow. But the only invariant measure for
the horocycle flow is the Liouville measure. Hence from
(\ref{eq3}) we obtain:
\begin{equation}
\lim_{R\to +\infty}\frac{\ell(R)}{\ell_{0}(R)}=\lim_{R\to +\infty}\frac{1}{\ell_{0}(R)}
\int_{0}^{\ell_{0}(R)}\widetilde{\rho}^{-1}(R,s)\,ds=\int_{M}\rho^{-1}\,d\mu_{0}=\rho_{g}.
\label{eq5}
\end{equation}
Similarly, we obtain:
\begin{equation}
F:=\lim_{R\to +\infty}\frac{1}{\ell_{0}(R)}
\int_{0}^{\ell_{0}(R)}\widetilde{f}(R,s)\,ds=\int_{M}f\,d\mu_{0}=
\frac{1}{a_{g}}\int_{M}\Omega.
\label{eq6}
\end{equation}
The last equality implies that given $\e>0$ there exists $m>0$ such that for all $R\geq m$
we have
\[(F+\e)\ell_{0}(R)\geq\int_{0}^{\ell_{0}(R)}\widetilde{f}(R,s)\,ds\geq (F-\e)\ell_{0}(R),\]
and hence from (\ref{eq1}) and (\ref{eq4}) we obtain:
\begin{align*}
\limsup_{R\to +\infty}\frac{a(R)}{\ell_{0}(R)}
&\geq (F-\e)\limsup_{R\to +\infty}\frac{1}{\ell_{0}(R)}\int_{m}^{R}\ell_{0}(r)\,dr\\
&=(F-\e)\limsup_{R\to +\infty}\frac{1}{\ell_{0}(R)}(a_{0}(R)-a_{0}(m)).
\end{align*}
Now observe that from (\ref{eq1}) and (\ref{eq2}) we get:
\begin{equation}
\lim_{R\to +\infty}\frac{a_{0}(R)}{\ell_{0}(R)}=1/k,
\label{eq7}
\end{equation}
and hence
\begin{equation}
\limsup_{R\to +\infty}\frac{a(R)}{\ell_{0}(R)}\geq (F-\e)/k.
\label{eq8}
\end{equation}
It follows from (\ref{eq5}), (\ref{eq7}) and (\ref{eq8}) that:
\begin{equation}
\limsup_{R\to +\infty}\frac{a(R)}{\ell(R)}=
\limsup_{R\to +\infty}\frac{a(R)}{\ell_{0}(R)}\frac{\ell_{0}(R)}{\ell(R)}\geq
\frac{F}{k\,\rho_{g}}.
\label{eq9}
\end{equation}
Combining Theorem A with (\ref{eq9}) yields:
\[c(g,\Omega)\geq \frac{F^{2}}{2k^{2}\,\rho_{g}^{2}}
=\frac{a_{g}\,F^{2}}{-4\pi\chi\,\rho_{g}^{2}}.\]
This inequality and equation (\ref{eq6}) complete the proof of Theorem B.

\qed

We conclude this section with the following remarks.
Let us normalize the metric $g$ so that $\rho^{2} g$ has curvature $-1$. 
Then $-2\pi\chi=a_{g}$ and hence
Theorem B says that:
\[c(g,\Omega)\geq \frac{\left(\int_{M}\Omega\right)^{2}}{2\,\rho_{g}^{2}\,a_{g}^{2}}
\geq \frac{\left(\int_{M}\Omega\right)^{2}}{2\,a_{g}^{2}}.\]
Assume further that the cohomology class of $\Omega$ is the same up to  sign 
as that of the area form $\Omega_{g}$  or equivalently
that
\[\int_{M}\Omega=\pm\int_{M}\Omega_{g}.\]
Then the last inequality implies that
\[c(g,\Omega)\geq 1/2,\]
and $g$ has constant curvature $-1$ if equality holds. It follows
that the functional
\[g\mapsto c(g,\Omega_{g}),\]
over the space of metrics $g$ with $a_{g}=-2\pi\chi$ achieves its
minimum $1/2$ if and only if $g$ has constant curvature $-1$.

In \cite[Theorem 1.3]{PP1/2} Paternain and Paternain proved that
if the magnetic flow of $(g,\la\Omega)$ is Anosov, then there is
an upper bound for $\lambda^{2}$ in terms of the curvature tensor
of $g$ and the form $\Omega$. When $M$ is a surface and
$\Omega\equiv \Omega_{g}$ this bound reads:
\[\la^{2}<\frac{-\int_{M}K(x)\,dx}{a_{g}}=\frac{-2\pi\chi}{A_{g}},\]
where $K$ is the Gaussian curvature.
On the other hand our Theorems B and C give:
\[\la^{2}< \frac{1}{2\,c(g,\Omega_{g})}\leq \frac{-2\pi\chi\,\rho_{g}^{2}}{A_{g}}\leq 
\frac{-2\pi\chi}{a_{g}},\]
It follows that the bound in Theorem C is always sharper than
Paternain and Paternain's bound unless $g$ has constant negative curvature in which
case they coincide.






\section{Proof of Theorem C} \label{proofC}

We  recall the following key fact proved
by G. Paternain and M. Paternain in \cite{pp1,pp2}.

\begin{Proposition}Suppose that the magnetic flow $\phi$ of the pair
$(g,\Omega)$ is Anosov. Then the weak stable foliation
${\mathcal W}^{s}$ of $\phi$ is
transverse to the fibres of the fibration $\pi:SM\to M$.
\label{transverse}
\end{Proposition}

Let $\M$ be the universal covering of $M$ with covering
projection $p:\M\to M$.
Let $\widetilde{{\mathcal W}}^{s}$ be the lifted foliation which
is in turn a weak stable foliation for the lifted magnetic flow.
The next observation appears also in \cite{CIPP,PP2}.

\begin{Lemma}For any $(x,v)\in S\M$,
$\widetilde{{\mathcal W}}^{s}(x,v)$ intersects
each fibre of the fibration $\widetilde{\pi}:S\M\to \M$ at just one point.
\label{1-1}
\end{Lemma}

\begin{proof}By Proposition \ref{transverse} the foliation
 $\widetilde{{\mathcal W}}^{s}$ is transverse to the fibration
$\widetilde{\pi}:S\M\to\M$.
Since the fibres are spheres (which are compact)
a result of Ehresman (cf. \cite{CLN}) implies that
for every $(x,v)\in S\M$ the map
\[\widetilde{\pi}|_{\widetilde{{\mathcal W}}^{s}(x,v)}:\widetilde{{\mathcal W}}^{s}(x,v)\to \M,\]
is a covering map. Since $\M$ is simply connected,
$\widetilde{\pi}|_{\widetilde{{\mathcal W}}^{s}(x,v)}$ is in fact
a diffeomorphism
and $\widetilde{{\mathcal W}}^{s}(x,v)$ is simply connected.
Consequently, $\widetilde{{\mathcal W}}^{s}(x,v)$ intersects
each fibre of the fibration
$\widetilde{\pi}:S\M\to \M$
at just one point.


\end{proof}

The lemma implies that each leaf $\widetilde{{\mathcal W}}^{s}(x,v)$
is diffeomorphic to $\M$ and hence $\M$ is diffeomorphic
to $\re^{n}$. This implies that $\wO$ is an exact
form, so we can write $\wO=d\Theta$ for some
smooth 1-form $\Theta$.
As in the introduction, let us consider the Lagrangian on $\M$ given by
\[L(x,v)=\frac{1}{2}|v|^{2}_{x}-\Theta_{x}(v).\]
The extremals of $L$, i.e., the solutions of the Euler-Lagrange
equations of $L$ coincide with the lift to $\M$ of the
magnetic geodesics.
The Hamiltonian associated with $L$ is
\[H(x,p)=\frac{1}{2}|p+\Theta_{x}|^{2}.\]
The Legendre transform
${\mathcal L}:T\M\to T^{*}\M$ takes orbits of $L$ to orbits
of the Hamiltonian flow of $H$ with respect to the canonical
symplectic form of $T^{*}\M$.

\begin{Lemma}There exists $\e>0$ such that for any
$k\in (1/2-\e,1/2+\e)$, there exists a smooth function
$u:\M\to\re$ such that for all $x\in\M$ we have
\[H(x,d_{x}u)=k.\]
\label{llave}
\end{Lemma}

\begin{proof}The Legendre transform
${\mathcal L}:T\M\to T^{*}\M$ associated to $L$ carries $S\M$
diffeomorphically onto the level set $\{H=1/2\}$. Using
${\mathcal L}$ we can push forward
the weak stable foliation $\widetilde{{\mathcal W}}^{s}$
to obtain the weak stable foliation
of the Hamiltonian flow of $H$ with respect to the canonical
symplectic form of $T^{*}\M$. 
By the last lemma, the leaves of this foliation, 
which we still denote by $\widetilde{{\mathcal W}}^{s}$,
project diffeomorphically onto $\M$; and hence, for any $(x,p)$
with $H(x,p)=1/2$,
$\widetilde{{\mathcal W}}^{s}(x,p)$
is the graph of a 1-form $\omega$. On the other hand it is well
known that the weak stable leaves are Lagrangian
submanifolds and hence $\omega$ must be closed.
Since any closed 1-form in the universal covering
must be exact, it follows that each
leaf $\widetilde{{\mathcal W}}^{s}(x,p)$ is the graph
of an exact 1-form. This means that there exists
a smooth function $u:\M\to\re$ such that
$H(x,d_{x}u)=1/2$. Finally by structural stability, if the magnetic
flow is Anosov, there exists $\e>0$ such that the restriction of the
Hamiltonian flow of $E$ with respect to $\omega_{1}$ to
the energy level $E^{-1}(k)$ is Anosov for any
$k\in (1/2-\e,1/2+\e)$. The previous argument can be reproduced
to obtain a smooth function $u:\M\to\re$ such that
$H(x,d_{x}u)=k$ for all $x\in \M$ and all $k\in (1/2-\e,1/2+\e)$
as desired.


\end{proof}

By Lemma \ref{llave} there exist $\e>0$ and a smooth function
$u:\M\to\re$ such that $d_{x}u$ is in the level set $H =  1/2-\e/2$
for all $x\in\M$. That is, for all $x\in \M$,
\[|d_{x}u+\Theta_{x}|^{2}=1-\e.\]
Recall that
\begin{align*}
c(g,\Omega)&=\inf_{u\in C^{\infty}(\M,\re)}\sup_{x\in\M}H(x,d_{x}u)\\
&=\inf_{u\in C^{\infty}(\M,\re)}\sup_{x\in\M}
          \frac{1}{2}|d_{x}u+\Theta_{x}|^{2}.
\end{align*}
It follows that
\[c(g,\Omega)\leq 1/2-\e/2<1/2,\]
which proves the first claim in Theorem C.


Given $(x,v)\in\M$, let $E^{s}(x,v)\subset T_{(x,v)}S\M$ be the
weak stable subspace at $(x,v)$ and let $E^{ss}(x,v)$ be
the strong stable subspace at $(x,v)$. Let $X(x,v)$ be the vector
field of the magnetic flow at $(x,v)$. The subspace
$E^{s}(x,v)$ is the direct sum of $E^{ss}(x,v)$ and the
1-dimensional subspace spanned by $X(x,v)$. Since $M$ is compact
and $E^{s}$ is transverse to the vertical subbundle
$V:=\mbox{\rm ker}\,d\widetilde{\pi}$ (cf. Proposition
\ref{transverse}), the angle between $E^{s}(x,v)$
and $V(x,v)$ is uniformly bounded away from zero.
Hence there exists a positive constant $R_{1}$ such that
for all $(x,v)\in S\M$ and all $\xi\in E^{s}(x,v)$ we have
\begin{equation}
|d_{(x,v)}\widetilde{\pi}(\xi)|_{\M}\leq
|\xi|_{SM}\leq
R_{1}\,|d_{(x,v)}\widetilde{\pi}(\xi)|_{\M},
\label{angulo1}
\end{equation}
where we consider in $S\M$ the Sasaki metric induced by
the metric in $\M$.

Given $w\in T_{x}\M$ we can write in a unique way
$w=z+u$ where $z\in d_{(x,v)}\widetilde{\pi}(E^{ss}(x,v))$
and $u$ is a vector collinear with $v$.
Since the angle between $E^{ss}(x,v)$
and the 1-dimensional subspace spanned by $X(x,v)$
is uniformly bounded away from zero there exists another positive
constant $R_{2}$ such that
\begin{equation}
|z|\leq R_{2}\,|w|.
\label{angulo2}
\end{equation}

Now fix a weak stable leaf $W$ in $S\M$. The restriction
of $\widetilde{\pi}$ to $W$ gives a diffeomorphism between
$W$ and $\M$ by Lemma \ref{1-1}. Using $W$ we can define
a smooth vector field $Z$ on $\M$ by setting
$Z(x):=v=d_{(x,v)}\widetilde{\pi}(X(x,v))$ where $v$ is the unique
unit vector such that $(x,v)\in W$. Let $\tau_{s}$ be the flow
of $Z$. Clearly
\[\tau_{s}(x)=\widetilde{\pi}\circ \phi_{s}(x,v).\]

\begin{Lemma}There exist positive constants
$C$ and $\kappa$ such that for all $x\in \M$
and all $z\in d_{(x,v)}\widetilde{\pi}(E^{ss}(x,v))$ we have
\[|d_{x}\tau_{s}(z)|\leq C\,e^{-\kappa s}|z|,\]
for all $s\geq 0$.
\label{contraccion}
\end{Lemma}

\begin{proof}Since the magnetic flow is Anosov, there exist
positive constants $C_{1}$ and $\kappa$ such that for all
$(x,v)\in S\M$ and all $\xi\in E^{ss}(x,v)$ we have
\[|d_{(x,v)}\phi_{s}(\xi)|_{S\M}\leq C_{1}\,e^{-\kappa s}|\xi|_{S\M},\]
for all $s\geq 0$.
Combining this inequality with (\ref{angulo1}) and the definition
of $\tau_{s}$ we obtain
\[|d_{x}\tau_{s}(z)|_{\M}\leq C_{1}\,R_{1}\,e^{-\kappa s}|z|_{M},\]
for all $s\geq 0$. Set $C:=C_{1}\,R_{1}$.

\end{proof}



Let us complete the proof of Theorem C. We follow
Pansu in  Proposition 7.1 of \cite{Pa}.
Let $\tau_{s}$ be the flow of the vector field $Z$ that we introduced
above.

Let $\alpha$ be a closed bounded $k$-form on $\M$ with $k\geq 2$.
To solve $d\beta=\alpha$ we use Poincar\'e's formula
\[\beta_{T}=-\int_{0}^{T}\tau_{s}^{*}(i_{Z}\alpha)\,ds.\]
Since $\alpha$ is closed, Cartan's formula $L_Z = i_Zd + di_Z$ and
the fact that $d$ is linear and commutes with pullbacks give us
\[
\alpha - \tau^*_T\alpha  = -\int_0^T \frac d{ds} \tau^*_s\alpha\,ds
   = -\int_0^T \tau^*_s L_Z\alpha\,ds
   = -\int_0^T \tau^*_s di_Z\alpha\,ds
   = d\beta_T.
   \]

%Since $\alpha$ is closed, Cartan's formula implies:
%\[\left.\frac{d}{ds}\right|_{s=0}(\tau_{s}^{*}\alpha)=d(i_{Z}\alpha).\]
%Then $d\beta_{T}=\alpha-\tau_{T}\alpha$.

Take $x\in\M$ and $w_{1},\dots,w_{k-1}\in T_{x}\M$.
Write
\[ w_{i}=z_{i}+u_{i}, \]
 where
$z_{i}\in d_{(x,v)}\widetilde{\pi}(E^{ss}(x,v))$ and $u_{i}$
is collinear with $v:=Z(x)$ for $i=1,\dots,k-1$.
Then
\begin{align*}
\left[\tau^{*}_{s}(i_{Z}\alpha)\right]_{x}(w_{1},\dots,w_{k-1})&=
\alpha_{\tau_{s}(x)}(d_{x}\tau_{s}(w_{1}),\dots,d_{x}\tau_{s}(w_{k-1}),
d_{x}\tau_{s}(Z(x))\\
&=\alpha_{\tau_{s}(x)}(d_{x}\tau_{s}(z_{1}),\dots,d_{x}\tau_{s}(z_{k-1}),
Z(\tau_{s}(x))).
\end{align*}
Hence using Lemma \ref{contraccion} and (\ref{angulo2})
we obtain
\begin{align*}
\left|\right.\left[\tau^{*}_{s}(i_{Z}\alpha)\right]_{x}(w_{1},\dots,w_{k-1})\left|\right.
&\leq \|\alpha\|_{\infty}\,|d_{x}\tau_{s}(z_{1})\wedge\dots\wedge d_{x}\tau_{s}(z_{k-1})|\\
&\leq \|\alpha\|_{\infty}\,C^{k-1}\,e^{-\kappa(k-1)s}|z_{1}\wedge\dots\wedge z_{k-1}|\\
&\leq \|\alpha\|_{\infty}\,C^{k-1}\,R_{2}^{k-1}\,e^{-\kappa(k-1)s}|w_{1}\wedge\dots\wedge w_{k-1}|.
\end{align*}
Hence, the form $\beta_{T}$ converges as $T\to +\infty$ to a form $\beta$
such that
\[|\beta_{x}(w_{1},\dots,w_{k-1})|
\leq \frac{\|\alpha\|_{\infty}\,C^{k-1}\,R_{2}^{k-1}}{(k-1)\kappa}|w_{1}\wedge\dots\wedge w_{k-1}|,\]
and hence $\beta$ is a bounded form.
Also $\tau^*_{T}\alpha$ tends to zero so $d\beta=\alpha$.

\qed






\section{Monotonicity of the volume of minimal balls} \label{monot}
We suppose in this section that $\wO$ has a bounded primitive $\Theta$.
We begin by showing that $x\mapsto\mbox{\rm Vol}\,B_{min}(x,\la,T)$ is
invariant under covering transformations. Let $\varphi:\M\to\M$ be
a covering transformation. Since $\varphi^{*}\Theta-\Theta$ is
closed and $\M$ is simply connected, there exists a smooth
function $f$ such that $\varphi^{*}\Theta-\Theta=df$. Hence
$$L_{\la}\circ d\varphi=L_{\la}+df,$$ and therefore if
$\gamma:[0,T]\to \M$ is a curve connecting $x$ and $y$ we have
\[A_{L_{\la}+1/2}(\varphi\,\ga)=A_{L_{\la}+1/2}(\gamma)+f(y)-f(x).\]
Hence $\varphi$ takes minimizing $\la$-magnetic geodesics to minimizing
$\la$-magnetic geodesics and $\la$-magnetic geodesics to
$\la$-magnetic geodesics.
It follows immediately that we have
$\varphi\,B_{min}(x,\la,T)=B_{min}(\varphi(x),\la,T)$ and
$\varphi\,B_{mag}(x,\la,T)=B_{mag}(\varphi(x),\la,T)$.





\begin{Lemma}Suppose $0 \leq \la_1 \leq \la_2 < \sqrt{2c(g,\Omega)}$. 
Let $T(\la_i)$ be the length
of a minimizing
 $\la_i$-magnetic geodesic $\ga_i$ from $x$ to $y$
for $i =1,2$. Then
\[T(\la_2)\geq T(\la_1).\]
\label{4.1}
\end{Lemma}


\begin{proof}Using the minimization property we get
\[A_{L_{\la_1}+1/2}(\ga_1)\leq
A_{L_{\la_1}+1/2}(\ga_2),\]
\[A_{L_{\la_2}+1/2}(\ga_2)\leq
A_{L_{\la_2}+1/2}(\ga_1).\]
These are equivalent to
\[T(\la_1)-\la_1\int_{\ga_1}\Theta\leq
T(\la_2)-\la_1\int_{\ga_2}\Theta,\]
\[T(\la_2)-\la_2\int_{\ga_2}\Theta\leq
T(\la_1)-\la_2\int_{\ga_1}\Theta.\]
Hence
\[(\la_2-\la_1)\left(\int_{\ga_1}\Theta
                -\int_{\ga_2}\Theta\right)\leq 0.\]
It follows that
\[T(\la_1)\leq T(\la_2)+
\la_1\left(\int_{\ga_1}\Theta-\int_{\ga_2}\Theta\right)
      \leq T(\la_2).\]
\end{proof}




\begin{Proposition}\label{4.2}
If $0 \leq \la_1 \leq \la_2 <    1/\sqrt{2c(g,\Omega)}$, then
\[B_{min}(x,\la_2,T)\subseteq B_{min}(x,\la_1,T).\]
\end{Proposition}

\begin{proof}Take $y\in B_{min}(x,\la_2,T)$. By definition,
there exists a minimizing $\la_2$-magnetic geodesic from $x$ to $y$
with length $T(\la_2)\leq T$.
A theorem due to Ma\~n\'e \cite{Ma1,CDI} ensures that
there exists a minimizing $\la_1$-magnetic geodesic
from $x$ to $y$ with length $T(\la_1)$.
By Lemma \ref{4.1} $T(\la_1)\leq T(\la_2)\leq T$, and hence
$y\in  B_{min}(x,\la_1,T)$.

\end{proof}

%The proposition implies right away that the function
%$[0,1/\sqrt{2\,c(g,\Omega)})\ni\la\mapsto h_{v}(\la)$$
%is nonincreasing.




\begin{Remark}{\rm It might seem more natural to consider the balls
\[\{y\in\M:\;\Phi_{1/2}(x,y)<T\},\]
where $\Phi_{1/2}$ is the action potential for the Lagrangian $L_{\la}$
as defined in Appendix A. 
But it is not clear how to relate the growth rate of 
volume of these balls to the entropy of the magnetic flow 
because there is no simple 
relationship beween time and the action potential.
Peyerimhoff and Siburg \cite{PS} have shown that the ratio between
the metric
$d^\la_{mag}(x,y) = \Phi_{1/2}(x,y) + \Phi_{1/2}(y,x)$, which is defined by the action 
potential when $\la < 1/\sqrt{2c(g,\Omega)}$, and the Riemannian distance 
approaches $1$ as the distance from $x$ to $y$ approaches $0$.
An immediate consequence is that $d^\la_{mag}$ and the Riemannian metric 
define the same inner metric.

Since it is the action rather than the length that is minimized
along a minimizing magnetic geodesic, it is possible that
$B_{min}(x,\la,T)$ is strictly contained in $B_{mag}(x,\la,T)$.
Note that we have Proposition~\ref{4.2}  only for minimal balls,
because Lemma \ref{4.1} applies to {\it minimizing} magnetic geodesics.

In the case of the geodesic flow Manning \cite{Man1} showed that
\[\lim_{T\to\infty}\frac{1}{T}\log \mbox{\rm Vol} B(x,0,T)=h_{v}(0)\]
for all $x\in\M$. The proof is based on the fact that if $d$ is the distance
between $x$ and $y$, then
\[B(x,0,T)\subset B(y,0,T+d),\]
which in turn comes from the triangle inequality. 
Since the triangle inequality applies only to geodesic triangles, 
it is not clear that Manning's result extends to the case when $\la\neq 0$.}
\end{Remark}





\section{Proof of Theorem D} \label{proofD}




To simplify the notation we omit the dependence
on $\la$ from the notation.


We use the following theorem, which is a variant of a theorem 
due to C. Niche \cite{N}. A proof is sketched in 
Appendix B for the sake of completeness. 
The theorem generalizes Ma\~n\'e's formula
for geodesic flows \cite{Ma2}.
 Given $\theta =(x,v)\in SM$, let $X(\theta)$ be
the vector field of the magnetic flow of the pair $(g,\Omega)$ and let
\[\alpha(\theta)=\mbox{\rm ker}\,d_{\theta}\pi\oplus\re\,X(\theta),\]
where $\pi:SM\to M$ is the canonical
projection and $d_\theta$ denotes the derivative at $\theta$. 
Let $h_{top}$ be the topological entropy of the magnetic flow.


\begin{Theorem}Let $(g,\Omega)$ be a $C^{\infty}$ pair. Then we always have:
\[h_{top}\geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta.\]
Suppose in addition that the magnetic flow $\phi$ admits a continuous
invariant distribution of codimension one transversal to $X$. Then
\[h_{top}=\lim_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta.\]

\label{mf}
\end{Theorem}

We remark that in this theorem it is not important which metrics we choose
in $M$ and in $SM$ to measure the absolute value of the determinant.
Also note that when the magnetic flow is Anosov, the sum of the strong stable
and strong unstable bundles provides a continuous
invariant distribution of codimension one transversal to $X$.



Given $x\in\M$, we define the exponential map
$$\exp_{x}:T_{x}\M\to\M$$
of the magnetic flow as follows. 
If $v\in T_{x}\M-\{0\}$, then 
$\exp_{x}(v)=\widetilde\pi(\phi_{t}(x,u))$
where $t=|v|$ and $u=v/|v|$, and $\exp_{x}(0)=x$.
This map is smooth \cite{CI}.

\begin{Proposition}If the magnetic flow is Anosov on $SM$, then
the exponential map $\exp_{x}$ is a diffeomorphism for all
$x\in\M$.
\label{diffeo}
\end{Proposition}

\begin{proof} As we already mentioned in the introduction,
the results in \cite{pp1,pp2} ensure that there are
no conjugate points. Since $S\M$ is a regular level set for 
the energy, Theorem~F of \cite{CI} tells us  that $\exp_x$ is
a diffeomorphism provided
 $$
 \inf_{(x,v) \in S\M} \Theta_{can}(d{\mathcal L}(X(x,v))) > 0,
 $$
 where $\Theta_{can}$ is the canonical $1$-form on $T^*\M$ 
and $X$ is the vector field on $T\M$ that generates the
magnetic flow, which is
Euler-Lagrange flow for the Lagrangian
 $$L(x,v) = \frac12|v|_x^2 - \Theta_x(v). $$ 
Since the projection of $X(x,v)$ to $\M$ is
$v$, we see that 
 \begin{eqnarray*}
 \Theta_{can}(d{\mathcal L}(X(x,v))) 
 &=& {\mathcal L}(x,v)(d\pi_{T^*\M}(d{\mathcal L}(X(x,v)))) \\
 &=& \frac{\partial L}{\partial v}(x,v)v \\
 &=& L(x,v) + E(x,v)   \\
 &=& L(x,v) + 1/2
 \end{eqnarray*}
 on $S\M = E^{-1}(1/2)$. Since the magnetic flow is Anosov on $SM$, 
it follows from Theorem~C that $1/2 > c(g,\Omega)$. As we explained in
 the introduction,
there must be  a smooth function $u: T\M \to \re$ such that the Lagrangian
$\overline{L}(x,v) = L(x,v) - d_xu(v)$ satisfies
 $$
\inf_{(x,v) \in T\M}\overline{L}(x,v) + 1/2 > 0.
$$
The Lagrangians $L$ and $\overline L$ have the same Euler-Lagrange equation,
the same energy function and the same minimal trajectories, so we may replace
$L$ by $\overline L$.

\end{proof}



We now show:

\begin{Lemma}For a suitably chosen Riemannian metric on $SM$ and
for all $T>0$ we have
\[\int_{M}\mbox{\rm Vol}\,B_{mag}(x,T)\,dx\leq \int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta\]
and equality holds if the magnetic flow is Anosov.
\end{Lemma}

\begin{proof}Take $x\in\M$. Let $B(0,T)$ be the ball of radius $T$ in $T_{x}\M$.
By the definition of $B_{mag}(x,T)$ it is clear
that
$$B_{mag}(x,T)= \widetilde{\pi}\left(\phi_{[0,T)}(S_{x})\right)=
\exp_{x}(B(0,T)),$$
where $S_{x}$ is the unit sphere in $T_{x}\M$.
Endow $SM$ with a Riemannian metric $g_{0}$ defined as follows:
\begin{enumerate}
\item on the subspace $S(\theta)$ given by those $\xi\in T_{\theta}SM$ for which
$\langle d_{\theta}\pi(\xi),v\rangle=0$, we let $g_{0}$ coincide
with the Sasaki metric of $SM$;
\item $S(\theta)$ is orthogonal to $X(\theta)$ for all $\theta$;
\item $X(\theta)$ has norm one.
\end{enumerate}
Let $\psi:[0,T)\times S_{x}\to \M$ be the map
\[\psi(t,\theta)=\widetilde{\pi}(\phi_{t}(\theta)).\]
We endow $[0,T)\times S_{x}$ with the product of the canonical metrics on its factors. 
Since
$\widetilde{\pi}\left(\phi_{[0,T)}(S_{x})\right)=\psi([0,T)\times S_{x})$ we have
\[\mbox{\rm Vol}\,\widetilde{\pi}\left(\phi_{[0,T)}(S_{x})\right)\leq
\int_{0}^{T}\,dt\int_{S_{x}}
\left|\det\,d_{(t,\theta)}\psi\right|\,d\theta,\]
with equality if the magnetic flow is Anosov since in that case $\exp_{x}$ is a
 diffeomorphism by Proposition \ref{diffeo}.
Now observe that using $g$ on $M$ and $g_{0}$ on $SM$
and their lifts $\widetilde{g}$ and $\widetilde{g}_{0}$ to $\M$ and $S\M$ we have:
\[\left|\det\,d_{(t,\theta)}\psi\right|=
\left|\det\,d_{\theta}(\widetilde{\pi}\circ\phi_{t})|_{T_{\theta}S_{x}
\oplus \re\,X(\theta)}\right|,\]
hence
\[\mbox{\rm Vol}\,\widetilde{\pi}\left(\phi_{[0,T)}(S_{x})\right)\leq
\int_{0}^{T}\,dt\int_{S_{x}}\left|\det\,d_{\theta}
    (\widetilde{\pi}\circ\phi_{t})|_{T_{\theta}S_{x}\oplus \re\,X(\theta)}\right|\,d\theta.\]
Using that the maps $p:(\M,\widetilde{g})\to (M,g)$ and
$dp:(S\M,\widetilde{g}_{0})\to (SM,g_{0})$ are local isometries and Fubini's theorem the lemma
follows.

\end{proof}


Let us complete the proof of Theorem D. By the previous lemma and Theorem \ref{mf}
we have:
\begin{align*}
h_{v}&=\limsup_{T\to\infty}\frac{1}{T}\log\int_{M}\mbox{\rm Vol}\,B_{mag}(x,T)\,dx\\
&\leq \limsup_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta\\
&\leq h_{top}.
\end{align*}

Suppose now that the magnetic flow is Anosov. Then by the previous lemma
and Theorem \ref{mf} we have:

\begin{align*}
h_{top}&= \liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta\\
&=\liminf_{T\to\infty}\frac{1}{T}\log\int_{M}\mbox{\rm Vol}\,B_{mag}(x,T)\,dx\\
&\leq h_{v}.
\end{align*}

\qed



We conclude this section with a discussion about the relation of
$h_{v}$ with $\pi_{1}(M)$. Let $(g,\Omega)$ be a pair with
$\wO=d\Theta$ and $\Theta$ a bounded 1-form. Suppose that
$1/2>c(g,\Omega)$.

\begin{Proposition} Suppose that
$1/2>c(g,\Omega)$. Then $h_{v}$ is positive if and only if $\pi_{1}(M)$ has exponential
growth.
\label{mlike}
\end{Proposition}

\begin{proof}Since $1/2 > c(g,\Omega)$, the discussion in the
  introduction tells us that
given a sufficiently small $\kappa>0$ there exists a smooth function 
$u:\M\to\re$ such that \[L+1/2-du>\kappa.\]
Replacing $\Theta$ by $\Theta+du$ gives another bounded form 
and a new Lagrangian with the same orbits as $L$ and the same energy function.
The balls $B_{mag}(x,T)$ are the same as before, so we might as well assume that $L$ satisfies:
\[L+1/2>\kappa.\]
Let $d$ be the distance function on $\M$ induced by the Riemannian metric and
let $B_{geo}$ denote a ball in this metric. Obviously given $y\in B_{mag}(x,T)$ we have
$d(x,y)\leq T$ and hence $B_{mag}(x,T)\subseteq B_{geo}(x,T)$.

On the other hand, suppose that $y\in B_{geo}(x,T)$ 
and let $\ga:[0,L]\to\M$ be a $d$-minimizing
geodesic connecting $x$ to $y$ with length $L\leq T$.
Let $\ga_{m}:[0,R]\to\M$ be a minimizing {\it magnetic} geodesic connecting $x$ to $y$;
such a minimizing geodesic exists by Ma\~n\'e's theorem since $1/2$ is bigger than
the critical value.
Hence:
\[\kappa R < A_{L+1/2}(\ga_{m})\leq A_{L+1/2}(\ga) \leq KL\leq KT,\]
where
\[K:=\max_{(x,v)\in S\M}\left(L(x,v)+1/2\right).\]
It follows that $B_{geo}(x,T)\subseteq B_{min}(x,KT/\kappa) 
  \subseteq B_{mag}(x,KT/\kappa)$. This implies
\[\mbox{\rm Vol}\,B_{mag}(x,T)\leq \mbox{\rm Vol}\,B_{geo}(x,T)\leq 
     \mbox{\rm Vol}\,B_{mag}(x,KT/\kappa),\]
and therefore
\[h_{v}\leq \limsup_{T\to\infty}\frac{1}{T}\log
   \int_{M}\mbox{\rm Vol}\,B_{geo}(x,T)\,dx\leq \frac{K}{\kappa}\,h_{v}.\]
But
\[\limsup_{T\to\infty}\frac{1}{T}\log\int_{M}\mbox{\rm Vol}\,B_{geo}(x,T)\,dx\]
is the volume entropy of the Riemannian metric 
and it is well known that this quantity is positive
if and only if $\pi_{1}(M)$ grows exponentially.

\end{proof}

\begin{Corollary}Suppose that the 1-parameter family of magnetic flows $\phi^{\la}$ exits the set of
Anosov magnetic flows at $\la = \la_{c}$ and $h_{top}(\la_{c})=0$.
Then
\[\la^{2}_{c}=\frac{1}{2\,c(g,\Omega)}.\]
\end{Corollary}

\begin{proof}This follows from Theorems C, D, Proposition
\ref{mlike} and the fact that Anosov flows have positive topological entropy.
\end{proof}




\section{Magnetic flows on surfaces}\label{S=surf}

Let $M$ be an oriented surface endowed with a Riemannian metric $g$.
Given $(x,v)\in TM$, let $iv$ be the unique vector in $T_{x}M$ such that
$\{v,iv\}$ is a positively oriented orthonormal basis
of $T_{x}M$. The area form $\Omega_g$ is given by
\[\Omega_g(u,v)=g(iu,v).\]
Any closed 2-form $\Omega$ can be written as $\Omega=f\,\Omega_g$ for some
smooth function $f:M\to\re$.
The Lorentz force $Y$ associated with $\Omega$ is given by
\[Y_{x}(v)=f(x)\,iv,\]
where $\pi:TM\to M$ is the canonical projection.
It follows from equation (\ref{1}) that $t\mapsto \gamma(t)$ is
 a $\la$-magnetic geodesic if and only if:
\[\frac{D\dot{\ga}}{dt}=\la\,f(\ga)\,i\dot{\ga}.\]
Note that if $f\equiv 1$, then $\ga$ is a $\la$-magnetic geodesic
if and only if $\ga$ has constant geodesic curvature $\la$.

%Given $(x,v)\in SM$ we define {\it the vertical subspace
%at} $(x,v)$ as $V(x,v):=\mbox{\rm ker}\,d_{(x,v)}\pi$.
%We say that the orbit of $(x,v)$ {\it does not have conjugate
%points} if for all $t\neq 0$,
%$$d_{(x,v)}\phi_{t}^{\la}(V(x,v))\cap V(\phi_{t}^{\la}(x,v))=\{0\}.$$

Given $(x,v)\in SM$ and $\xi\in T_{(x,v)}TM$, let
\[J_{\xi}(t)=d_{(x,v)}(\pi\circ\phi_{t}^{\la})(\xi).\]
We call $J_{\xi}$ a {\it $\la$-magnetic Jacobi field} with initial
condition $\xi$. It was shown in \cite{PP1/2} that $J_{\xi}$
satisfies the following Jacobi equation:
\begin{equation}
\ddot{J_{\xi}} + R(\dot{\gamma},J_{\xi})\dot{\gamma} - \la[Y(\dot{J_{\xi}})+
(\nabla_{J_{\xi}}Y)(\dot{\gamma})] = 0,  \label{jacobi}
\end{equation}
where $\ga(t)=\pi\circ\phi_{t}^{\la}(x,v)$ and $R$ is the curvature tensor
of $g$ with sign convention used by Milnor in \cite{Mil}.

Let us express $J_{\xi}$ as follows:
\[J_{\xi}(t)=x(t)\dot{\ga}(t)+y(t)i\dot{\ga}(t),\]
and suppose in addition that $\xi\in T_{(x,v)}SM$, which implies
\begin{equation}
g_{\ga}(\dot{J}_{\xi},\dot{\ga})=0.\label{tangente}
\end{equation}
A straightforward computation using (\ref{jacobi}) and (\ref{tangente}) shows that
$x$ and $y$ must satisfy the scalar equations:

\begin{align}
&\dot{x}=\la\,f(\ga)\,y   \label{magJac1}\\
&\ddot{y}+\left[K(\ga)-\la\left\langle\nabla
f(\ga),i\dot{\ga}\right\rangle+\la^{2}\,f^{2}(\ga)\right]y=0.
\label{magJac2}
\end{align}
We  call the last equation the {\it scalar Jacobi equation}
of $\ga$ and we define the {\it magnetic curvature} as
\[K_{mag}^\la(x,v)=K(x)-\la\left\langle\nabla f(x),i\,v\right\rangle+\la^{2}\,f^{2}(x).\]



We say that equation~(\ref{magJac2}) has an {\it exponential dichotomy} if there exist
constants $C,\mu>0$, and solutions
$y^{s}$, $y^{u}$ of the scalar Jacobi equation (\ref{magJac2}) 
along each $\la$-magnetic geoedesic such that
\[|y^{s}(t)|\leq C\,e^{-\mu t}, \quad\hbox{\rm  for all $t\geq 0$},\]
\[|y^{u}(t)|\leq C\,e^{\mu t},\quad\hbox{\rm  for all $t\leq 0$}.\]


It is easily seen that the magnetic flow is Anosov if such an exponential dichotomy holds.
Since $f$ is uniformly bounded, we can associate to each pair of
 solutions $y^{s}$, $y^{u}$ as above, a pair of solutions
$x^{s}$, $x^{u}$ of equation (\ref{magJac1}) by setting
%\[x^{s}(0)\df -\int_{0}^{\infty}\la f(\ga(\tau))y^{s}(\tau)\,d\tau,\]
%\[x^{u}(0)\df \int_{0}^{\infty}\la f(\ga(-\tau))y^{u}(-\tau)\,d\tau.\]
%It is easily seen that
\[x^{s}(t)= -\int_{t}^{\infty}\la f(\ga(\tau))y^{s}(\tau)\,d\tau,\]
\[x^{u}(t)= \int_{-\infty}^{t}\la f(\ga(\tau))y^{u}(\tau)\,d\tau.\]
Then $x^{s}(t)$ and $x^{u}(-t)$ converge exponentially fast to $0$
as $t\to\infty$. Let $J^{s}$ denote the unique Jacobi field
determined by the initial conditions $$(x^{s}(0),y^{s}(0),f(\pi
(x,v))y^{s}(0),\dot{y}^{s}(0))$$ and let $J^{u}$ denote the unique
Jacobi field determined by the initial conditions
$$(x^{u}(0),y^{u}(0),f(\pi (x,v))y^{u}(0),\dot{y}^{u}(0)).$$ Then
\[E^{ss}(v)=\re(J^{s}(0),\dot{J}^{s}(0)),\]
\[E^{su}(v)=\re(J^{u}(0),\dot{J}^{u}(0))\]
are clearly the strong stable and unstable spaces.

The existence of an exponential dichotomy can be verified using the cone method pioneered by Alexeev and Lewowicz and refined by Wojtkowski. In our context it involves studying the magnetic Riccati equation
\begin{equation}\label{magRic}
\dot u_{mag}(t) + u_{mag}^2(t) + K_{mag}^{\la}(\dot\ga(t)) = 0
\end{equation}
along a $\la$-magnetic geodesic $\ga(t)$.

\begin{Lemma}\label{L=criterion}
Suppose there are constants $T > 0$ and $ H > 1$ such that $1/H \leq u_{mag}(T) \leq H$ whenever $u_{mag}(t)$ is a solution of (\ref{magRic}) with $u_{mag}(0) \geq 0$. Then equation (\ref{magJac2}) has an exponential dichotomy and the $\la$-magnetic geodesic flow is Anosov.
\end{Lemma}

\begin{proof} Consider the quadratic form $Q(y,\dot{y})=y\dot{y}$. The hypotheses of the lemma imply that there is an $\eta > 0$ such that $Q(y(T),\dot y(T)) > \eta[y(0)^2 + \dot y(0)^2]$ whenever $y(t)$ is a solution of the scalar Jacobi equation (\ref{magJac2}) along a $\la$-magnetic geodesic~$\ga(t)$ with $Q(y(0),\dot y(0)) \geq 0$. Wojtkowski \cite{W1} showed that this condition in turn implies that there is an $\eta' > 0$ such that
\[Q(y(T),\dot{y}(T))-Q(y(0),\dot{y}(0))>\eta'[y(0)^{2}+\dot{y}(0)^{2}]\]
for {\it any} solution $y(t)$ of (\ref{magJac2}) along any $\la$-magnetic geodesic~$\ga(t)$. It is well known that this last condition implies the existence of an exponential dichotomy.
\end{proof}

The magnetic Riccati equation has the same  geometric significance as the Riccati equation for the geodesic flow. Suppose that $(x(t),y(t))$ is the solution of the magnetic Jacobi equations (\ref{magJac1}) and (\ref{magJac2}) along a $\la$-magnetic geodesic $\ga_0$ that is defined by a $1$-parameter family $\ga_s(t)$ of $\la$-magnetic geodesics. Then $u_{mag}(t) = \dot y(t)/y(t)$ is a solution of the magnetic Riccati equation~(\ref{magRic}) and $u_{mag}(t)$ is the geodesic curvature at $\ga_0(t)$ of the curve through $\ga_0(t)$ orthogonal to the family~$\ga_s$.

\subsection{Rotationally symmetric surfaces} \label{SS=rotsym}
Suppose now that
$M=\re\times S^{1}$ is a rotationally symmetric surface and $\Omega = \Omega_g$, i.e.~$f \equiv 1$.
If $(s,\vr)$ are the obvious coordinates on $M$,
then the Riemannian metric of $M$ in these coordinates
has the expression:
\[g=ds^{2}+r(s)^{2}d\vr^{2},\]
where $r:\re\to (0,\infty)$ is a smooth function. 
Note that $r$ satisfies the scalar Jacobi equation
\begin{equation}\label{SJ}
r''(s) + K(s)r(s) = 0,
\end{equation}
where $K(s)$ is the Gaussian curvature on $\{s\} \times S^1$. The
$\la$-magnetic curvature on $\{s\} \times S^1$ is $K^\la_{mag}(s) = K(s) + \la^2$.

The function $u(s) = r'(s)/r(s)$ satisfies the Riccati equation
\begin{equation} \label{Riccati}
u'(s) + u^2(s) + K(s) = 0.
\end{equation}
The geometric significance of $u(s)$ is that the geodesic curvature  of the parallel of  latitude $\{s\} \times S^1$ is $\pm u(s)$ depending on the direction in which we traverse it.


We orient $M$ so that $\{\partial/\partial s, \partial/\partial\varphi\}$ 
is a positively oriented basis of $M$ and the
area form $\Omega_g$ is given by
\[\Omega_g(s, \vr  = r(s)\,ds\wedge d\vr.\]
We saw earlier that the $\la$-magnetic geodesics defined by $\Omega_g$ have constant geodesic curvature~$\la$. This means that the parallel $\{s\} \times S^1$ can be the trace of a $\la$-magnetic geodesic only if $\la = \pm u(s)$. More precisely we have:

\begin{Lemma}The parallel $t\mapsto (s,at)$
is a unit speed $\la$-magnetic geodesic if and only~if
$$
|a|= r(s)^{-1} \qquad\text{and}\qquad u(s)=\pm\la,
$$
where the positive sign holds if $a>0$ and the negative
sign holds if $a<0$. If $t\mapsto (s,at)$ is a $\la$-magnetic geodesic,
then $K^\la_{mag}(s) = -u'(s)$.
\label{pmg}
\end{Lemma}

Before proving this lemma, we introduce the Clairaut integral for the magnetic flow. Define
\[R(s):=\int_{0}^{s}r(\sigma)\,d\sigma.\]

\begin{Proposition}[Clairaut integral]If
$t\mapsto (s(t),\vr(t))$ is a $\la$-magnetic geodesic,
then
\[t\mapsto r^{2}(s(t))\dot{\vr}(t)-\la\,R(s(t))\]
is constant.
\label{cl}
\end{Proposition}

The proposition is an easy consequence of the following:

\begin{Lemma}A curve
$t\mapsto (s(t),\vr(t))$ is a $\la$-magnetic geodesic if and only if
$$
\ddot{s}(t) =r(s(t))\dot{\vr}(t)\,[r'(s(t))\dot{\vr}(t)-\la]\quad\text{and}\quad
 \frac{d}{dt}\left(r^{2}(s(t))\dot{\vr}(t)-\la\,R(s(t))\right)=0,
 $$
where dot indicates derivative with respect to $t$
and prime indicates derivative with respect to the $s$-parameter.
\label{eqns}
\end{Lemma}

\begin{proof}Note that $R\circ s \,d\varphi$ is a primitive of $\Omega$, since
$d(R\circ s\,d\vr)=r\circ s\,ds\wedge d\vr=\Omega$.
This means that the $\lambda$-magnetic geodesics are extremals for the
Lagrangian
$$L(x,v) = \frac12 |v|^2_x - R(s(x))\,d\varphi(v).$$
In terms of our local coordinates,
\[L(s,\vr,\dot{s},\dot{\vr})=
\frac{1}{2}\left(\dot{s}^{2}+r(s)^{2}\dot{\vr}^{2}\right)
-\la\,R(s)\,\dot{\vr}.\]
The two equations in the lemma are just the Euler-Lagrange equations for $L$,
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{s}}\right)-
\frac{\partial L}{\partial s}=0
\qquad\text{and}\qquad
 \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\vr}}\right)-
\frac{\partial L}{\partial \vr}=0.
$$
\end{proof}



\begin{proof}[Proof of Lemma \ref{pmg}]The parallel has unit speed if and only if
$r^{2}(s)\,a^{2}=1$, which is equivalent to $|a| = r(s)^{-1}$.
Using Lemma \ref{eqns}, we see
that the parallel is a $\la$-magnetic geodesic if and only if
$r'(s_{0})\,a=\la$. When $\la = \pm u(s)$, we have
$$
K^\la_{mag}(s) = K(s) + \la^2 = -u'(s) - u^2(s) + u^2(s) = -u'(s)
$$
 by~(\ref{Riccati}).
\end{proof}

The functions $$r_{+,\la}(s) = r(s) - \la R(s) \qquad\text{and}\qquad  r_{-,\la}(s) = -r(s) - \la R(s)$$ govern the behaviour of the $\la$-magnetic geodesics in much the same way that the function $r(s)$ governs the behaviour of the geodesics. If $r_{-,\la}(s) < c < r_{+,\la}(s)$, there will be two unit vectors at each point of $\{s\} \times S^1$ tangent to $\la$-magnetic geodesics along which the Clairaut integral is equal to~$c$; these vectors will make equal angles with the positive direction along the parallel. The following is an easy consequence of Lemma~\ref{pmg}:

\begin{Lemma}\label{pmg2}
 The parallel $t \mapsto (s,r(s)^{-1}t)$ is a unit speed $\la$-magnetic geodesic if and only if $r'_{+,\la}(s) = 0$. If $r'_{+,\la}(s) = 0$, the $\la$-magnetic curvature along $\{s\} \times S^1$ is $-r''_{+,\la}(s)/r(s)$.
\end{Lemma}

\begin{proof} Since $r'_{+,\la}(s) = r'(s) - \la r(s) = r(s)[u(s) - \la]$, we have $r'_{+,\la}(s) = 0$ if and only if $u(s) = \la$. Differentiating again gives
$$
-\frac{r''_{+,\la}(s)}{r(s)} = -\frac{r''(s)}{r(s)} + \la\frac{r'(s)}{r(s)} = K(s) + \la^2 = K^\la_{mag}(s)$$
when $\la = u(s)$.
\end{proof}

In particular, if $r'_{+,\la}(s) = 0$ and $r''_{+,\la}(s) > 0$, then $t \mapsto (s,r(s)^{-1}t)$ is a closed $\la$-magnetic geodesic along which the $\la$-magnetic curvature is negative; the corresponding periodic orbit of the $\la$-magnetic flow is hyperbolic. Suppose now that we have $s' < s''$ and $c$ with the following properties, which are indicated in Figure \ref{fig2}:
\begin{enumerate}
\item $r_{+,\la}(s') = r_{+,\la}(s'') = c$;
\item $r_{-,\la}(s) < c < r_{+,\la}(s)$ for $s'< s < s''$;
\item $r'_{+,\la}(s') = 0$ and $r'_{+,\la}(s'') < 0$;
\item $r''_{+,\la}(s') > 0$.
\end{enumerate}

\begin{figure}

\special{psfile=figure2.eps voffset=-200 hoffset=40 hscale=70 vscale=70 angle=0}

\vspace{7cm}

\caption{Properties of a homoclinic interval.}

\label{fig2}

\end{figure}





There will be two unit vectors at each point of $(s',s'') \times S^1$ that are tangent to $\la$-magnetic geodesics with Clairaut integral $c$. These geodesics will be tangent to the parallel $\{s''\} \times S^1$ and both forward and backward asymptotic to the $\la$-magnetic geodesic $t \mapsto (s',r(s')^{-1}t)$.
This situation is analogous to what happens for the geodesic flow. The hyperbolic closed orbit of the $\la$-magnetic flow corresponding to
$t \mapsto (s',r(s')^{-1}t)$ has a homoclinic connection.

We say that $[s',s'']$ is a {\it homoclinic interval} for the
$\la$-magnetic flow on $M$ if  properties 1--4 hold.






\section{Examples with Anosov intervals that do not contain $0$} \label{S=egs}

Let $M$ be an oriented surface and let $\Omega_g$ be the area
form of the Riemannian metric~$g$. At the level of the differential of the magnetic flow
$\phi^{\la}$, increasing the intensity $\la$ of the field amounts
to increasing the curvature like term $K_{mag}^\la=K+\la^{2}$ in the corresponding Jacobi
equation.
This effect is obviously monotonic as $\la$ increases and works to make the magnetic flow non Anosov.
It can be overcome, however, by a second effect: the geodesic curvature of the $\la$-magnetic geodesics increases as $\la$ increases and
consequently the location of the $\la$-magnetic geodesics changes.

This possibility can be realized in the following way.
Consider a rotationally symmetric surface
as shown in Figure \ref{fig3}.

\begin{figure}

\special{psfile=figure3.eps voffset=-250 hoffset=70 hscale=50 vscale=50 angle=0}

\vspace{9cm}

\caption{Magnetic geodesics move as intensity increases.}

\label{fig3}

\end{figure}

The Gaussian curvature of the surface is everywhere negative. It is less strongly negative in the annulus indicated in the figure and, as the intensity $\la$ increases, $K_{mag}^\la$ becomes positive in this annulus, but stays negative elsewhere.
The figure shows how the shape of a $\la$-magnetic geodesic changes as $\la$ increases and the geodesic curvature of the $\la$-magnetic geodesics increases. When $\la = \la_1$, the magnetic curvature in the annulus has become positive  and one of the parallels of latitude in the annulus is a closed $\la$-magnetic geodesic along which the $\la$-magnetic curvature is constant and positive. This ensures that the magnetic flow is not Anosov.
 When $\la = \la_2$, the magnetic curvature in this annulus has become more positive, but the $\la_2$-magnetic geodesics have stronger geodesic curvature than the parallels of latitude in the annulus, and so the $\la_2$-magnetic geodesics only stay in the annulus for a short time. The negative magnetic curvature outside the annulus
is then able to overcome the effects of the positive magnetic curvature and the magnetic flow is again Anosov.

It is also possible to start with positive Gaussian curvature everywhere along a closed geodesic and make the magnetic flow become Anosov as the magnetic intensity increases. Of course in both examples the magnetic flow will eventually be non Anosov if the magnetic intensity is increased enough.




We now describe these examples in detail.
Given a smooth function $u: \mathbb R \to \mathbb R$ and a constant $r_0 > 0$, we define the function $r: \mathbb R \to \mathbb R^+$ to be the solution of  $r'(s) = u(s)r(s)$ with $r(0) = r_0$. In other words,
\[r(s)=r_{0}\exp\left(\int_{0}^{s}u(t)\,dt\right).\]
Let $M = \re\times S^{1}$ be the rotationally symmetric surface
determined by $r$. As we saw in Section~\ref{SS=rotsym}, $M$ has Gaussian curvature $K(s) = -u'(s) - u^2(s)$ on the parallel $\{s\} \times S^1$ and this parallel has geodesic curvature $\pm u(s)$ depending on the direction in which it is traversed.

We choose $u(s) = \tanh(s)$ except for a $C^0$ small perturbation in a very short interval $[s_1,s_2] \subset [-1/8,1/8]$. We require that $|u(s)| < \tanh(1/4)$ and $K(s) = -u'(s) -u^2(s) < 1/4$ when $s_1 \leq s \leq s_2$. These properties ensure that the $\la$-magnetic geodesic flow is Anosov for $\la = 1/2$.
In order to prove this, we check that the hypothesis of Lemma~\ref{L=criterion} holds. It suffices to show that there is a $T > 0$ such that every $1/2$-magnetic geodesic segment with length $T$ has the property that the magnetic curvature is at most $-3/4$ except for a sufficiently short subset where the magnetic curvature is at most $1/2$.




 When $\la = 1/2$, we have $K_{mag}^{\la} = K + 1/4$. Hence the $\la$-magnetic curvature is at most $-3/4$ outside $A = [s_1,s_2] \times S^1$ and is at most $1/2$ in $A$. Let $B = [-1/4,1/4] \times S^1$. Since the geodesic curvatures of the parallels in $B$ lie in the interval $[-\tanh(1/4),\tanh(1/4)] \subset [-1/4,1/4]$ and the $\la$-magnetic geodesics have geodesic curvature $1/2$ when $\la = 1/2$, we see that a $1/2$-magnetic geodesic segment in $B$ can be tangent to the $s$-direction at most once and cannot be asymptotic to the $s$-direction. A maximal $1/2$-magnetic geodesic segment in $B$ must begin and end with a segment that crosses one of the two components of~$B \setminus A$ and
can contain at most two connected subsegments that lie in $A$.

Any curve that crosses a component of $B \setminus A$ has length at least $1/8$.
 We now choose $T = 1/8$. Let $t_A$ be the length of the longest magnetic geodesic segment in $A$.   A $1/2$-magnetic geodesic with length $1/8$ must have magnetic curvature at most $-3/4$ everywhere except for a subset with length at most $2t_A$ in which the curvature is at most $1/2$.
 It follows easily from the next lemma, with $\la = 1/2$ and $\delta = 1/4$, that we can make $t_A$ as small as we wish     by making $s_2 - s_1$ small enough. Thus $T = 1/8$ has the desired property described above.






\begin{Lemma}There is a constant $C > 0$ with the following property. Suppose that for some $\la$ and $\delta > 0$ the geodesic curvature of the parallel of latitude
$\{s\}\times S^1$ lies in the interval $[-\la+\delta,\la-\delta]$ for $s_1 \leq s \leq s_2$. Let $\ga$ be a connected $\la$-magnetic geodesic segment that  lies in the region $[s_2-s_1]\times S^1$.
Then the length of $\ga$
is at most $C\,\sqrt{(s_2 - s_1)/\delta}$.
\end{Lemma}

\begin{proof}
Let $\psi(t)$ be the angle between $\dot{\ga}(t)$ and the unit vector field
$U:=\partial/\partial s$. It will suffice to show that $|\dot{\psi}(t)|>\delta$
when $\ga(t)$ lies in the region $s_1<s<s_2$.
But
\[|\dot{\psi}(t)|\geq \left|\Big|\frac{D\dot{\ga}}{dt}\Big|-\Big|\frac{DU}{dt}\Big|\right|,\]
and $|\frac{D\dot{\ga}}{dt}|=\la$ because $\la$ is the geodesic curvature of $\ga$.
Since the vector field $U$ is tangent to the meridian geodesics, $\nabla_{U}U=0$ and
\[\frac{DU}{dt}=\nabla_{\sin \psi(t)V(t)}U,\]
where $V(t)$ is a unit vector orthogonal to $U(t)$ and therefore tangent to the parallel
through $\ga(t)$. The latter means that $|\nabla_{V(t)}U|$ is the absolute value of the
geodesic curvature of the parallel through $\ga(t)$. Hence
\[\left|\Big|\frac{D\dot{\ga}}{dt}\Big|-\Big|\frac{DU}{dt}\Big|\right|
\geq\la-|\sin\psi(t)|(\la-\delta)\geq \delta\]
in the region $[s_1,s_2]\times S^1$.

\end{proof}



We now choose the interval $[s_1, s_2]$ and  the behaviour of the function $u$ in this interval.


For the first example,  in which the geodesic flow is Anosov and the magnetic flow becomes non Anosov and switches back to Anosov as the intensity increases, we choose $[s_1,s_2]$ to be a small neighbourhood of $1/16$ and arrange that $u'(1/16) < 0$ but that $-u'(s) < u^2(s)$ for all $s \in [s_1,s_2]$. Since $K(s) = -u'(s) - u^2(s)$ by the Riccati equation (\ref{Riccati}), we have $K(s) < 0$ for all $s \in [s_1,s_2]$. Outside of $[s_1,s_2]$ we have $u(s) = \tanh(s)$ and  (\ref{Riccati}) gives $K(s) = -1$.
When $\la = u(1/16)$ the parallel $t \mapsto (1/16, r(1/16)^{-1}t)$  is a magnetic geodesic along which the magnetic curvature is identically $$K(1/16) +  \la^2 = -u'(1/16) - u^2(1/16) + \la^2 = -u'(1/16) > 0.$$
The existence of a closed magnetic geodesic along which the magnetic curvature is constant and positive means that the $u(1/16)$-magnetic geodesic flow is not Anosov.

For the second example, in which the geodesic flow is non Anosov and the magnetic flow is Anosov for some positive intensity, we choose $[s_1,s_2]$ to be a small neighbourhood of $0$ and arrange that $u(0) = 0$ and $u'(0) < 0$. This ensures that $\{0\} \times S^1$ is a closed geodesic and $K(0) = -u'(0) - u^2(0) > 0$, so the geodesic flow is not Anosov.

In both examples the argument given earlier in this section shows that the $1/2$-magnetic flow is Anosov.

It is possible to compactify the examples.
     We  use the same construction as in \cite{BBB} and \cite{Burns}. Let us cut off an end of $M$ along a parallel of latitude and then slit the end along a meridian geodesic. We obtain a fan shaped subset of the Poincar\'e disc. This subset is bounded by two geodesic rays $\eta'$ and
$\eta''$ and a curve corresponding to the parallel of latitude. For
any large enough $n$, it is possible to draw a sequence  of $4n +
1$ hyperbolic geodesic segments $$d_{n}',c_{n}',d_{n-1}', \ldots
,d_{1}',c_{1}', d_{0},c_{1}'',d_1'', \ldots ,c_n'',d_n''$$ such
that
    \begin{itemize}
    \item $d_n'$and $d_n''$ are orthogonal to $\eta'$ and $\eta''$
respectively,

    \item adjacent geodesics in the sequence are orthogonal,

    \item $c_i'$ and $c_i''$ have the same length for $1  \le  i  \le
n$.
\end{itemize}







     By identifying $\eta'$ with $\eta''$ and $c_i'$ with $c_i''$, $1  \le  i
\le  n$, we obtain a hyperbolic metric on the sphere with $n + 2$ punctures.
One of the holes is bounded by a copy of the parallel of latitude where we cut off the end and the others
by closed geodesics $d_0, \ldots ,d_n$. Now we can adjoin handles  with
curvature $-1$ along $d_0, \ldots ,d_n$.



It is easy to modify the examples so that there is an arbitrarily long finite sequence $ \la_1 < \la_2 < \dots < \la_n$ such that the $\la$-magnetic flow is Anosov when $\la = \la_k$ for even $k$ and is non Anosov when $\la = \la_k$ for odd $k$. However we do not know how to arrange for an infinite number of changes from Anosov to non Anosov and back again.


\section{Increasing topological entropy between Anosov intervals}\label{S=hup}

The compactified version of the first example of the previous section can be modified so that the topological entropy for the $u(1/16)$-magnetic flow,  which is not Anosov, is greater than the topological entropy for the geodesic flow.


We again choose $[s_1,s_2]$ to be a small neighbourhood of $1/16$. Now we also choose an even smaller neighbourhood of $1/16$, \ $[s_3,s_4] \subset (s_1,s_2)$.
We want to create a homoclinic interval $[s',s'']$ as defined at the end of Section~\ref{S=surf} inside the interval $(s_3,s_4)$. To this end we set $\la_1 = \tanh(1/16)$ and consider a function $w(s)$ such that $w(s) = \tanh(s)$ for $s \notin [s_1,s_2]$, \ $w(s) = \la_1$ for $s \in [s_3,s_4]$, and $w$ is monotonic in the intervals $[s_1,s_3]$ and $[s_2,s_4]$. We also ensure that $-w'(s) - w^2(s) < 0$ for all $s$.

 We also choose a function $G(s)$ whose graph has the form shown in Figure \ref{fig4} and
  let $g(s) = G'(s)$. The function $G$ is constant and positive on $(-\infty,s_3]$ and is
   constant and negative on $[s_4,\infty)$. Inside $(s_3,s_4)$ are two points $s' < s''$
    where $G$ vanishes; we have $G >0$ on $(s_3,s') \cup (s',s'')$ and $G < 0$ on $(s'',s_4)$.
     Also $G''(s') > 0$ and $G'(s'') < 0$.
  Both $G(s)$ and $g(s)$ are small for all $s$. In particular, we ensure that $G(s) \leq r_0$
  for all $s$, where $r_0$ is the radius for the surface at $s=0$ that was chosen at the
   beginning of Section~\ref{S=egs}.

\begin{figure}

\special{psfile=figure4.eps voffset=-200 hoffset=40 hscale=50 vscale=50 angle=0}

\vspace{7cm}

\caption{Graph of $G$.}

\label{fig4}

\end{figure}



We choose $R(s)$ to be the solution of the differential equation
\begin{equation} \label{Req}
R''(s) - w(s)R'(s) = g(s)
\end{equation}
with $R(0) = 0$ and $R'(0) = r_0$. We set $r(s) = R'(s)$ and
consider the surface of revolution defined by the function $r(s)$. In terms of $r$, equation~(\ref{Req}) becomes
\begin{equation}\label{req}
r'(s) - w(s)r(s) = g(s).
\end{equation}
Inside $[s_3,s_4]$ we have $r'(s) - \la_1R'(s) = g(s)$, which
gives $r_{+,\la_1}(s) := r(s) - \la_1R(s) = G(s) + c$ for some
constant~$c$. We see immediately from the graph of $G$ that
$[s',s'']$ satisfies all of the properties of a homoclinic
interval, except for the condition $r_{-,\la_1}(s) < c$ for $s' <
s < s''$. But it is clear from (\ref{req}) that $r(s) \geq r_0$ for
$s \geq 0$ and we have $r_{+,\la_1}(s) \leq c+r_0$ when $s_3 \leq
s \leq s_4$ because $G(s) \leq r_0$. Hence $r_{-,\la_1}(s) =
r_{+,\la_1}(s) -2r(s) \leq c - r_0$ when $s_3 \leq s \leq s_4$.
Thus $[s',s'']$ satisfies all of the conditions to be a homoclinic
interval. As explained in Section~\ref{S=surf}, the orbit of the
$\la_1$-magnetic flow corresponding to the parallel $\ga_0(t) =
(s',r(s')^{-1}t)$ is a hyperbolic closed orbit with a homoclinic
connection.

Both the geodesic flow and the $1/2$-magnetic flow are Anosov. Recall that the curvature is  $K(s) = -u'(s) -u^s(s)$ where $u(s) = r'(s)/r(s)$.
Outside $[s_1,s_2]$ equation~(\ref{req}) is $r'(s) - \tanh(s)r(s) = 0$ and hence $K(s) = -1$ for $s \notin [s_1,s_2]$, as in the examples of the previous section.
The argument used in the previous section applies to show that the $1/2$-magnetic flow is Anosov. The geodesic flow will be Anosov if we have $K(s) < 0$ for $s \in [s_1,s_2]$ also. If we had $g(s) = 0$ for all $s$, equation~(\ref{req}) would give us $u = w$ and $K(s)$ would be everywhere negative since we chose $w$ so that $-w'(s) - w^2(s) < 0$  for all~$s$. As long as we choose $g$ close enough to $0$, we will have  $K(s) < 0$ for all $s \in [s_1,s_2]$.




Now we make a small change in the magnetic field in order to break the homoclinic connection for the $\la_1$-magnetic flow and create a transverse homoclinic point. Let $\ga_{su}$ be a $\la_1$-magnetic geodesic that is both positively and negatively asymptotic to the hyperbolic closed geodesic~$\ga_0(t) = (s',r(s')^{-1}t)$.
Parametrize $\ga_{su}$ so that it does not have a self intersection at $\ga_{su}(0)$. Then we can choose a function $f$ on $M$ with the following properties:
\begin{enumerate}
\item $f=1$ outside a small neighbourhood of $\ga_{su}(0)$ that does not intersect $\ga_0$;
\item $f=1$ at all points on $\ga_{su}$;
\item $\nabla f \neq 0$ at $\ga_{su}(0)$;
\item $\nabla f$ points to the same side of $\gamma_{su}$ at all points on $\gamma_{su}$ where $\nabla f \neq 0$.
\end{enumerate}
 The new magnetic field is $f\Omega_g$. Both $\ga_0$ and $\ga_{su}$ are  $\la_1$-magnetic geodesics for the new field because $f=1$ along them. The stable and unstable manifolds for $\ga_0$ will have a {\it transverse} intersection at $\dot\ga_{su}(0)$. The argument was given by Donnay \cite{Don} in the context of geodesic flows and used in \cite{BG}. Let $u^-$ and $u^+$ be the solutions to the magnetic Riccati equation along $\ga_{su}$ that give the geodesic curvatures of the curves orthogonal to the $\la_1$-geodesics that respectively forwards and backwards asymptotic to~$\ga_0$. Before the perturbation we have $u^-_{old} \equiv u^+_{old}$. Let $t_1$ be the time when $\gamma_{su}$ enters and $t_2$ the time when $\gamma_{su}$ leaves the support of the perturbation to the magnetic field. Then $u^-_{new}(t) =  u^-_{old}(t)$ for $t \geq t_2$ and  $u^+_{new}(t) =  u^+_{old}(t)$ for $t \leq t_1$. All that one needs to arrange is that $u^+_{new}(t_2) \neq u^-_{new}(t_2) = u^+_{old}(t_2)$. It is clear from the magnetic Jacobi equation~(\ref{magJac2}) that this will be the case if $f$ is close to $1$ and property 4 above holds.

By Smale's theorem, the existence of a transverse homoclinic orbit means that there is a horseshoe in the $\la_1$-magnetic geodesic flow.
Let $h$ be the entropy of this horseshoe.

We now compactify the example in such a way that the geodesic flow has topological entropy less than $h$ and it is still true that the geodesic and $1/2$-magnetic flows are Anosov. Let $u(s) = r'(s)/r(s)$ where $r(s)$ is the function chosen above. Then $u(s) = \tanh(s)$ if  $|s| \geq 1$. We choose a large constant $R$. We change $u(s)$ for $|s| > 2R$ in such a way that $K(s) = -u'(s) -u^2(s) < 0$ for all $s$ and $u(s) \equiv h/2$ if $|s| \geq 3R$.
We also change the function $f$ in the region where $|s| > R$ in such a way that $f$ is constant on the parallels of latitude and $f$ decreases slowly from $1$ to a very small positive value $f_0$ as $|s|$ increases from $R$ to $2R$ and  $f \equiv f_0$ when $|s| \geq 2R$. If we make $R$ large enough, $f_0$ small enough, and $|\nabla f|$ small enough, both the geodesic flow and the $1/2$-magnetic flow will still be Anosov.

We now use the procedure explained in the previous section to compactify the ends $(-\infty,-3R] \times S^1$ and $[3R,\infty)\times S^1$ by adding handles with constant curvature $-h^2/4$ to obtain a compact Riemannian surface with everywhere negative curvature. The geodesic flow of this surface is Anosov. If $R$ is large enough, most of the surface will have curvature $-h^2/4$ and the average value of the square root of minus the curvature will be less than $3h/4$ and, by a result of Manning \cite{Man2},  the topological entropy of the geodesic flow will be less than $3h/4$. It follows from Theorem~E in the introduction that the $\la$-magnetic flow also has topological entropy less than $3h/4$ for all $\la \geq 0$ such that the $\la$-magnetic flow is Anosov. In particular the $1/2$-magnetic flow has topological entropy less than $3h/4$.













\section{Appendix A} \label{AppA}

Let $(g,\Omega)$ be a pair with $\wO=d\Theta$ with $\Theta$ a smooth
1-form not necessarily bounded.
Recall that
\[L(x,v)=\frac{1}{2}|v|^{2}_{x}-\Theta_{x}(v),\]
and
\[H(x,p)=\frac{1}{2}|p+\Theta_{x}|^{2}.\]
We also recall the two definitions of critical value:
\[c(L):=\inf\{k\in\re:\;A_{L+k}(\ga)\geq 0\;\mbox{\rm  for any absolutely continuous
closed curve $\ga$}\}\]
and
\[c(g,\Omega):= \inf_{u\in
  C^{\infty}(\M,\re)}\max_{x\in \M}H(x,d_{x}u).\]



\begin{Proposition} $c(L) = c(g,\Omega)$.
\end{Proposition}

\begin{proof}We begin by showing that $c(L) \leq c(g,\Omega)$. 
This is obvious if $c(g,\Omega)=\infty$. 
 It therefore suffices to show that $c(L) \leq k$ if
there exists a smooth function $u:\M\to\re$
such that $H(x,d_{x}u)\leq k<\infty$ for all $x\in \M$.
Observe that
\[H(x,p)=\max_{v\in T_{x}\M}\{p(v)-L(x,v)\}.\]
Since $H(x,d_{x}u)\leq k$ for all $x\in \M$ it follows that for all $(x,v)\in T\M$,
\[d_{x}u(v)-L(x,v)\leq k.\]
Therefore, along any absolutely continuous closed curve $\ga:[0,T]\to
\M$, we have
 \[ A_{L+k}(\ga) = \int_{0}^{T}(L(\ga,\dot{\ga})+k)\,dt
= \int_{0}^{T}(L(\ga,\dot{\ga})+k-d_{\ga}u(\dot{\ga}))\,dt\geq 0,\]
and thus $k\geq c(L)$ as desired.


We now turn to proving the reverse inequality $c(L) \geq c(g,\Omega)$. 
Since this is obvious if $c(L) = \infty$, we may assume that $c(L)$ is 
finite.
For each $k\in \re$ we define the {\it action potential}
$\Phi_{k}:\M\times \M\to \re$ by
\[\Phi_{k}(x,y)=\inf\{A_{L+k}(\ga) : \hbox{\rm $\gamma$ is an absolutely continuous curve $x$ to $y$}\}.\]
It is obvious from its definition that
\[\p(x_{1},x_{3})\leq \p(x_{1},x_{2})+\p(x_{2},x_{3})\]
for all $x_1,x_2,x_3 \in \M$.
When $k\geq c(L)$, we have $\Phi_{k}(x,y) > - \infty$ for all $x$ and $y$ and the function  $\Phi_{k}$ is locally Lipschitz. 
This is proved in the case of a Lagrangian on a closed manifold in \cite{Ma1,CDI,CI2}. The only difference in our situation is that
we cannot claim that $\Phi_{k}$ is uniformly
Lipschitz since we are not assuming that $\Theta$ is bounded.



\begin{Lemma}Suppose that $c(L)$ is finite and $k\geq c(L)$.
If $u:\M\to\re$ is differentiable at $x\in \M$ and satisfies
$$
u(y)-u(x)\le\Phi_{k} (x,y)
$$
for all $y$ in a neighbourhood of $x$, then $H(x,d_{x}u)\le k$.
\end{Lemma}

\begin{proof} Let $\gamma(t)$ be a differentiable curve on $\M$ with
$(\ga(0),\dot{\ga}(0))=(x,v)$. Then
$$
\limsup_{t\to 0^+} \frac{u(\ga(t))-u(x)}{t}
\leq \limsup_{t\to 0^+} \frac{\Phi_{k} (\gamma(0),\gamma(t))}{t}
= \limsup_{t\to 0^+} \frac{1}{t}\int_0^t\bigl[L(\ga,\dot{\ga})+k\bigr]\, ds.
$$
Since the integral on the right is a differentiable function of $t$, we obtain
$$d_{x}u(v) \le L(x,v)+k.$$ 
Since $v$ was an arbitrary vector in $T_x\M$, it follows that
\[H(x,d_{x}u)=\max_{v\in T_{x}\M}\{d_{x}u(v)-L(x,v)\}\leq k.\]


\end{proof}

We now complete the proof that $c(L) \geq c(g,\Omega)$.
Fix a point $x_0 \in \M$
and define a function $u:\M \to \re$ by $u(x) = \Phi_{c(L)}(x_0,x)$.
By the previous lemma, $H(x, d_xu) \leq c(L)$ at every point $x \in \M$
where $u(x)$ is differentiable. Since $u$ is locally Lipschitz, $u$ is 
differentiable almost everywhere by Rademacher's theorem. We now need to
show that for any $k > c(L)$ 
 there is a smooth function $\widehat u: \M \to \re$ 
that approximates $u$ well enough  so that
$H(x, d_x\widehat u) \leq k$ for all $x \in \M$.
The necessary approximation argument is
presented in detail in
Section~6 of~\cite{FaMa}.
The first step
is to choose a locally finite $U_i$ covering of $\M$ by relatively compact 
open sets that are the domains of charts. Next one uses a convolution argument
to create a smooth approximation $\widehat u_i$ to $u_i$ on each~$U_i$. The convexity of the Hamiltonian $H(x,p)$ in the second variable and Jensen's inequality allow one to show that for any given $\epsilon_i > 0$ we can choose $\widehat u_i$ so that $H(x,d_x\widehat u_i) \leq c(L) + \epsilon_i$ for all $x \in U_i$. Then these 
local approximations are combined using a partition of unity subordinate to the cover to form $\widehat u$. The convexity of $H(x,p)$ as a function of $p$ is used again in showing that if the $\epsilon_i$ are chosen small enough, then one can obtain $H(x, d_x\widehat u) \leq k$ for all $x \in \M$.

\end{proof}

  


\section{Appendix B} \label{AppB}

In this appendix we sketch a proof of the following theorem which is 
a variation of a theorem in \cite{N}.
This theorem was used for the proof of Theorem D and generalizes
Ma\~n\'e's formula for geodesic flows \cite{Ma2}.

 Given $\theta=(x,v)\in SM$, let $X(\theta)$ be
the vector field of the magnetic flow $\phi_t$ of the pair $(g,\Omega)$ and let
\[\alpha(\theta)=\mbox{\rm ker}\,d_{\theta}\pi\oplus\re\,X(\theta),\]
where $\pi:SM\to M$ is the canonical
projection. Let $h_{top}$ be the topological entropy of the magnetic flow.


\begin{Theorem}Let $(g,\Omega)$ be a $C^{\infty}$ pair. Then we always have:
\[h_{top}\geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta.\]
Suppose in addition that the magnetic flow $\phi$ admits a continuous
invariant distribution of codimension one transversal to $X$. Then
\[h_{top}=\lim_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta.\]

\label{mfa}
\end{Theorem}

A crucial ingredient in the proof of Theorem \ref{mfa} is Kozlovski's formula
which we now recall.
Given a linear map $L:E\rightarrow F$ between finite dimensional vector spaces
with inner products, we define its {\it expansion} $\mbox{ex}(L)$ by
\[\mbox{ex}(L)=\max_{S}|\det(L|_{S})|,\]
where the maximum is taken over all subspaces $S\subset E$.

\begin{Theorem}[\cite{K}] Let $X$ be a closed Riemannian manifold and let
$\phi_{t}:X\to X$ be a flow 
of class $C^{\infty}$. Then
\[h_{top}=\lim_{T\to\infty}\frac{1}{T}\log\int_{X}\mbox{\rm ex}(d_{x}\phi_{T})\,dx,\]
where $h_{top}$ is the topological entropy of the flow $\phi$.
\label{koz}
\end{Theorem}

The predecesor of this formula is {\it Przytycki's inequality} \cite{Pr}. Kozlovski's proof in \cite{K}
for the equality case is based on Yomdin's work \cite{Y}.

The first inequality in Theorem \ref{mfa} is an immediate consequence of Kozlovski's formula
since $\left|\det\,d_{\theta}\pi|_{E}\right|\leq 1$ for any subspace $E\subset T_{\theta}SM$
and hence
\begin{align*}
\limsup_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}&
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta\\
&\leq\limsup_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}\phi_{t}|_{\alpha(\theta)}\right|\,d\theta\\
&\leq \max\left\{0,\limsup_{T\to\infty}\frac{1}{T}\log\left.\int_{SM}\right.
\left|\det\,d_{\theta}\phi_{T}|_{\alpha(\theta)}\right|\,d\theta \right\}\\
&\leq h_{top}.
\end{align*}
To prove the second statement in the theorem we certainly need more work.

Let $X$ be a closed Riemannian manifold and let
$\phi_{t}:X\to X$ be a flow without singularities
of class $C^{\infty}$ which preserves a volume form $dx$.
Let $\tau:S\to X$ be a symplectic vector
bundle over $X$ and let $\phi_{t}^{*}:S\to S$ be a {\it symplectic cocycle over}
$\phi$. By this we mean:
\begin{enumerate}
\item $\tau\circ\phi_{t}^{*}=\phi_{t}\circ\tau$;
\item for all $x\in X$ and $t\in\re$, $\phi_{t}^{*}(x):=\phi_{t}^{*}|_{S(x)}:S(x)\to S(\phi_{t}(x))$ is
a symplectic linear isomorphism;
\item for all $s$ and $t$ in $\re$ we have $\phi_{t+s}^{*}(x)=\phi_{s}^{*}(\phi_{t}(x))\circ \phi_{t}^{*}(x)$.
\end{enumerate}


Let $\Lambda\to X$ be the Grassmannian bundle of Lagrangian subspaces, i.e., for each $x\in X$,
the fibre $\Lambda(x)$ consists of all Lagrangian subspaces of $S(x)$.
Suppose that the symplectic vector bundle $S$ is endowed with a smooth Lagrangian
distribution $\alpha^{*}$. This means that we have a smooth map $x\mapsto \alpha^{*}(x)$
which is a section of the bundle $\Lambda\to X$.

The following definitions are taken from \cite{BP}. Given a Lagrangian subspace
$\la\in\Lambda$, we identify $T_{\la}\Lambda$ with the space ${\mathcal S}(\la)$
of symmetric bilinear forms in the following way. 
Every curve $\lambda(t)$ in $\Lambda$
with $\lambda(0)=\lambda$ can be written as $\lambda(t)=\Phi_{t}\lambda$ 
where $\Phi_{t}$ is a
path of linear symplectic transformations with $\Phi_{0}=Id$.
Then the symmetric bilinear form corresponding to the vector $\lambda'(0)$ is given by
\[(\xi,\eta)\rightarrow \omega\left(\right.\xi,\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}\eta\left.\right),\]
for all $\xi,\eta\in\lambda$. One easily checks that this correspondence is well defined.

Given a pair $(\phi, \phi^{*})$ where $\phi^{*}$ is a symplectic cocycle over $\phi$,
let us consider the tangent vector
\[l_{x}\df\left.\frac{d}{dt}\right|_{t=0}\phi^{*}_{t}(\alpha(\phi_{-t}x))\in T_{\alpha(x)}\Lambda.\]
We shall say that the pair
$(\phi, \phi^{*})$ is $\alpha^{*}${\it -optical} if the bilinear form in ${\mathcal S}(\alpha(x))$ that corresponds to $l_{x}$ by the identification described above is positive definite for all $x\in X$.


Now choose an almost complex structure $J:S\to S$ which is compatible with the symplectic structure
$\omega$, i.e., for each $x\in X$, $\omega_{x}(\cdot,J_{x}\cdot)$ defines a Riemannian metric
on the bundle $S$. We shall measure the expansion of $\phi^{*}$ with respect to this metric.

The next proposition is proved exactly as Proposition 4.18 in \cite{P3}. Our framework has been
set up precisely for the proof to go through.

\begin{Proposition}Let $(\phi, \phi^{*})$ be an $\alpha^{*}$-optical pair.
Then there exists a constant $D>0$ such that for all $t$ we have
\[D\int_{X}\mbox{\rm ex}(\phi_{t}^{*}(x))\,dx\leq \int_{X} \left|\det\,\phi_{t}^{*}(x)|_{\alpha^{*}(x)}\right|\,dx.\]
\label{ex-al}
\end{Proposition}

Now let $\tau:S\to SM$ be the symplectic vector bundle given by
$$S(\theta)=T_{\theta}SM/\re\,X(\theta),$$
with the symolectic structure induced by $\omega_1 = \omega_0 + \pi^*\Omega$.
The derivative $d_{\theta}\phi_{t}$ factors naturally to the quotient spaces
and induces a symplectic cocycle $\phi^{*}$ over $S$. If we take the projection
of $\alpha(\theta)$ to $S(\theta)$ we obtain a smooth Lagrangian distribution that we
denote by $\alpha^{*}$ and it is easy to check that the pair $(\phi, \phi^{*})$ is $\alpha^{*}$-optical \cite{BP}.




We shall also need the following proposition which is proved exactly as Lemma 4.7
in \cite{P3}.

\begin{Proposition}There exists a constant $C>0$ such that for all
$T>0$ we have:
\[\int_{0}^{T+1}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta
\geq C\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}\phi_{t}|_{\alpha(\theta)}\right|\,d\theta.\]
\label{pro}
\end{Proposition}


We now show:

\begin{Lemma}Suppose that $\phi$ admits a continuous
invariant distribution of codimension one transversal to $X$. Then there exist
positive constants $A$ and $B$ such that for all $\theta\in SM$ and all $t\in\re$ we have:
\begin{align*}
& \mbox{\rm ex}(d_{\theta}\phi_{t})\leq A\,\mbox{\rm ex}(\phi_{t}^{*}(\theta))\\
& \left|\det\,\phi^{*}_{t}(\theta)|_{\alpha^{*}(\theta)}\right|
\leq B\,\left|\det\,d_{\theta}\phi_{t}|_{\alpha(\theta)}\right|.
\end{align*}
\label{cociente}
\end{Lemma}

\begin{proof} Let $\theta\mapsto T(\theta)$ be the continuous
invariant distribution of codimension one transversal to $X$. Define a continuous
Riemannian metric $g_{T}$ on $SM$ as follows:
\begin{enumerate}
\item on the subspace $T(\theta)$, we let $g_{T}$ coincide with the Sasaki
metric of $SM$,
\item $T(\theta)$ is $g_{T}$-orthogonal to $X(\theta)$ for all $\theta$,
\item $X(\theta)$ has $g_{T}$-norm 1.
\end{enumerate}
Let $\mbox{\rm ex}^{g}$ denote expansion measured with respect to a Riemannian metric
$g$. We clearly have
\[\mbox{\rm ex}^{g_{T}}(d_{\theta}\phi_{t})=\mbox{\rm ex}^{g_{T}}(d_{\theta}\phi_{t}|_{T(\theta)}),\]
because $T(\theta)$ and $X(\theta)$ are $d_{\theta}\phi_{t}$-invariant and $T(\theta)$ is
the $g_{T}$-orthogonal complement of $\re X(\theta)$.
Now induce a continuous Riemannian metric $\widetilde{g}_{T}$ on $S$ in such a way that the restriction
of the projection map to $T(\theta)$ is an isometry. Then clearly:
\[\mbox{\rm ex}^{g_{T}}(d_{\theta}\phi_{t}|_{T(\theta)})=\mbox{\rm ex}^{\widetilde{g}_{T}}(\phi_{t}^{*}(\theta)).\]
The first inequality in the lemma follows from the following easy lemma whose proof we omit:

\begin{Lemma}Let $V\to X$ be a vector bundle over a compact manifold $X$. Given two
continuous Riemannian metrics $g_{1}$ and $g_{2}$ on $V$ there exists a constant $c>0$
such that for any two points $x$ and $y$ in $X$ and any linear map $L:V(x)\to V(y)$ we have
\[\mbox{\rm ex}^{g_{1}}(L)\leq c\,\mbox{\rm ex}^{g_{2}}(L).\]
\end{Lemma}

The second inequality in Lemma \ref{cociente} is proved in a similar way.

\end{proof}

Let us complete now the proof of Theorem \ref{mfa}. Suppose that $\phi$ admits a continuous
invariant distribution of codimension one transversal to $X$. Combining Theorem \ref{koz},
Lemma \ref{cociente}, Proposition \ref{ex-al} and Proposition \ref{pro} we obtain:
\begin{align*}
h_{top}&= \lim_{T\to\infty}\frac{1}{T}\log\int_{SM}\mbox{\rm ex}(d_{\theta}\phi_{T})\,d\theta\\
&\leq \liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}\mbox{\rm ex}(d_{\theta}\phi_{t})\,d\theta\\
&\leq \liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}\mbox{\rm ex}(\phi^{*}_{t}(\theta))\,d\theta\\
&\leq\liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,\phi^{*}_{t}(\theta)|_{\alpha^{*}(\theta)}\right|\,d\theta\\
&\leq \liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}\phi_{t}|_{\alpha(\theta)}\right|\,d\theta.\\
&\leq \liminf_{T\to\infty}\frac{1}{T}\log\int_{0}^{T}\,dt\int_{SM}
\left|\det\,d_{\theta}(\pi\circ\phi_{t})|_{\alpha(\theta)}\right|\,d\theta
\end{align*}
thus concluding the proof of Theorem \ref{mfa}.

\qed


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\end{document}
\end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%what follows is Manning's argument that shows that the exponential
%growth rate of each ball is less than or equal to $h_top$.
%The problem with this argument is doing the Freire-Ma\~n\'e, the advantage
%is that one gets it for class C3. I keep it in case we need it.

%We consider in $M$ and $\M$ the distance induced
%by the Riemannian metric and in $SM$ and $S\M$ the distance
%induced by the Sasaki metric.

%Without loss of generality we can assume
%that $h_{v}(x)>0$.
%For given small $\delta>0$ we consider
%$B(x,T+\delta/2)-B(x,T)$.
%Take any small $\varepsilon$
%with $h_{v}(x)-\varepsilon>0$. Let $V(x,T)=\mbox{\rm Vol}(B(x,T))$.
%Suppose that there exists $T_{0}>0$ such that
%\[V(x,T+\delta/2)-V(x,T)\leq \exp(h_{v}(x)-\varepsilon)T,\]
%for all $T\geq T_{0}$. Then by induction on $N$
%\[V(x,T_{0}+N\delta/2)\leq
%A\,\exp\left\{(T_{0}+N\delta/2)(h_{v}(x)-\varepsilon)\right\}+
%V(x,T_{0}),\]
%for all integers $N\geq 0$, where $A$ is a positive constant such that
%\[\exp\{-\delta/2(\lambda-\varepsilon)\}\leq A/(1+A).\]
%This would contradict the fact that the exponential growth rate of $V(x,T)$
%is $h_{v}(x)$. Then,
%\[V(x,T+\delta/2)-V(x,T)\geq \exp(h_{v}(x)-\varepsilon)T,\]
%for a sequence $T_{1},T_{2},\dots$ of values of $T$ tending to $\infty$.
%For such a value of $T$ we take a maximal subset $Q_{T}$ of
%$B(x,T+\delta/2)-B(x,T)$ whose points are pairwise $2\delta$
%apart in the distance $d$ induced by the Riemannian metric
%on $\M$. Then
%\[\#\,Q_{T}\geq \frac{V(x,T+\delta/2)-V(x,T)}{\sup_{z\in M}\mbox{\rm Vol(
%$d$-ball of center $z$ and radius $2\delta$)}}
%\geq c'_{\delta}\exp(h_{v}(x)-\varepsilon)T,\]
%for a constant $c'_{\delta}$. Any two points in $\widetilde{M}$ can
% be joined by a minimizing magnetic
%geodesic so each $x$ can be joined to each $q\in Q_{T}$ by a magnetic geodesic
%of length between $T$ and $T+\delta/2$. The unit initial vectors
%of these magnetic geodesics give a set $Y_{T}\subset S\widetilde{M}$ that
%is $(T,\delta)$-separated for $\widetilde{\phi}_{t}$, the magnetic
%flow of $\widetilde{M}$.
%This is because $d(\pi\widetilde{\phi}_{T}y,q)<\delta/2$,
%$d(\pi\widetilde{\phi}_{T}y',q')<\delta/2$ and $d(q,q')\geq 2\delta$
%imply
%\[d_{S\M}(\widetilde{\phi}_{T}y,\widetilde{\phi}_{T}y')\geq
%d(\pi\widetilde{\phi}_{T}y,\pi\widetilde{\phi}_{T}y')>\delta.\]
%Now observe that the set $dp(Y_{T})\subset SM$ is
%$(T,\delta')$-separated, where $\delta'=\frac{1}{2}\min\{\delta,r(M)\}$ and
%$r(M)$ is the injectivity radius of $M$.
%Thus
%\[h_{top}\geq \limsup_{n\to\infty}\frac{1}{T_{n}}\log\#\,Y_{T_{n}}
%\geq h_{v}(x)-\varepsilon,\]
%and $\varepsilon$ is arbitrary so $h_{top}\geq h_{v}(x)$ as required.

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