$C^1$-Genericity of Symplectic Diffeomorphisms and Lower Bounds for Topological Entropy
Thiago Catalan
We show that there is a $C^1$-generic (Baire second class) subset $\mathcal{R}$ of symplectic diffeomorphism on $2d$-dimensional manifold, $d\ge 1$ such that every $f$ in $\mathcal{R}$ satisfies a trichotomy: or $f$ is Anosov or $f$ is robustly transitive partially hyperbolic with {\em unbreakable center} of dimension $2m$, $1 \le m < d$ or $f$ has totally elliptic periodic points dense on $M$. In the second case, we also show the existence of a sequence of $m$-{\em elliptic} periodic points converging to $M$. Moreover, we provide lower bounds for non-Anosov diffeomorphisms in $\mathcal{R}$ by means of theirs largest central Lyapunov exponent.
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\title[$C^1$-Genericity of Symplectic Diffeomorphisms]{$C^1$-Genericity of Symplectic Diffeomorphisms and Lower Bounds for Topological Entropy}

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\author[T. Catalan]{Thiago Catalan}
\address{}
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\author[V. Horita ]{Vanderlei Horita}
\address{}
\email{}

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\begin{document}

\begin{abstract}
We show that there is a $C^1$-generic (Baire second class) subset
$\mathcal{R}$ of symplectic diffeomorphism on $2d$-dimensional manifold,
$d\ge 1$ such that every $f$ in $\mathcal{R}$ satisfies a trichotomy:
or $f$ is Anosov or $f$ is robustly transitive partially hyperbolic with
{\em unbreakable center} of dimension $2m$, $1 \le m < d$ or $f$ has
totally elliptic periodic points dense on $M$.
In the second case, we also show the existence of a sequence of
$m$-{\em elliptic} periodic points converging to $M$.
Moreover, we provide lower bounds for non-Anosov diffeomorphisms in
$\mathcal{R}$ by means of theirs largest central Lyapunov exponent.
\end{abstract}

\let\thefootnote\relax\footnote{2000 {\it Mathematics Subject Classification}. 
Primary 37J10, 37D30, 37B40, 37C20.}
\let\thefootnote\relax\footnote{{\it Key words and phrases}.
Partially hyperbolic symplectic systems, topological entropy, elliptic periodic points,
homoclinic tangency, generic properties.}
\let\thefootnote\relax\footnote{Work partially supported by CNPq, FAPEMIG, 
FAPESP, and PRONEX.}

\maketitle

\section{Introduction}
\label{s.introduction}

The concept of topological entropy of a dynamical system provides
information about its complexity and it is invariant by conjugacy.
Topological entropy is a positive real number that, roughly, measures
the rate of exponential growth of the number of distinguishable orbits
with finite but arbitrary precision as time advances.
Precisely, let $(X, d)$ be a compact metric space and $f \colon
X \to X$ be a continuous map. For each natural number $n$, we define
the metric
$$
    d_n(x,y)=\max\{d(f^i(x),f^i(y)): 0\leq i\leq n\}.
$$
Note that, given any $\varepsilon > 0$ and $n \ge 1$, two points of
$X$ are $\varepsilon$-close with respect to this metric if their
first $n$ iterates are $\varepsilon$-close.
A subset $E$ of $X$ is said to be $(n, \varepsilon)$-{\em separated} if
each pair of distinct points of $E$ is at least $\varepsilon$ apart
in the metric $d_n$. Denote by $N(n, \varepsilon)$ the maximum
cardinality of an $(n,\varepsilon)$-separated set.
The {\em topological entropy} of the map $f$ is defined by
$$
    h_{top}(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
    \frac{1}{n}\log N(n,\varepsilon)\right).
$$
Recall that the limit defining $h_{top}(f)$ always exists in the extended
real line (but could be infinite).

Lyapunov exponents are another useful tool to measure complexity
of a dynamical system. They are important constants to measure the asymptotic
behavior of dynamics in the tangent space level. Positive Lyapunov exponents indicate orbital divergence and
long-term unpredictability of a dynamical system because the
omnipresent uncertainty in determining its initial state
grows exponentially fast in time. In other words, Lyapunov
exponents tells us the rate of divergence of nearby trajectories. To be more precisely, given a diffeomorphism $f$ over a manifold $M$, we say that a real number $\lambda(x)$ is a {\it Lyapunov exponent} of $x\in M$ if there exists a nonzero vector $v\in T_xM$ such that
$$
\lim_{n\rightarrow \pm\infty} \frac{1}{n}\log \|Df^n(x) \ v\|=\lambda(x).
$$


One of the main results in this work relates these two different ways to
measure the complexity of a system.
We give lower bounds for the topological entropy for a class of symplectic
systems using Lyapunov exponents of the hyperbolic periodic points.
In other words, we obtain an estimate to topological complexity of the
system via differential properties of its hyperbolic periodic points.
%Moreover, we can not expect better results, at least in a
%residual subset of symplectic $C^1$-diffeomorphisms.

We can also infer if some kind of hyperbolicity (uniform or partial) comes
from the intrinsic symplectic structure of the diffeomorphisms just looking
to differential properties of its periodic points.
%In this paper we also use the analysis of the periodic points to obtain informations about the level of hyperbolicity existent for generic symplectic diffeomorphisms. That is, we %the other main result in this work
%use the particular intrinsic structure of a symplectic diffeomorphism to decide when generically such diffeomorphism is Hyperbolic (Anosov), or Partially Hyperbolic; just looking to differential properties of its periodic points.
Let us make precise these statements.

We say that a diffeomorphism $f \colon M \to M$ is {\it partially hyperbolic}
if there exists a continuous $Df$-invariant splitting
$TM=E^s\oplus E^c\oplus E^u$ with non trivial extremal sub-bundles
$E^s$ and $E^u$, such that for every $x\in M$ and every $m$ large enough:
$$
\|Df^m|E^i(x)\| \ \|Df^{-m}| E^j(f^m(x))\|\leq \displaystyle\frac{1}{2},
\ \text{ for any } (i,j)=(s,c),\ (s,u), \ (c,u);
$$
$$
 \|Df^m|E^s(x)\|\leq \displaystyle\frac{1}{2} \quad \text{ and }\quad
 \|Df^{-m}|E^u(x)\|\leq\displaystyle\frac{1}{2}.
$$
If the center bundle is trivial then $f$ is hyperbolic and we say that
$f$ is an {\it Anosov diffeomorphism.}

Now, let $(M,\omega)$ be a compact, connected, and boundaryless symplectic
manifold with dimension 2d.
We denote the set of symplectic $C^1$-diffeomorphisms on $M$ that preserve
the symplectic form $\omega$ by $\dw$.

%Let $f$ be a partially hyperbolic diffeomorphism with tangent bundle
%splitting $TM = E^s \oplus E^c \oplus E^u$.
We say that a partially hyperbolic diffeomorphism $f \in \dw$ has
{\em unbreakable center} if the splitting has minimal center dimension.
Recall that a partially hyperbolic symplectic diffeomorphism has even
center dimension, and denote by $\ph^1(m) \subset \diff^1_\omega(M)$,
$0 \le m < d$, the subset of partially hyperbolic $C^1$-diffeomorphisms
having unbreakable center of $\dim(E^c) = 2m$.
Note that $\mathcal{PH}_{\omega}^1(i)$ and $\mathcal{PH}_{\omega}^1(j)$
are disjoint subsets for every distinct $0<i, j<d$.
For convenience, we denote by $\mathcal{PH}_{\omega}^1(0)=\mathcal{A}$
the subset of Anosov diffeomorphisms and by $\mathcal{PH}^1_{\omega}(d)$
the complement of the closure of the set formed by all Anosov and partially
hyperbolic diffeomorphisms.

Let $\per_h(f)$ be the set of hyperbolic periodic points of $f$.
We say that $p \in \per_h(f)$ is a {\em $m$-elliptic periodic point}
if $Df^{\tau(p,f)}(p)$ has exactly $2m$ modulus one non real and
simple eigenvalues, where $\tau(p,f)$ is the period of $p$.

Recall that if $f$ is partially hyperbolic and has a $m$-elliptic
periodic point then $\dim E^c$ should be larger than $2m$.
Moreover, $m$-elliptic periodic points are robust for symplectic
diffeomorphisms.
In \cite{ArBC}, Arnaud, Bonatti, and Crovisier show that a
partially hyperbolic diffeomorphism $f\in \ph^1(m)$ on $4$-dimensional
manifold should have $m$-elliptic periodic points dense on $M$ (see
also \cite{A}).
They also conjectured that the same is true in any dimension.
In this direction it is worth to point out that Newhouse \cite{N2}
prove that in the complement of the set of Anosov symplectic
diffeomorphisms (in $\dw$) there are generically symplectic diffeomorphisms
exhibiting $1$-elliptic periodic points. %See also \cite{M-C}.

The next result provides a positive answer to the conjecture of Arnaud,
Bonatti, and Crovisier.

\begin{main}
\label{t.trichotomy}
There exists a residual subset $\mathcal{R}_1
\subset \dw$, such that if $f\in \mathcal{R}_1$ one of the following properties happens:
\begin{itemize}
\item[a)] $f$ is an Anosov symplectic diffeomorphism,

\item[b)] $f$ is a robust transitive partially hyperbolic symplectic
diffeomorphism in $\ph^1(m)$, for some $0 < m < d$, and there is a sequence
of $m-$elliptic periodic points converging to $M$ (in the Hausdorff
topology).

\item[c)] $f$ is not partially hyperbolic and has a sequence of totally
elliptic periodic points converging to $M$ (in the Hausdorff topology).
\end{itemize}
\end{main}


%The previous result answer a question posed by Arnaud, Bonatti, and Crovisier in
%\cite{ABC05}.

\medskip


The bounds for topological entropy mentioned early are for diffeomorphims in items (b) and (c)
in previous theorem.
Before make precise this statement, let us recall some previous related
results.

Given $p \in \per_h(f)$, we denote $\tau(p,f)$ the period of $p$
and $\lambda_i(p,f)$ stands for the eigenvalues of $Df^{\tau(p,f)}(p)$.
Let
$$
\lambda_{min}(p,f) = \min \{ |\lambda_i(p,f)| \colon |\lambda_i(p,f)| >1\}
$$
and
\begin{equation}
s(f) = \sup \left\{ \frac{1}{\tau(p,f)} \log \lambda_{min}(p,f), p\in
\per_h(f) \right\}.
\end{equation}
So, $s(f)$ is the supremum over the smallest positive Lyapunov exponents within the Lyapunov exponents of the hyperbolic periodic points of $f$ .

Newhouse \cite{N3} gives a lower bound for the topological entropy
$h_{top}(. )$ in a 2-dimensional setting: there is a residual subset
of $C^1$-diffeomophisms in surfaces $\mathcal{R} \subset \diff^1_\omega(M^2)$
such that for every non Anosov diffeomorphism $f \in \mathcal{R}$, we have
$$
h_{top}(f) \ge s(f).
$$
Recently Catalan and Tahzibi \cite{CT} generalize this result
to any dimension.
Furthermore, they also prove the equality holds for surface Anosov
diffeomorphism.
It is worth to point out that for hyperbolic systems the bound
is opposite.
In fact, there exists a $C^1$-open and dense set of surface Anosov
diffeomorphisms such that $h_{top}(f) < s(f)$.
See \cite{N3} and also \cite{CT}.

\smallskip

For a linear operator $A: V \rightarrow V$ and an $A$-invariant
subspace $E\subset V$ the spectrum radius of $A$ restrict to $E$ is
denoted by $\sigma(A|E)$.
Hence, given a partially hyperbolic diffeomorphism $f\in \ph^1(m)$
we define
$$
S_m(f):=\sup\left\{\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c(p))
\colon p\in \per_h(f)\right\},
$$
the supremum over the largest Lyapunov exponents within the central
bundle of the hyperbolic periodic points of $f$.
It is straightforward to see that $S_m(\cdot)$ is a lower
semicontinuous function.

\begin{main}
\label{t.entropy}
There exists a residual subset $\mathcal{R}\subset
\dw$ such that if $f\in\mathcal{R} \cap \ph^1(m)$, $0< m \le d$, then
$$
h_{top}(f)\geq S_m(f).
$$
\end{main}

Note that in $\ph^1(d)$, the previous result yields the best lower bound
to topological entropy in terms of single Lyapunov exponents of hyperbolic
periodic points. %Moreover, we have the following directly consequence of Theorem \ref{t.trichotomy} and Theorem \ref{t.entropy}.
Furthermore, if we know the value of the entropy of $f \in \mathcal{R}$
and $f$ has a hyperbolic periodic point with a Lyapunov exponent $\lambda_p$
such that $h_{top}(f)< \lambda_p$, then $f$ is either an Anosov or a partially
hyperbolic diffeomorphism.


%\begin{main}
%\label{t.entropy}
%There exists a residual subset $\mathcal{R}\subset
%\dw$ such that if $f\in\mathcal{R}$ is a non Anosov diffeomorphism
%then
%$$
%h_{top}(f)\geq S_m(f), \text{ if }\;f\in \ph^1(m) \text{ for some }\;
%0<m\leq d.
%$$
%\end{main}
%
%Note that in $\ph^1(d)$,
%the previous result yields the best lower bound possible to topological entropy by means of a Lyapunov exponent of a hyperbolic periodic point of a generically symplectic diffeomorphism. %Moreover, we have the following directly consequence of Theorem \ref{t.trichotomy} and Theorem \ref{t.entropy}.

%\begin{corollary}
%If $f$ is a  generic symplectic diffeomorphism, and has a hyperbolic periodic point $p$ having a Lyapunov exponent $\lambda_p$  such that $h_{top}(f)< \lambda_p$, then $f$ is either an Anosov diffeomorphism, or a partially hyperbolic diffeomorphism.
%\end{corollary}

%Also, we remark that it's possible to given an example  where the estimative given by Theorem \ref{t.entropy} is also  the best one for a partially hyperbolic diffeomorphism. More precisely, let $f$ be an Anosov symplectic diffeomorphism  over a 2-dimensional symplectic manifold $M$, satisfying $h_{top}(f)<s(f)$, and define a partially hyperbolic symplectomorphism $g=f\times Id$ over $M\times M$, where $Id$ is the identity map on $M$.  Thus, provided the topological entropy of $g$ is equal to $f$, and we have only zero central Lyapunov exponents: $S_2(g)\leq h_{top}(g)< s(g)$. However, this is not a generic example.


% and moreover, it's not possible to use this argument to find an example in higher dimension. This is because, for higher dimension there exists an open and dense subset inside the set formed by Anosov symplectomorphisms where the topological entropy is upper bound by the supremum over the sum of all positive Lyapunov exponents of a hyperbolic periodic point. We believe,

%In particular, $g$ satisfies the thesis of Theorem \ref{t.entropy}, and $h(g)<\lambda$ for




The paper is organized as follows: in Section 2 we prove some useful perturbative results in the symplectic scenario, recall some known results as connecting lemma, and we use the theory of periodic linear systems to find periodic points with nice eigenvalues from a previous fixed periodic point. In Section 3, we prove that $S_m(f)$ could be reached by diagonalizable periodic points, and thus using a technical result we prove Theorem 2. In Section 4, we prove Theorem 1, and moreover we get nice intersections between the strong stable and unstable  manifolds of diagonalizable periodic points, which will be used to prove the technical proposition in Section 5.



%\textcolor[rgb]{0.98,0.00,0.00}{Resume of contents, comments on higher topology-Turaev}


\section{Preliminaries: Perturbative lemmas and periodic linear systems in the symplectic scenario}

First, let us recall some basic facts about symplectic vector spaces.

Let $(V, \omega)$ be a symplectic vector space of dimension $2d$.
For any subspace $W \subset V$ we define its {\it symplectic orthogonal}
vector space as
$$
  W^{\omega} = \{  v \in V; \omega(v, w) = 0 \quad \text{for all}
  \quad w \in W \}.
$$
The subspace $W$ is called {\it symplectic} if $W^{\omega} \cap
W = \{0\}$.
We say $W$ is {\it isotropic} if $W \subset W^{\omega},$ that is
$\omega|(W\times W) =0$.
When $W = W^{\omega}$ we say that $W$ is a {\it Lagrangian}
subspace.


For a symplectic form $\omega$ in $V$, there is a symplectic basis
$\mathcal{B}=\{e_1,\ldots, e_{2d}\}$ of $V$ such that, with respect
to this basis, $\omega$ is in the standard form,
$\omega=\sum_{i=1}^d de_i \wedge de_{i+d}$, i.e.,
$\omega(e_i, e_{d+i})=1$ and $\omega(e_i,e_j)=0$ if $j\neq d+i$,
for every $1\leq i\leq d$.
Now, if $J$ is the canonical map on $V$, with respect to $\mathcal{B}$,
such that $J^2=-Id$,  we say that a {\it linear map $A$ is symplectic}
if $A^*JA=J$.
In particular, $A$ is a symplectic map if and only if $A^*\omega=\omega$.
Note, if we take an inner product on $V$ %$<\ ,\ >$ is the inner product in $V$
for which $\mathcal{B}$ is an orthonormal basis, then $\omega(u,v)=<u ,\ Jv>$.

Given two vector subspaces $E$ and $E'$ of same dimension, it is trivial
to find a linear isomorphism $A$ such that $A(E') = E$.
Moreover, if $E$ and $E'$ are close (that is to say there is
orthonormal basis of $E$ and $E'$ that are close in Hausdorff metric)
then $A$ is close to the identity.
Next lemma asserts that $A$ can be taken symplectic and preserving
a complementary space of $E$.

%What follows are some symplectic linear perturbative lemmas, which %ones together with Franks Lemma
%allow us to perform nice local perturbations of symplectic systems.
%For example,  in general is somehow easy to find a linear map $A$ close to identity such that $A(E)=F$, if $E$ is a subspace close to $F$. But here, we need that this kind of map $A$ should also be symplectic.
%The following lemma manages this in a special case. Recall that two subspaces $E$ and $F$ of $V$ with same dimension, say $n$, are close if they have orthonormal basis close, with respect to a fixed inner product on $V$.  %linear immersions of $R^n$ in $V$ which have images $E$ and $F$ are close linear maps.
%
%%However, we need here that this kind of map $A$ should also be symplectic. The following lemma manages this in a special case.
%
%
%%in the symplectic scenario.



\begin{lemma}
Let $V=E\oplus F$, where $E\subset (V,\omega)$ is an isotropic subspace. For any $\eps>0$, there exists $\delta>0$ such that
if $W\subset (V,\omega)$ is an isotropic subspace $\delta-$close to $E$,  $dim \ W=dim \ E$,  then there exists a symplectic linear map $B$ in $V$ $\eps-C^1$ close to $Id$ such that $B(W)=E$ and $B|F=Id$.
\label{isotropic2}\end{lemma}

\proof
We use here coordinates $(x,y)$ in $V$ with respect to the decomposition $V=E\oplus F$, and we fix an arbitrary norm $\| \ \|$ in $V$.

Since $W$ is close enough to $E$, there exists a linear map $A:E\rightarrow F$  such that $W=\{(x,Ax);\ x\in E\}$. Moreover, given $\eps>0$ we can chose $\delta>0$ small enough such that if $W$ is $\delta-$close to $E$, then $\|A\|<\eps$. Thus if we define $j:E \rightarrow V$ by $j(x)=(x,A(x))$, since $W$ is an isotropic subspace, we have $j^*\omega=0$,  where $j^*\omega$ is the pull-back of the symplectic form $\omega$ by $j$.  Analogously, if
$i:E\rightarrow V$ is the natural inclusion, $i(x)=(x,0)$,  we have $i^*\omega=0$.

Finally, if we define $B:V\rightarrow V$ by $B(x,y)=(x,y-A(x))$, then to conclude the lemma we just need to show that $B$ is indeed symplectic, since $\|B-Id\|\leq \|A\|<\eps$, $B(W)=E$ and $B|F=Id$.

Hence, let $\pi:V\rightarrow E$ be the projection on the first coordinate, i.e.,  $\pi(x,y)=x$. Now, we can rewrite $B$ in the following way $
B=Id+i\circ\pi-j\circ\pi. $ And therefore, we finally can verify that
$$B^*\omega=\omega+\pi^*i^*\omega-\pi^*j^*\omega=\omega,$$ where we use that $i^*\omega=j^*\omega=0$ in the second equality.
%Therefore, the lemma is proved.

$\hfill\square$



The next lemma allows to perform perturbations inside a symplectic subspace, keeping invariant its symplectic orthogonal space.

\begin{lemma}
Let $W\subset (V,\omega)$ a symplectic subspace. For any $\eps>0$, there exists $\delta>0$, such that if $A: W\rightarrow W$ is a symplectic linear map %i.e., $U^*\omega|W\times W=\omega|W\times W$,
$\delta-C^{1}$ close to the identity map $Id|W$, then there exists a symplectic linear map $B$  over $V$, $\eps-C^1$ close to $Id$ such that $B|W=A$ and $B| W^{\omega}=Id$.
\label{symplectic}\end{lemma}

{\it Proof: } We suppose $dim \ W=2m$,  $m<d$, and we take $\eps>0$ arbitrarily. %since $W$ is a symplectic subspace of $V$.

Now, let $\{e_1,\ldots, e_{2m}\}$ be a symplectic basis of $W$, % i.e., $\omega(e_i, e_{m+i})=1$ and $\omega(e_i,e_j)=0$ if $j\neq m+i$, for every $1\leq i\leq m$.
and  $\{e_{2m+1}, \ldots, e_{2d}\}$ be a symplectic basis to $W^{\omega}$. Hence, $\{e_1,\ldots, e_{2d}\}$ is a symplectic basis to $V$. Let us consider now the inner product  for which  this basis is orthonormal,  and thus $\omega(u,v)=<u,Jv>$, where
$$
J=\left[\begin{array}{cc}
J_W& 0 \\ 0 & J_{W^{\omega}}
\end{array}\right],
\text{ where }
J_{i}=\left[\begin{array}{cc}
0& Id_i \\ -Id_i & 0
\end{array}\right], \ i= W, \ W^{\omega},
$$
being $Id_i$ the identity matrix of order $m\times m$ if $i=W$, or order $d-m\times d-m$ if $i=W^{\omega}$.

Provided $A$ is a symplectic linear map over $W$, we have $A^* J_W A=J_W$. Hence, defining
$$
B=\left[\begin{array}{cc}
A& 0 \\ 0 & Id_{W^{\omega}}
\end{array}\right],
$$
we have that $B^* J B=J$, which implies that $B$ induces a symplectic linear map on $V$, where $B|W=A$ and $B|W^{\omega}=Id$.
%using in $V$ coordinates given by de fixed symplectic basis.

Therefore, we can choose $\delta>0$ small enough depending on the symplectic basis fixed at begin, such that if $A$  is $\delta-$close to Id|W, then the linear map $B$  is  $\eps-$close to $Id$, which finishes the proof.

$\hfill\square$







%{\color{red} NAO SEI SE VAMOS USAR O LEMA SEGUINTE}



%\begin{lemma}
%Let $W\subset (V,\omega)$ an isotropic subspace. For any $\eps>0$, there exists $\delta>0$ such that  if $A:W\rightarrow W$ is a linear map $\delta-C^1$ close to $Id$, then there exists a symplectic linear map $B$ in $V$ $\eps-C^1$ close to $Id$ such that $B|W=A$.
%\label{isotropic}\end{lemma}

%\proof
%We fix first a basis $\mathcal{B}$ of $V$ for what $\omega$ is in the standard form. Now, let $\tilde{W}$ a Lagrangian subspace containing $W$, and let $\mathcal{\tilde{C}}=\{u_1,\ldots, u_n\}$ an orthonormal basis of $\tilde{W}$ with respect the inner product induced by $\mathcal{B}$. Moreover, we can assume that $\{u_1,\ldots, u_k\}$ is a basis for $W$, for some $1\leq k\leq n$.

%Using $\mathcal{\tilde{C}}$, we can obtain a symplectic basis $\mathcal{C}=\{u_1,\ldots,u_n, -J(u_1),\ldots, -J(u_n)\}$ of $V$.

%Since, $A$ is an automorphism over $W$, there exists a linear map $\tilde{A}: \tilde{W}\rightarrow \tilde{W}$ such that $\tilde{A}|W=A$ and $\tilde{A}(u_i)=u_i$, for $k+1\leq i\leq n$. Note that $\tilde{A}$ is also $\delta-C^1$ close to Id. In particular, if we define $H=A-Id$ then $\|H\|<\delta$. %$A=Id+H$.

%Finally, we denote by $B$ the linear map on $V$ with the following matrix  with respect the basis $\mathcal{C}$:
%$$
%\left(\begin{array}{cc}
%Id+H & 0 \\ 0 & (Id+H^*)^{-1}
%\end{array}\right).
%$$
%Hence, taking $\eps=\delta$, since $Id+H=A$ and $B^* J B=J$,  $B$ is the linear symplectic map we wanted.

%$\hfill\square$



%{\color{red} AQUI COLOCAR OS LEMAS DE PERTURBA��O, FRANKS LEMMA, CONECTING LEMMA E BLA}


Now,  we recall  the famous Franks' Lemma which one enable us to perform non-linear perturbations along a finite piece of an orbit from linear perturbations, in particular the above lemmas. See \cite{F} and \cite{HT}.
 %for the symplectic version.  %{\color{red} REFERENCIA PARA ESTE RESULTADO}

For that, we consider $(M,\omega)$ be a symplectic manifold as in the introduction. As in the linear case,  a diffeomorphism $f$ over $M$ is a symplectic diffeomorphism iff $f^*\omega=\omega$. Recall we have denoted by $\dw$ de set formed by symplectic diffeomorphisms on $M$.


%simply performing arguments from Linear Algebra.
\begin{lemma}[Symplectic Franks' Lemma] Let $f\in \dw$ and $B$ a finite $f$-invariant set. Assume that $A$ is a small symplectic perturbation of $Df$ along $B$. Then for every neighborhood $U$ of $B$ there is a symplectic diffeomorphism $g$ arbitrarily $C^1$-close to $f$ coinciding with $f$ on $B$ and out of $U$, and such that $Dg$ is equal to $A$
on $B.$
\end{lemma}


\subsection{Connecting invariant submanifolds.}
%Another important perturbative lemma that will be necessary in this work is the Hayashi connecting Lemma \cite{HAY}.

Given a hyperbolic periodic point $p$ of a symplectic diffeomorphism $f$ we define its {\it stable (unstable)  manifold}, $W^s(p,f)$ $(\ W^u(p,f) \ )$, by the subset of points in $M$ whose forward (past) orbit by $f^{\tau(p,f)}$ %\ (f^{-\tau(p, f)})$
accumulates on $p$.


\begin{remark} If $f$ is a symplectic diffeomorphism over $M$, and $p$ is a hyperbolic periodic point of $f$, the stable and unstable manifolds of $p$, $W^{s}(p,f)$ and $W^u(p,f)$, are Lagrangian sub manifolds of $M$, that is $T_xW^{s(u)}(p,f)$ is a Lagrangian subspace for every $x\in W^{s(u)}(p,f)$. In particular, $E^s(p,f)$ and $E^u(p,f)$ are Lagrangian subspaces of $T_pM$.
\label{dec.}\end{remark}







We define the \emph{homoclinic class} of a hyperbolic periodic point, $H(p,f)$, as the closure of transversal intersections between the stable manifold and unstable manifold  of points in the orbit of $p$: $H(p,f)=\overline{W^s(o(p))\pitchfork W^u(o(p))}.$
A homoclinic class is a transitive set, and moreover coincides with the closure of the hyperbolic periodic points homoclinically related to $p$, where we say that a hyperbolic periodic point $q$ is \emph{ homoclinically related} to $p$ if $W^s(p)\pitchfork W^u(q)\neq \emptyset$ and $W^u(p)\pitchfork W^s(q)\neq \emptyset$.



Now, given a periodic point  $p$ of $f\in\dw$, if there exists a partially hyperbolic splitting $T_pM=E^{ss}_k\oplus E^c_k\oplus E^{uu}_k$ for $Df^{\tau(p,f)}$, with $dim(E^{ss}_k)=dim (E^{uu}_k)=k$ and $dim E^c_k=2d-2k$, then by \cite{HPS}, %the strong directions %$E^{ss}_k(p)$ and $E^{uu}_k(p)$ directions
%are uniquely integrable,
there exists a local $f^{\tau(p,f)}$ $(f^{-\tau(p,f)})$-invariant {\it $k-$dimensional strong
stable ( unstable ) manifold} $W_{k, loc}^{ss} (p)$ ( $W^{uu}_{k,loc}(p)$ ) %and a  local $f^{-\tau(p,f)}$-invariant {\it $k-$dimensional strong unstable manifold}
%$W^{uu}_{k,loc}(p)$, both of them
varying $C^1-$continuously  with respect
to the diffeomorphism, tangentt to $E^{ss}_k\ (E^{uu}_k)$ at $p$.  Hence, we define the $k-$dimensional strong stable and unstable manifolds of $p$, respectively, by
$$
W_{k}^{ss} (p)=\bigcup_{n\in \N} f^{-n}(W_{k, loc}^{ss} (p)) \text{ and } W_{k}^{uu} (p)=\bigcup_{n\in \N} f^{n}(W_{k, loc}^{uu} (p)).
$$
Note that $W_{d}^{ss} (p)=W^{s} (p)$ and $W_{d}^{uu} (p)=W^{u} (p)$.

%In this work, by {\it strong homoclinic intersections} we mean intersections between $W_{k}^{ss} (p)$ and $W_{k}^{uu} (p)$, $k\neq d$.

\begin{remark} More general, if $f$ is a partially hyperbolic symplectic diffeomorphism over $M$, $TM=E^{ss}\oplus E^c\oplus E^{uu}$, then by \cite{HPS} there are strong stable and unstable invariant manifolds for every point $x\in M$. Moreover, provided these sub manifolds are strictly contained in the stable or unstable manifold, they are isotropic sub manifolds. That is, their tangent space are isotropic subspaces. Also, we would like to observe that a partially hyperbolic splitting is such that $E^c(x)$ is a symplectic subspace and moreover $(E^c(x))^{\omega}=E^{ss}\oplus E^{uu}(x)$, for every $x\in M$. In particular, $E^{ss}\oplus E^{uu}(x)$ is a symplectic subspace.
\label{decomp.}\end{remark}


For a special kind of periodic points, namely diagonalizable hyperbolic periodic points, we can find $k-$strong manifolds for any $0<k<d$. A periodic point $p$ of $f\in\dw$ is {\it diagonalizable} if $Df^{\tau(p,f)}(p)$ has only real positive eigenvalues with multiplicity one. Hence, for any diffeomorphism $f\in \dw$ and any dia\-gonalizable hyperbolic periodic point $p$ of $f$,  denoting by $\lambda_{1,p},\ldots, \lambda_{2d,p}$ the distinct
simple eigenvalues of $Df^{\tau(p,f)}(p)$ and by $E_{\lambda_{j,p}}\subset T_pM$,
$1\leq j\leq 2d$, the respective eigenspaces, we can define the $k$-dimensional strong stable  and  strong unstable
directions of $p$ for any $0<k<d$, in the following way %More precisely, for a diagonalizable periodic point $p$ we define
$$
E^{ss}_k(p)=\bigoplus_{1\leq j\leq k} E_{\lambda_{j,p}} \, \text{ and }
E^{uu}_k(p)=\bigoplus_{2d-k+1\leq j\leq 2d} E_{\lambda_{j,p}}.
$$
%for every $1\leq k\leq d$. For convenience,
We also define
$$
E^{c}_k(p)=\bigoplus_{k+1\leq j\leq 2d-k} E_{\lambda_{j,p}}.
$$
but we remark that in this case $k$ has no relation with the center dimension.
%By \cite{HPS77}, the strong directions %$E^{ss}_k(p)$ and $E^{uu}_k(p)$ directions
%are uniquely
%integrable, which implies the existence of a  local $f$-invariant {\it $k-$dimensional strong
%stable manifold} $W_{k, loc}^{ss} (p)$ and a  local $f^{-1}$-invariant {\it $k-$dimensional strong unstable manifold}
%$W^{uu}_{k,loc}(p)$, both of them varying $C^1-$continuously  with respect
%to the diffeomorphism for every $1\leq k\leq d$. Hence, we define the $k-$dimensional strong stable and unstable manifolds of $p$, respectively, by
%$$
%W_{k}^{ss} (p)=\bigcup_{n\in \N} f^{-n}(W_{k, loc}^{ss} (p)) \text{ and } W_{k}^{uu} (p)=\bigcup_{n\in \N} f^{n}(W_{k, loc}^{uu} (p)).
%$$
%Note that $W_{d}^{ss} (p)=W^{s} (p)$ and $W_{d}^{uu} (p)=W^{u} (p)$.

In this work, by {\it strong homoclinic intersections} we mean intersections between $W_{k}^{ss} (p)$ and $W_{k}^{uu} (p)$, $k\neq d$. Moreover if $q$ is a strong homoclinic intersection such that $T_qW_{k}^{ss} (p)\cap T_qW_{k}^{uu} (p)=0$, then we say $q$ is a {\it strong transversal homoclinic intersection}.


To create such intersections we can use a connecting lemma. See \cite{H}.  Here we will use the following one, which is a symplectic version, proved by  Xia and Wen \cite{XW}. %We also refer the reader to \cite{H}


\begin{proposition}[Theorem E in \cite{XW}]
Let $z\in M$ be a non-periodic point of $f\in \dw$. For any $C^1$ neighborhood $\mathcal{U}$ of $f$, there are $\rho>1$, $L\in \N$ and $\delta_0>0$ such that for any $0<\delta<\delta_0$, and for any
point p outside the tube $\Delta=\cup_{n=1}^L f^{-n} B(z, \delta)$ and any point $q\in B(z,\delta /\rho)$, if the positive $f$-orbit of $p$ hits $B(z, \delta / \rho)$ after $p$, then there is a symplectic diffeomorphism $g\in \mathcal{U}$ such that $g$ = $f$ off $\Delta$ and $q$ is on the positive g-orbit of $p$.
\label{xw}\end{proposition}



We remark that
symmetricly, we can restate the previous proposition for a tube along the positive orbit of $z$, and require that the negative orbit of $ q$ goes through the tube at least once before the hit.


%\begin{theorem}[Theorem A in \cite{Xia-Wen}] Let $x$ and $y$ be two points of $M$ with $\omega(x)\cap \alpha(y)\neq \emptyset$. We also assume that $\omega(x)\cap \alpha(y)$ contains some non-periodic point $z$. Then for any $C^1$ neighborhood
%$\mathcal{U}$ of $f$, there is $g \in \mathcal{U}$ such that $y$ is on the positive $g$-orbit of $x$. More precisely, for any $C^1$ neighborhood $\mathcal{U}$ of $f$, there is a positive integer $L$ such that for any $\delta>0$, there is a $g \in  \mathbb{U}$ such that $g= f$ outside the tube $\cup{i=1}^L B(f^i(z), \delta)$ and such that $y$ is on the positive $g$-orbit of x.
%\end{theorem}



%In this work we will use this result mainly to create strong homoclinic intersections. Let us explain this in more details.



\begin{lemma} Let $f\in \dw$ and $p$  be a hyperbolic or a $m$-elliptic periodic point of $f$ having $k$-dimensional strong stable and unstable manifolds.  For any neighborhood $\mathcal{U}$ of $f$ there exists a diffeomorphism $g\in \mathcal{U}$ such that there exists a $k-$strong homoclinic intersection for the analytic continuation $p(g)$ of the periodic point $p$ for $g$. That is, there exists an intersection between $W_{k}^{ss} (p(g))$ and $W_{k}^{uu} (p(g))$.
\label{conexao}\end{lemma}



%\begin{lemma} Let $f\in \dw$ and $p$ be a diagonalizable hyperbolic periodic point of $f$. For any $1\leq k\leq d$, and any neighborhood $\mathcal{U}$ of $f$ there exists a difeomorphism $g\in \mathcal{U}$ such that $g=f$ in a small neighborhood of the orbit of $p$ and there exists an intersection between $W_{k}^{ss} (p)$ and $W_{k}^{uu} (p)$.
%\label{conexao}\end{lemma}


Using the following result of Arnaud, Bonatti and Crovisier, which is a symplectic version of the result in \cite{BC}, the proof of the previous lemma is similar to the proof of Theorem D in \cite{XW}. %However, for sake of completeness
We will put here a sketch of the proof.

%  To prove this lemma we will also need the following result of ARNAUD-BONATTI-CROVISIER, which is a symplectic version of the result in \cite{BC}.

\begin{proposition}[Theorem 1 in \cite{ArBC}]
There exists a residual subset $\mathcal{R}$ of $\dw$ such that if $f\in \mathcal{R}$ then there exists a hyperbolic periodic point $p$ of $f$ such that $M=H(p,f)$. In particular, $f$ is transitive. \label{ABC}\label{abc}\end{proposition}



{\it Proof of Lemma \ref{conexao}:}

By Proposition \ref{abc}, after a perturbation, we can suppose that $f$ is transitive. Now, let $z^s\in W_k^{ss}(p)$ and $z^u\in W_k^{uu}(p)$. Then, we can choose $\rho>1$, $L\in \N$ and $\delta_0$ such that Proposition \ref{xw} and its symmetrically version are true when we take $z=z^s$ or $z=z^u$. Hence, we can consider $\delta>0$ small enough such that if we denote by $\Delta^s=\cup_{n=1}^L f^{-n} B(z^s, \delta)$ and $\Delta^u=\cup_{n=1}^L f^n B(z^u, \delta)$ then
$$
f^n(z^s)\not\in \Delta^s\cup \Delta^u \text{ and } f^{-n}(z^u)\not\in \Delta^s\cup \Delta^u, \text{ for every } n\in \N.
$$
Now, since $f$ is transitive there exists $x\in B(z^u,\delta/\rho)$ such that $f^m(x)\in B(z^s, \delta / \rho)$. Thus, by Proposition \ref{xw}, there exists a symplectic diffeomorphism $g$ $C^1$ close to $f$ such that $g=f$ outside $\Delta^s$ and $z^s$ belongs to the positive orbit of $x$. In particular, $x\in W_k^{ss}(p,g)$. Applying Proposition \ref{xw} once more, since $x=f^{-n}(z^s)\in B(z^u,\delta/\rho)$ for some $n>0$, there exists a symplectic diffeomorphism $h$, $C^1-$close to $g$ such that $h=g$ outside $\Delta^u$, and $z^u$ belongs to the negative orbit of $z^s$. Therefore, $h=f$ outside $\Delta^s\cup \Delta^u$ which implies $z^s, z^u\in W_k^{ss}(p)\cap W_k^{uu}(p)$.

$\hfill\square$

\subsection{Periodic symplectic linear systems}


We will define the concept of periodic linear systems with transitions in the symplectic scenario as done in \cite{HT}. For the original definition see \cite{BDP}.


A {\it periodic symplectic linear system( or linear symplectic cocycle over $f$)} is a 4-tuple $\mathcal{P}=(\Sigma, f, \mathcal{E},A)$, where $f$ is a homeomorphism,
$\Sigma$ is an infinite set of periodic points of $f$, $\mathcal{E}$ is a symplectic vector bundle defined over $\Sigma$ (i.e., each fiber $\mathcal{E}(x)$ is a symplectic vector space), and $A:\mathcal{E}\rightarrow \mathcal{E}$ is   such that $A(x):\mathcal{E}_x\rightarrow \mathcal{E}_{f(x)}$ is a symplectic linear isomorphism for each $x$.
Considering  an orthonormal system of coordinates of the linear bundle $\mathcal{E}$, the periodic symplectic linear system  $(\Sigma, f, \mathcal{E},A)$ can  be considered as a system of matrices $(\Sigma, f, A)$. Hence, given an element $x\in \Sigma$, we denote by
$$
M_A (x) = A(f^{n-1}(x)) A(f^{n-2}(x)) \ \dots \   A(x),
$$
where $n$ is the period of the periodic point $x$.
As before,  a
periodic linear system is {\it diagonalizable} at the point $x\in \Sigma$ if  $M_A(x)$
has only real eigenvalues of multiplicity one.

Given a set $\mathcal{B}$, a {\em word} with letters in $\mathcal{B}$
is a finite sequence of elements of $\mathcal{B}$.
The product of the word $[a]= [a_1, \dots , a_n]$ by
$[b]= [b_1, \dots , b_m]$ is the word $[a_1, \dots , a_n, b_1, \dots, b_m]$.
We say a word is {\em not a power} if $[a] \neq [b]^k$ for every word
$[b]$ and $k > 1,$ and we say that two words $[a]$ and $[b]$ are $\eps-$close if they have the same length, and the correspondents letters are $\eps-$close.  With this notation, for a periodic symplectic linear system $(\Sigma, f, \mathcal{E}, A)$ if we consider the word
$[M_A (x)] = (A(f^{n-1}(x)), \dots , A(x)),$
where $n$ is period of $x\in \Sigma$, then
the matrix $M_A(x)$ is the product of the letters of the word $[M_A (x)]$.

\begin{definition}[Definition 1.6 of \cite{BDP}]
\label{d.transition} Given $\varepsilon > 0$, a periodic symplectic linear
system $(\Sigma, f, \mathcal{E},A)$ {\em admits
$\varepsilon$-transitions} if for every finite family of points
$x_1, \dots , x_n = x_1 \in \Sigma$ there is an orthonormal system
of coordinates of the linear bundle $\mathcal{E}$ so that
$(\Sigma, f, \mathcal{E},A)$ can now be considered as a system of
matrices $(\Sigma, f, A)$), and for any $(i, j) \in \{1,\dots
,n\}^2$ there exist $k(i, j) \in \mathbb{N}$ and a finite word
$[t^{i,j}]= (t_1^{i,j}, \dots , t_{k(i,j)}^{i,j})$ of symplectic matrices, satisfying the following properties:
\begin{enumerate}
\item For every $m \in \mathbb{N}$, $\imath = (i_1, \dots, i_m) \in \{1,\dots,n\}^m$,
and $\alpha = (\alpha_1,\dots ,\alpha_m) \in \mathbb{N}^m$
consider the word
\begin{align*}
[W(\imath, \alpha)] & = [t^{i_1,i_m}][M_A(x_{i_m})]^{\alpha_m}
[t^{i_m,i_{m-1}}][M_A(x_{i_{m-1}})]^{\alpha_{m-1}} \dots \\
& \quad \dots [t^{i_2,i_1}][M_A(x_{i_1})]^{\alpha_1},
\end{align*}
where the word
$w(\imath, \alpha) = ((x_{i_1}, \alpha_1) ,\dots,(x_{i_m}, \alpha_m))$
with letters in $M \times \mathbb{N}$ is not a power.
Then there is $x(\imath, \alpha) \in \Sigma$ such that
\begin{itemize}
\item The length of $[W(\imath, \alpha)]$ is the period of
$x(\imath,\alpha)$.
\item The word $[M_A (x(\imath, \alpha))]$ is $\varepsilon$-close to
$[W(\imath, \alpha)]$ and there is an $\varepsilon$-symplectic perturbation
$\tilde A$ of $A$ such that the word $[M_{\tilde{A}} (x(\imath,\alpha))]$
is $[W(\imath, \alpha)]$.
\end{itemize}

\item One can choose $x(\imath, \alpha)$ such that the distance between
the orbit of $x(\imath, \alpha)$ and any point $x_{i_k}$ is bounded by some
function of $\alpha_k$ which tends to zero as $\alpha_k$ goes to infinity.
\end{enumerate}
\end{definition}

Given $\imath ,\alpha$ as above, the word $[t^{i, j}]$ is an
$\varepsilon$-transition from $x_j$ to $x_i$.
We call {\em $\varepsilon$-transition matrices} the matrices $T_{i,j}$
which are the product of the letters composing $[t^{i, j}]$.
We say a periodic linear system system {\em admits transitions} if
for any $\varepsilon > 0$ it admits $\varepsilon$-transitions.


A nice property of periodic linear systems admitting transitions is the fact that we have densely defined on $\Sigma$ diagonalizable perturbations of $A$,  as we can see in Lemma 4.16 in \cite{BDP} (see Lemma 4.6 in \cite{HT} for a symplectic version). Moreover, the next result says that these diagonalizable perturbations has also ``nice''  extremal Lyapunov exponents.

%Here, we will need that these diagonalizable periodic points satisfies specific properties.

\begin{lemma} Let $(\Sigma, f, \mathcal{E},A)$ be a periodic symplectic linear system. For any $\eps>0$, and $x\in \Sigma$, there exists  $y\in \Sigma$ and  a symplectic perturbation $\tilde{A}$ of $A$ defined on the orbit of $y$, such that $M_{\tilde{A}}(y)$ is diagonalizable and moreover if $\lambda_x$ and $\lambda_y$ denote the largest absolute eigenvalue of $M_A(x)$ and $M_{\tilde{A}}(y)$, respectively, then
$$
\left|\frac{1}{\tau(x)}\log |\lambda_x|- \frac{1}{\tau(y)}\log |\lambda_y|\right|<\eps,
$$
where $\tau(x)$ and $\tau(y)$ denote the period of $x$ and $y$, respectively.
\label{exp}\end{lemma}

\begin{remark}
In the previous lemma  if we have an additional hypothesis that the linear bundle $\mathcal{E}$ has a dominated splitting for $f$, $\mathcal{E}=\mathcal{E}_1\oplus \ldots \oplus\mathcal{E}_n$, then the diagonalizable periodic point $y$ could be found such that $M_{\tilde{A}}(y)$ keeps invariant the sub bundles $\mathcal{E}_i$ and moreover
$$
\left|\frac{1}{\tau(x)}\log |\lambda_{x,i}|- \frac{1}{\tau(y)}\log |\lambda_{y,i}|\right|<\eps,
$$
where $\lambda_{x,i}$ and $\lambda_{y,i}$ denotes the largest Lyapunov exponent of $M_A(x)|\mathcal{E}_i$ and $M_{\tilde{A}}(y)|\mathcal{E}_i$, respectively, for every $1\leq i\leq n$.
\label{reexp}\end{remark}


It's important to observe that although  Lemma \ref{exp} does not appears in \cite{BDP} neither in \cite{HT}, it is in fact a consequence of their construction of the diago\-nalizable hyperbolic periodic point. The proof of this result in the symplectic scenario requires more sophisticated technics provided we need to keep the structure after the perturbation, as we can see in \cite{HT}. Here, we will use Lemma \ref{symplectic} and Lemma \ref{isotropic2} to exhibit a proof.

 %as we can see bellow.

\vspace{0,1cm}
{\it Proof: }

By a symplectic perturbation of $A$ along the hyperbolic periodic orbit of $x$, we can  assume that $M_A(x)$ has only simple eigenvalues, and moreover such that any pair of eigenvalues with the same modulus are complex and conjugated, and have rational argument.  Hence we consider the following partially hyperbolic splitting $\R^{2d}=E_1\oplus\ldots \oplus E_n$ given by eigenspaces of $M_A(x)$ (suppose $\mathcal{E}$ is  a 2d-dimensional vector bundle), which implies $1\leq dim\ E_i\leq 2$,  and as a consequence of the symplectic structure we have  for every distinct $1\leq i, j\leq n$:
\begin{itemize}
\item $E_i$ is an isotropic subspace,

\item $E_i\oplus E_j$ is a symplectic subspace if $i+j=n+1$, and moreover

\item $(E_i\oplus E_j)^{\omega}=\bigoplus_{k\neq i,j} E_k$, if $i+j=n+1$.
\end{itemize}

Note there exists a positive integer $k$ such that $(M_A(x))^k$ has only real eigenvalues.  However, $(M_A(x))^k|E_i$ has a real  eigenvalue with multiplicity two if $dim \ E_i=2$.
%if for some $1\leq i\leq n/2$, $dim \ E_i=2$ then $(M_A(x))^k|E_i$ has a real  eigenvalue with multiplicity two.
 In this situation, we can use Lemma \ref{symplectic} to find a symplectic linear map $H_i$ arbitrary close to identity, such that $H_i (M_A(x))^k|E_i\oplus E_{n+1-i}$ has four non-complex eigenvalues. Moreover,  considering the word
$$
[M_1]=(H_iA(f^{\tau(x)-1}(x)), A(f^{\tau(x)-2}),\ldots, A(x))[M_A(x)]^{k-1},
$$
and denoting by $M_1$ the matrix obtained as the product of  letters of $[M_1]$, we have $M_1=H_i (M_A(x))^k$.  Note that the word $[M_1]$ is so close to $[M_A(x)]^k$  as $H_i$ is close to $Id$, and for any $\eps>0$ we can choose $H_i$ such that %When we say that two words are near if they have the same length and the correspondents letters are near.
if we denote by $\xi_i$ the complex eigenvalue of $M_A(x)|E_i$, any eigenvalue  $\lambda$ of
$M_1|E_i$ is such that
\begin{equation}
\left|\frac{1}{\tau(x)}\log |\xi_i| -  \frac{1}{(\tau(x))^k}\log |\lambda|\right|<\frac{\eps}{2}.
\label{eigen}\end{equation}



Hence, given $\eps>0$, after finitely many symplectic perturbations using maps close to Identity, if necessary, restrict to each $E_i\oplus E_{n+1-i}$, we can find a word $[M_1$],  $\eps/2-$close to $[M_A(x)]^k$, such that $M_1$ has only real eigenvalues with multiplicity one, and equation (\ref{eigen}) is true for every $1\leq i\leq n$. %we denote by $\lambda_i$ the complex eigenvalue of $M_A(x))|E_i$, any eigenvalue  $\lambda$ of
%$M_1|E_i$ is such that
%\begin{equation}
%\left|\frac{1}{\tau(x)}\log |\lambda_i| -  \frac{1}{\tau(x)^k}\log |\lambda|\right|<\eps.
%\label{eigen}\end{equation}
%depending on $H_i$. %We also note that the words $[M_1]$ and $[M_A(x)]^k$ are near. When we say that two words are near if they have the same length and the correspondents letters are near.
%Hence, by above we can suppose $k$ such that $M_1$ has only real eigenvalues with multiplicity one.
To not complicate the notation we will continue denoting the eigenspaces of $M_1$ by $E_i$, which now are all one-dimensional subspaces.% and thus $n=2d$ in this case. %Also,  given $\eps>0$, we can assume $[M_1$] $\eps/2-$close to $[M_A(x)]^k$.

Provided the linear system has transitions, there exists a word $[t]=(t_1,\ldots, t_{r})$  of symplectic matrices which is a $\eps/2-$transition from $x$ to itself. We denote by $T$ the symplectic matrix obtained by the product of letters of $[t]$.  After a symplectic perturbation of the matrix $t_{1}$, if necessary, we can suppose that
$$
T(E_n)\cap E_1\oplus\ldots\oplus E_{n-1}=\{0\}\text{ and } T^{-1}(E_1)\cap E_2\oplus\ldots\oplus E_{n}=\{0\}.
$$
Thus by the choice of the partially hyperbolic splitting on $\R^{2d}$, $(M_1)^jT(E_n)$ converges to $E_n$ when $j$ goes to infinity. Hence, for $j_n$ large enough, we can use Lemma \ref{isotropic2} to find a symplectic linear map $L_n$, $\eps/2-$close to $Id$, such that $L_n(M_1)^{j_n}T(E_n)=E_n$ and $L_n|(E_1\oplus \ldots \oplus E_{n-1})=Id$.

Analogously, provided  $(M_1)^{-j}T^{-1}(E_1)$ converges to $E_1$ when $j$ goes to infinity we can choose $j_1>0$ and find a symplectic $\eps/2-$perturbation $L_1$ of $Id$, such that $L_1(E_1)=(M_1)^{-j_1}T^{-1}(E_1)$ and $L_1|(E_2\oplus\ldots\oplus E_n)=Id$. Therefore, defining $M_2=L_n(M_1)^{j_n}T(M_1)^{j_1}L_1$ we have that
$M_2(E_1)=E_1$ and $M_2(E_n)=E_n$. Once again, as a consequence of the symplectic structure, $M_2$ also satisfies  the following:
$$M_2(E_2\oplus\ldots\oplus E_{n-1})=E_2\oplus\ldots\oplus E_{n-1}.$$
In fact, if this is not true, then there exist $v\in E_2\oplus\ldots\oplus E_{n-1}$ and $u\in E_1\oplus E_{n}$ such that $\omega(M_2(v), u)\neq 0$. On the other hand, by construction, $M_2^{-1}(u)\in E_1\oplus E_n$ and then $\omega(v, M_2^{-1}(u))=0$, which gives  a contradiction since $M_2$ is symplectic.




%On the other hand, by construction of the symplectic map $M_2$, $M_2^{-1}(u)\in E_1\oplus E_n$, and then $\omega(v, M_2^{-1}(u))=0$, which gives  a contradiction since $\omega(M_2(v), u)=\omega(v, M_2^{-1}(u))$.

Proceeding similarly as in the construction of the word $[M_1]$,  $M_2$ could be seen as a matrix induced by a word $[M_2]$ $\eps/2-$near $[M_1]^{j_n}[t][M_1]^{j_1}$.


Therefore, repeating finitely many times the above process we can find a positive integer $j$ and a word $[\tilde{M}]$ of symplectic linear matrices $\eps/2-$close to $[M_1]^{j}[t][M_1]^{j}$, such that  $\tilde{M}(E_i)=E_i$ for every $1\leq i\leq n$.

Now, since for every $1\leq i\leq n$, $E_i$ are one-dimensional subspaces, we can choose $l>0$ large enough and define $M=(M_1)^l \tilde{M}$, such that if $\mu_i$ and $\lambda_i$ denote the eigenvalues of $M|E_i$ and $M_1|E_i$, respectively, thus we have
\begin{equation}
\left|\frac{1}{\tau}\log |\mu_i |-\frac{1}{(\tau(x))^k}\log |\lambda_i|\right|<\frac{\eps}{2},
\label{eigen2}\end{equation}
where $\tau$ is the length of the word $[M_1]^{j+l}[t][M_1]^{j}$.


Finally, by choice of $[t]$ there exists $y\in \Sigma$ such that $[M_A(y)]$ is $\eps/2-$close to $[M_A(x)]^{k(l+j)}[t][M_A(x)]^{j}$, which by construction is $\eps/2-$close to $[M]=[M_1]^l [\tilde{M}]$.  Hence, by equations (\ref{eigen}) and (\ref{eigen2}) if we denote by $\mu_M$ the largest absolute eigenvalue of $M$, we have
$$
\left|\frac{1}{\tau(y)}\log |\mu_M |-\frac{1}{\tau(x)}\log |\lambda_x|\right|<\eps.
$$

Therefore, there exists a symplectic $\eps-$perturbation $\tilde{A}$ of $A$ defined on the orbit of $y$, such that $M_{\tilde{A}}(y)=M$, which implies $y$ is diagonalizable and has the largest Lyapunov exponent close to the $x$, provided $\lambda_y=\lambda_M$.

$\hfill\square$



% the letters of the word $[M]$ is a $\eps-$perturbation of $[M_A(y)]$, diagonalizable and satisfies the hypothesis!

%we have that $M_2(E_i)=E_i$ for every $1\leq i\leq n$. Now, if $dim \ E_i=2$ for some $1\leq i\leq n/2$, this implies that $E_i$ is an eigenspace to $M_2$ related to a complex eigenvalue $\lambda_i$ which we can also assume  having rational argument for $M_2$. Hence, we can choose $k_i$, and find a small perturbation $H_i$ of $Id$, using Lemma \ref{symplectic}, such that $H_i (M_A(x))^{k_i}  M_2(E_i\oplus E_{n+1-1})$ has only real eigenvalues and  $ H_i (M_A(x))^{k_i}  M_2$ keeps invariant the other directions $E_j$, $j\neq i,n+1-1$. Moreover, note that if $\lambda$ is an eigenvalue of  $H_i (M_A(x))^{k_i}  M_2(E_i\oplus E_{n+1-1})$, then
%$$
%\frac{1}{k_i+1}
%





The following lemma gives an example of linear systems having  symplectic transitions. It is a symplectic version of [Lemma 1.9 in \cite{BDP}].


\begin{lemma}[Lemma 4.5 in \cite{HT}]  Let $f$ be a symplectic diffeomorphism and let $p$ be a hyperbolic periodic point of $f$. The derivative $Df$ induces a continuous periodic symplectic linear system on the set $\Sigma$ formed by hyperbolic periodic points homoclinically related  to $p$. %$H(p,f)$. of index k and homoclinically related to P .
\label{ex}\end{lemma}




\section{Bounds for entropy}
\label{s.bounds-entropy}

%In this section we will prove Theorem~\ref{t.entropy}.
%\textcolor[rgb]{0.98,0.00,0.00}{Write a sketch of the proof}
To prove Theorem~\ref{t.entropy} we need some
results.
%First, we recall a straightforward consequence of arguments present
%in \cite[Section ?????]{HT06}.

%\begin{lemma}
%\label{l.identity}
%Let $f\in \mathcal{PH}_{\omega}^1(m)$. For any small neighborhood
%$\mathcal{U}\subset \ph^1(m)$ of $f$ there exists a diffeomorphism
%$g\in U$ and a periodic point $p$ of $g$ such that
%$Df^{\tau(p,g)}|E^c=Id$.
%\end{lemma}

%An alternative proof can also be found in \cite{ABW09}.

First, we will use periodic symplectic linear systems with transitions to prove that the estimate on the largest central Lyapunov
exponent can take account just diagonalizable hyperbolic
periodic points.


\begin{lemma}
\label{l.diagonalizable}
There exists a residual subset
$\mathcal{R}\subset \dw$ such that if $f\in \mathcal{R}\cap \ph^1(m) $ then
$$
S_m(f)=\sup\left\{\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c(p));\,\;
p\in \per_h(f) \text{ is diagonalizable}\right\}.
$$
\end{lemma}

\begin{proof}

Let us define
$$
\tilde{S}_m(f) = \sup\left\{\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c(p)),\;
p\in \per_h(f) \text{ is diagonalizable}\right\}.
$$

As $S_{m}(\cdot)$, the funcion $\tilde{S}_m(\cdot)$ is also a
lower semicontinuous function, provided diagonalizable hyperbolic periodic points are robust.  Hence there exists a residual
subset $\mathcal{R}\subset \dw$ where both maps are continuous.
Taking $f\in \mathcal{R}$, for any $\eps>0$ there exists a small neighborhood $\mathcal{U}$ of $f$ such that
$$
\tilde{S}_m(g)<\tilde{S}_m(f)+\frac{\eps}{3}, \text{ for every }
g\in \mathcal{U}.
$$


By definition of $S_m(f)$, there exists  a hyperbolic
periodic point $p$ of $f$ such that
$$
S_m(f)<
\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c(p))+\frac{\eps}{3}.
$$

Now, by Lemma \ref{ex}, we can consider the following periodic symplectic linear system with transition $(\Sigma, f, TM, Df)$, where $\Sigma$ is the set of hyperbolic periodic points of $f$ homoclinicaly related to $p$, which one we can suppose non trivial since $f$ belongs to a residual subset (see \cite{X}).

Let $\delta>0$ be the small constant given by Franks Lemma for $f$ and the neighborhood $\mathcal{U}$.
Provided $TM=E^s\oplus E^c\oplus E^u$, by Lemma \ref{exp} and Remark \ref{reexp} there exists $\tilde{p}\in \Sigma$ and a symplectic $\delta-$perturbation $\tilde{A}$ of
$Df$ along the orbit of $\tilde{p}$ such that $\tilde{p}$ is diagonalizable and moreover
$$
 \frac{1}{\tau(p,f)}\sigma(Df^{\tau(p,f)}|E^c(p))< \frac{1}{\tau(\tilde{p})}\sigma(M_{\tilde{A}}(\tilde{p})|E^c)+\frac{\eps}{3}.
$$
Hence, %provided $E^c$ is a symplectic vector bundle, by Lemma \ref{symplectic}  and
by Franks Lemma we can find a symplectic diffeomorphism $g\in \mathcal{U}$ %close to $f$,
such that $\tilde{p}$ is a diagonalizable hyperbolic periodic point of $g$ satisfying
$$
 \frac{1}{\tau(p,f)}\sigma(Df^{\tau(p,f)}|E^c(p))< \frac{1}{\tau(\tilde{p},g)}\sigma(Dg^{\tau(\tilde{p},g)}|E^c(\tilde{p}))+\frac{\eps}{3}.
$$
Using all above inequalities we have the following:
\begin{align*}
S_m(f)&<
\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c(p))+\frac{\eps}{3}
\\
&<\frac{1}{\tau(\tilde{p},g)}\log\sigma(Dg^{\tau(\tilde{p},g)}|E^c(\tilde{p}))+\frac{2\eps}{3}
\\
&\leq \tilde{S}_m(g)+\frac{2\eps}{3}
\\
&< \tilde{S}_m(f)+\eps.
\end{align*}

Therefore, since $\eps>0$ is arbitrary, we have $S_m(f)\leq
\tilde{S}_m(f)$. Which finishes the proof provided
$\tilde{S}_m(f)\leq S_m(f)$ by definition.

\end{proof}


The next proposition is the main technical result in this paper.

\begin{proposition}
\label{p.estimate} Let $0<m\leq d$ and $f\in int(\ph^1(m))$.  If $p$ is
a diagonali\-zable hyperbolic periodic point of $f$,  then for any
neighborhood $\mathcal{U}$  of $f$ and any large positive integer
$n$, there exists a diffeomorphism $g\in \mathcal{U}$, such that $p$
still is a diagonalizable hyperbolic periodic point of $g$,  and
moreover there exists a hyperbolic basic set $\Lambda(p,g)\subset
H(p,g)$ such that
$$
h_{top}(g|\Lambda(p,g))>\frac{1}{\tau(p,g)}
\log\sigma(Df^{\tau(p,g)}|E^c)-\frac{1}{n}.
$$
\end{proposition}

We postpone the proof of this proposition to Section
\ref{section5}. Now, let us prove
Theorem~\ref{t.entropy}.


\vspace{0,2cm}

\begin{proof}[Proof of Theorem~\ref{t.entropy}]
For any positive integer $n>0$, and every $0<m\leq d$ we define
$\mathcal{B}_{m,n}\subset \ph^1(m)$ the subset of diffeomorphisms
$g$ such that
$$
h_{top}(g)>S_m(g)-\frac{1}{4n}.
$$

Now, since $S_m(\cdot)$ is a lower semicontinuous map defined in the open set $\ph^1(m)$,
there is a residual subset  $\mathcal{R}_m\subset \ph^1(m)$ where
$S_m(\cdot)$ is continuous. %{\color{red} tomar cuidado que para m igual a n, o conjunto nao e aberto}
Let us fix now some $0<m\leq d$, and let $f\in \mathcal{R}_m$.
Given $n>0$, we can choose a small neighborhood $\mathcal{U}$ of $f$ such that
\begin{equation}
S_m(f)>S_m(\xi)-\frac{1}{4n}, \text{ for every } \xi\in \mathcal{U}.
\label{t.eq1}\end{equation}

By Lemma \ref{l.diagonalizable}, there exists a
diagonalizable hyperbolic periodic point $p$ of $f$ such that
\begin{equation}
\frac{1}{\tau(p,f)}\log\sigma(Df^{\tau(p,f)}|E^c)>S_m(f)-\frac{1}{4n}.
\label{t.eq2}\end{equation}

% every diffeomorphism $g\in \mathcal{U}$ is such that $s_m(f)>s_m(g)-\frac{1}{k}$,
%since $f$ is a continuity point of $S_m(\cdot)$.

Now, by Proposition \ref{p.estimate} there exists a diffeomorphism
$g\in \mathcal{U}$ and a hyperbolic basic set $\Lambda(p,g)\subset H(p,g)$
such that
\begin{equation}
h_{top}(g|\Lambda(p,g))>\frac{1}{\tau(p,g)} \log\sigma(Dg^{\tau(p,g)}|E^c)-\frac{1}{4n}.
\label{t.eq3}\end{equation}

Therefore, if $\tilde{g}\in \mathcal{U}$ is a diffeomorphism $C^1$-close to $g$ then
there is a continuation of the hyperbolic basic set $\Lambda(p,g)$ which
we denote by $\Lambda(p(\tilde{g}),\tilde{g})$, where $p(\tilde{g})$ is
a continuation of $p$, and thus using equation (\ref{t.eq3}),  continuity of the Lyapounov exponents, and finally equations (\ref{t.eq2}) and (\ref{t.eq1},), we have
\begin{align*}
h_{top}(\tilde{g}) \geq h_{top}(\tilde{g}|\Lambda(p(\tilde{g}), \tilde{g}))
&= h_{top}(g|\Lambda(p,g))
\\
&\geq \frac{1}{\tau(p,g)} \log\sigma(Dg^{\tau(p,g)}|E^c)-\frac{1}{4n}
\\
&\geq \frac{1}{\tau(p,f)} \log\sigma(Df^{\tau(p,f)}|E^c)-\frac{1}{2n}
\\
&\geq S_m(f)-\frac{3}{4n}
\\
&\geq S_m(\tilde{g})-\frac{1}{n}.
\end{align*}
Hence, every $\tilde{g}$ $C^1-$close to $g$ belongs to $\mathcal{B}_{m,n}$.
Which implies that $\mathcal{B}_{m,n}$ contains an open and dense subset in
$\ph^1(m)$ since $\mathcal{R}_m$ is a dense subset in $\ph^1(m)$.
%When $m=n$, we remark since $\ph^1(n)$ is not an open subset of $\dw$, that
%$\mathcal{B}_{n,k}$ is  an open and dense subset in
%$(\overline{\mathcal{PH}})^c$.

Thus, defining $$\mathcal{B}_n=\bigcup_{m=1}^d \mathcal{B}_{m,n}\cup
\mathcal{A},$$ we have that $\mathcal{B}_n$ is an open and dense subset
of $\dw$.
Therefore,
$$
\mathcal{R}=\bigcap_{n\in \N} \mathcal{B}_n
$$
is a residual subset in $\dw$, and satisfies the properties required.
In fact, if $f\in \mathcal{R}$ and is non Anosov, then there exists
$0<m\leq d$ such that $f\in \mathcal{B}_{m,n}$ for every $n>0$.
Therefore,
$$
h_{top}(f)\geq S_m(f).
$$

\end{proof}


\section{Nice properties of the strong stable and unstable manifolds and a proof of Theorem \ref{t.trichotomy}}


%Now, we recall a straightforward consequence of arguments present
%in \cite[Section 5]{HT06}.

The first result of this section is related to results in \cite{BDP}, more precisely we obtain the existence of a nice non hyperbolic periodic point for partially hyperbolic diffeomorphisms in $\ph^1(m)$. After, we will use this result to prove Theorem \ref{t.trichotomy} and also to find hyperbolic periodic points having strong homoclinic intersections with small angles. % between the tangent spaces of the strong manifolds.

%in the lack of the existence of a special kind of dominated splitting.% for an specific sub bundle.

\begin{proposition}
\label{l.identity}
Let $f\in \mathcal{PH}_{\omega}^1(m)$. For any small neighborhood
$\mathcal{U}\subset \ph^1(m)$ of $f$ and $\eps>0$ there exists a diffeomorphism
$g\in \mathcal{U}$ and a periodic point $p$ of $g$ such that
$Df^{\tau(p,g)}(p)|E_{m}^c=Id$, and moreover the orbit of $p$ is $\eps-$dense in $M$.
\label{identity}\end{proposition}

We would like to remark that part of this proposition appears in \cite{ABW}, Theorem 3.5.  However, this proposition is in fact a directly consequence of the following result of Horita and Tahzibi, together with Proposition \ref{abc}. %Hence, we will put here this shorter proof.



%To prove this Proposition, we will use the follow technical result of the second author of this paper and A. Tahzibi:

%\begin{proposition}[Proposition 5.3 in \cite{HT06}]
%For any $\eps>0$,  $N\in \N$ and $K>0$ there is $l>0$ such that any continuous periodic $2N$-dimensional linear system $(\Sigma,f, E, A)$ bounded by $K$ (i.e. $\|A\|<K$) and having symplectic transitions satisfies the following,
%\begin{itemize}
%\item  either $A$ admits an $l$-dominated splitting,

%\item or there are a symplectic $\eps$-perturbation $\tilde{A}$ of $A$ and a point $x\in \Sigma$ such that $M_{\tilde{A}}(x)$ is the identity matrix.
%\end{itemize}
%\label{HT}\end{proposition}

\begin{proposition}[Proposition 5.3 in \cite{HT}]
For any $\eps>0$,  and $K>0$ there is $l>0$ such that any symplectic periodic $2d$-dimensional linear system $(\Sigma,f, E, A)$ bounded by $K$ (i.e. $\|A\|<K$) and having symplectic transitions satisfies the following,
\begin{itemize}
\item  either $A$ admits an $l$-dominated splitting,

\item or there are a symplectic $\eps$-perturbation $\tilde{A}$ of $A$ and a point $x\in \Sigma$ such that $M_{\tilde{A}}(x)$ is the identity matrix.
\end{itemize}
\label{HT}\end{proposition}

\begin{remark}
We would like to note that the periodic point $x$ found in the second item of the previous Proposition could be found $\eps-$dense in $\Sigma$ for any $\eps>0$.  %{\color{red} Moreover, arguing as in the proof of Lemma \ref{exp}, we can also assume that $x$ is such that $Df^{\tau(x)}|E^s(x)$ has simple eigenvalues. VERIFICAR ISTO, SE PRECISA ARGUMENTAR}
\label{rmkHT}\end{remark}

%\begin{remark}
%We would like to note that the periodic point $x$ found in the second item of the previous Proposition could be assumed $\eps-$dense in $\Sigma$ for any $\eps>0$.
%\label{rmkHT}\end{remark}


%An alternative proof can also be found in \cite{ABW09}.

{\it Proof of Proposition \ref{identity}:}

Let $TM=E^s\oplus E^c\oplus E^u$ be the partially hyperbolic splitting given by $f$ over $M$. %Provided $f$ is a symplectic diffeomorphism %note that %we have that the invariant sub bundle
%$E^c$ is a symplectic vector bundle over $M$.
After a perturbation, we can suppose by Proposition \ref{abc} that $M=H(p,f)$, %which implies in particular that $f$ is transitive.
and thus we denote by $\Sigma$ the set formed by hyperbolic periodic points homoclinically related to $p$, considering then the following periodic symplectic linear system with symplectic transitions $(\Sigma,f, E^c, Df)$.


Provided $f\in \mathcal{PH}_{\omega}(m)$, the vector bundle $E^c$ admits no dominated splitting for $Df$. In fact, since $E^c$ is a symplectic vector bundle (Remark \ref{decomp.}), if  admits a dominated splitting, then admits a partially hyperbolic splitting for $Df$ (see \cite{BV}, section 4, for a proof of this), which contradicts the fact of $f$ has an unbreakable $2m-$dimensional center bundle. %$f\in \mathcal{PH}_{\omega}(m)$, .
 Therefore, by Proposition  \ref{HT} and Remark \ref{rmkHT} there exists $x\in \Sigma$, $\eps-$dense in $M$ since $\Sigma$ is dense in $M$, and a symplectic perturbation $\tilde{A}$ of  $Df|E^c$ along the orbit of $x$, such that $M_{\tilde{A}}(x)=Id$. %If we define $A_i=\tilde{A}_i\circ Df^{-1}|E^c(f^{i}(x))$, then $A_i$ is so close to identity $Id|E^c$ as $\tilde{A}_i$ is close to $Df|E^c(f^{i-1}(x))$.

Hence, taking $\delta>0$  given by the Franks Lemma for $f$ and the neighborhood $\mathcal{U}$, we can use Lemma \ref{symplectic} to find symplectic linear maps $A_i$, $\delta-$close to $Df(f^i(x))$, $0\leq i<\tau(x)$, such that $A_i|E^c=\tilde{A}(f^i(x))$ and $A_i|E^{ss}\oplus E^{uu}=Df$. Thus, by Franks Lemma there exists $g\in \mathcal{U}$ such that $x$ still is a periodic point of $g$,  and $Dg(g^{i}(x))=A_i$, for any $0\leq i <\tau(x)$, which implies %$Dg|E^c(g^{i}(x))=\tilde{A}_i$, and then
$$
Dg^{\tau(x)}|E^c(x)=M_{\tilde{A}}(x)=Id|E^c(x).
$$


%Provided $f\in \mathcal{PH}_{\omega}(m)$, the vector bundle $E^c$ admits no dominated splitting for $Df$. In fact, since $E^c$ is a symplectic vector bundle, if  admits a dominated splitting, then in fact admits a partially hyperbolic splitting for $Df$ (see BOCHI-VIANA), which contradicts the fact of $f\in \mathcal{PH}_{\omega}(m)$. Therefore, by Proposition  \ref{HT} and Remark \ref{rmkHT} there exists $x\in \Sigma$, $\eps-$dense in $M$ since $\Sigma$ is dense in $M$, and symplectic maps $\tilde{A}_i$ is arbitrary close to  $Df|E^c(f^{i-1}(x))$, for any $0\leq i\leq \tau(f,x)$, such that $M_{\tilde{A}}(x)=Id$. If we define $A_i=\tilde{A}_i\circ Df^{-1}|E^c(f^{i}(x))$, then $A_i$ is so close to identity $Id|E^c$ as $\tilde{A}_i$ is close to $Df|E^c(f^{i-1}(x))$.

%Let $\delta>0$ be given by the Franks Lemma considering $f$ and the neighborhood $\mathcal{U}$. Now, by Lemma \ref{symplectic} there is $\rho>0$ such that if $A_i$ is $\rho-$close to $Id|E^c(x)$, then there exists a symplectic linear map $B_i:T_{f^{i}(x)}M\rightarrow T_{f^{i}(x)}M$, $\delta-$close to $Id$,  such that $B_i|E^c=A_i$. Finally, taking $C_i=B_i\circ Df(f^{i}(x))$, which is $\delta-$close to $Df(f^{i}(x))$, we can use Franks Lemma to find a diffeomorphism $g\in \mathcal{U}$ such that $g^i(x)=f^i(x)$, for any $1\leq x\leq <\tau(x)$, and $Dg(g^{i}(x))=C_i$, which implies $Dg|E^c(g^{i}(x))=\tilde{A}_i$, and then
%$$
%Dg^{\tau(x)}|E^c(x)=M_{\tilde{A}}(x)=Id|E^c(x).
%$$


%Therefore,  by Lemma BLA (ESTE LEMMA TEM QUE TER, QUE E O Q SE O CARA E SIMPLETICO ENTAO PODEMOS FAZER UMA PERTURBACAO PERTURBANDO APENAS NELE) we can perform a small perturbation of $f$ in order to find a diffeomorphism $g\in \mathcal{U}$ such that $x\in Per(g)$, $Dg^{\tau}|E^c(x)=Id$ and moreover the orbit of $x$ is $\eps-$dense in $M$.

$\hfill\square$




%Let $TM=E^s\oplus E^c\oplus E^u$ be the partially hyperbolic splitting given by $f$ over $M$. Provided $f$ is a symplectic diffeomorphism we have that the invariant sub bundle $E^c$ is a symplectic vector bundle over $M$. Hence, we can consider the following linear symplectic system $(M,f, E^c, Df)$  (a linear symplectic cocycle over $f$). Moreover, after a perturbation if necessary, we can suppose by BONATTI-CROVISIER that $M=H(p,f)$, which implies in particular that $f$ is transitive. Hence, by LEMA BLA IN HT, this linear symplectic system also has transition.










After Proposition \ref{identity} we be able to give a proof of Theorem \ref{t.trichotomy}.


\vspace{0,2cm}
{\it Proof of Theorem \ref{t.trichotomy}:}

%We suppose $dim \ M=2d$, and
%We remember first that
By Dolgopyat and Wilkinson \cite{DW}, there exists  $\displaystyle\tilde{\mathcal{PH}}_{\omega}(m)\subset \mathcal{PH}_{\omega}(m)$ an open and dense subset of robust transitive partially hyperbolic symplectic diffeomorphisms, for any $1\leq m<d$.

%To simplify the notation, we replace the set $\mathcal{PH}_{\omega}(m)$ by the one formed by robust transitive partially hyperbolic diffeomorphisms. % will denote the set formed by given by DW.

Now, denoting by $\mathcal{A}$ the set formed by Anosov symplectic diffemorphisms we define
$$\tilde{\mathcal{PH}}_{\omega}(d)=\dw-cl\left(\mathcal{A}\bigcup_{1\leq m<d}\tilde{ \mathcal{PH}}_{\omega}(m)\right).$$ Note that $\tilde{\mathcal{PH}}_{\omega}(d)$ is also an open and dense subset of $\mathcal{PH}_{\omega}(d)$. %  and as before we replace $\mathcal{PH}_{\omega}(d)$ by $\mathcal{D}$. % and thus as an open set in $\dw$.

%  $m=n$, we take $f$ in the open set $\mathcal{D}$

Hence, for any $1\leq m\leq d $ and $n\in \N$ we denote by $\mathcal{B}_{n,m}\subset \tilde{\mathcal{PH}}_{\omega}(m)$ the subset of diffeomorphisms  $g$ having a $m-$elliptic periodic point, $1/n-$dense in $M$. Since $m-$elliptic periodic points are robust %in $\mathcal{PH}_{\omega}(m)$
we have that $\mathcal{B}_{n,m}$ is an open set.

Let $f\in \tilde{\mathcal{PH}}_{\omega}(m)$ and $n\in \N$.  %and Let we fix an arbitrary $0\leq m\leq d$ and $n\in \N$, and let $f\in \mathcal{PH}_{\omega}(m)$. %( if $m=n$, we take $f$ in the open set $\mathcal{D}$).
By Proposition \ref{identity}, there exists a diffeomorphism $g$ $C^1-$close to $f$, having a periodic point $p$ $1/n-$dense in $M$, such that $Dg^{\tau(p,g)}|E^c(p)=Id$. Now, let $\{e_1, \ldots, e_{2m}\}$ be a symplectic basis in $E^c(p)$. Chan\-ging   the order of the elements inside this basis we can choose another basis for $E^c(p)$:  $B=\{e_1, \ e_{m+1},$ $ \ldots, e_{i},\ e_{m+i}, \ldots, e_{m},\ e_{2m}\} $, and thus for any small positive values $\alpha, \beta>0$ we can define a symplectic linear map in $T_pM$ induced by the following matrix with respect to basis $B$:
$$
A=\left[\begin{array}{ccccc}
\tilde{A}& 0 & 0 & \ldots & 0
\\ 0 & \tilde{A}& 0&\ldots &0
\\
\vdots &&&&
\\0&\ldots&&& \tilde{A}
\end{array}\right],
\text{ where }
\tilde{A}=\left[\begin{array}{cc}
\alpha& \beta \\ -\beta & \alpha
\end{array}\right].
$$
Note this symplectic linear map restrict to the symplectic plane generated by $\{e_i,\ e_{m+i}\}$ is  a small rotation, for any $1\leq i\leq m$.

Thus, taking $\alpha$ and $\beta$ small enough, we can suppose $A$ arbitrary close to $Id$. Hence, by Lemma \ref{symplectic} and Remark \ref{decomp.} we can find a symplectic linear map $B:T_pM\rightarrow T_pM$ arbitrary close to $Id$, such that $B|E^c=A$ and $B|E^{ss}\oplus E^{uu}=Id$. Taking $C=B \circ Dg(g^{\tau(p)-1})$ which is a symplectic linear map close to $Dg(g^{\tau(p)-1})$, we can use Franks Lemma to perform a local perturbation and find a difeomorphism $h$ $C^1-$close to $g$, such that $p$  still is a periodic point of $h$ and $Dh^{\tau(p,h)}(p)=B\circ Dg^{\tau(p,g)}(p)$. Then $Dh^{\tau(p,h)}|E^c(p)=A$, which implies $p$ is a $m-$elliptic periodic point. Since this perturbation keeps the orbit of $p$, this $m-$elliptic periodic point still is $1/n-$dense in $M$, which implies $h\in \mathcal{B}_{n,m}$. Note, when $m=d$, $p$ is a totally elliptic periodic point.

Therefore the sets  $\mathcal{B}_{n,m}$ are open and dense inside $\tilde{\mathcal{PH}}_{\omega}(m)$, which implies
$$
\mathcal{R}=\mathcal{A}\cup \left(\bigcup_{1\leq m\leq d} \bigcap_{n\in \N} \mathcal{B}_{n,m}\right)
$$
is a residual subset of $\dw$.

To finish, we only observe that diffeomorphisms in $\mathcal{R}$ satisfies one, and only one, of the three items in the Theorem \ref{t.trichotomy}.

 $\hfill\square$

To continue, %Before we state the next lemma,
we need to define {\it angle between vector
subspaces}.
Given a Riemaniann manifold $M$, and vectors $v,w\in T_qM$, we define the  angle between $v$ and $w$ by
$$
\ang(v,w)=\left|\tan\left[\arccos\left(\frac{<v,w>}{\|v\|\|w\|}\right)\right]\right|.
$$
%be the angle between $v$ and $w$.
If $E$ is a vector subspace of $T_qM$, the angle between a vector $u\in T_qM$ and
$E$ is defined by
$$
\ang(v,E)=\min_{w\in E,\, |w|=1}\, \ang(v,w).
$$
Finally if $E,F\subset T_qM$ are subspaces we define
$$
\ang(E,F) =\min_{w\in E,\, |w|=1} \, \ang(w,F).
$$
It's worth to point out the previous definition is equivalent
to the following more frequent definition of angle between two
transversal vector subspaces\;: given a decomposition $T_qM=E\oplus F$,
let $L: E^{\perp}\rightarrow E$ be the linear map such that
$F=\{w+Lw; \; w\in E^{\perp}\}$ then the angle between $E$ and $F$
is defined as $\|L\|^{-1}$.







%Before we state the next lemma, we need to define {\it angle between vector
%subspaces}.
%Given vectors $v,w\in\R^{2d}$, we define the  angle between $v$ and $w$ by
%$$
%\ang(v,w)=\left|\tan\left[\arccos\left(\frac{<v,w>}{\|v\|\|w\|}\right)\right]\right|.
%$$
%%be the angle between $v$ and $w$.
%If $E$ is a vector subspace of $\R^{2d}$, the angle between a vector $u$ and
%$E$ is defined by
%$$
%\ang(v,E)=\min_{w\in E,\, |w|=1}\, \ang(v,w).
%$$
%Finally if $E,F\subset\R^{2d}$ are subspace we define
%$$
%\ang(E,F) =\min_{w\in E,\, |w|=1} \, \ang(w,F).
%$$
%It's worth to point out the previous definition is equivalent
%to the following more frequent definition of angle between two
%transversal vector subspaces\;: given a decomposition $\R^{2d}=E\oplus F$,
%let $L: E^{\perp}\rightarrow E$ be the linear map such that
%$F=\{w+Lw; \; w\in E^{\perp}\}$ then the angle between $E$ and $F$
%is defined as $\|L\|^{-1}$.












%Before we state the next lemma, we need to define angle between subspaces. If $E$ and $F$ are subspaces of $\R^{2n}$ with same dimension, then we define the angle between $E$ and $F$ by  $ang(E,F)=\|L\|$, {\color{red} ESTA DEFINI��O N�O EST� BOA, N�O FUNCIONA. USAR A DEFINI��O DE �NGULO COMO M�NIMO DE VETORES. CONTAS TEM NA TESE} where $L: E\rightarrow E^{\bot}$ is such that $F=\{w+Lw; \, w\in E\}$. If $E$ and $F$ are subspaces such that $\R^{2n}=E\oplus F$, then we define the angle between $E$ and $F$ by  $ang(E,F)=\|L\|^{-1}$, where $L: E^{\bot}\rightarrow E$ is such that $F=\{w+Lw; \, w\in E^{\bot}\}$.


%TEM QUE DEFINIR O �NGULO MAIS GERAL, CONSIDERANDO ESPA�OS COM DIMENS�ES MAIORES.


%{\color{red} seguinte, aqui n�o precisa pedir q o ponto seja diagonaliz�vel. Consegue um ponto peri�dico que tem estas variedades k-dimensionais com interse��o pequena. Pronto, agora qq ponto diagonaliz�vel tem este angulo pequeno tamb�m. Ent�o no lema depois deste, faz o lemma para dois pontos q tem mesmas dimens�es de variedades k-fortes, a menos de perturba��o podemos ver estes angulos pequenos. Pronto, fica tudo mais simples! :)}
\begin{lemma}
\label{lemma1}
Let $1\leq m\leq d$ and $f\in int(\ph^1(m))$. %and $p$ be a diagonalizable hyperbolic periodic point of $f$.
For any $\eps>0$ and any neighborhood $\mathcal{U}$ of $f$,
there exists a diffeomorphism $g\in \mathcal{U}$ with a
hyperbolic periodic point $p$ having $d-m+1-$strong stable and unstable
manifolds $W^{ss}_{d-m+1}(p)$ and $W^{uu}_{d-m+1}(p)$ such
that $\ang(E_{d-m+1}^{ss}(p),E_{d-m+1}^{uu}(p))<\eps$. %AQUI N�O TEM INTERSE��O TRANSVERSAL, AS DIRE��ES NO PONTO Q S�O PEQUENAS, DE FATO FAZ COMO USAMOS L� EMBAIXO, EXISTE U E V NAS VARIEDADES FORTES, ESTES TAIS QUE O OMEGA � DIFERENTE DE ZERO E O ANGULO � PEQUENO. (ISTO N�O � NECESS�RIO, POIS SE O �NGULO DAS DIRE��ES � PEQUENO QUER DIZER QUE EXISTE ESTES VETORES (OU SEJA, TEMOS QUE DEFINIR O �NGULO COMO NA TESE SENDO MINIMO DE VETORES, AGORA QUE OMEGA � DIFERENTE DE ZERO NESTES VETORES POR CONTINUIDADE :)))
\end{lemma}


%The previous lemma is also a consequence of Proposition \ref{identity}.

\begin{proof}
The proof is similar in spirit to the proof of Theorem \ref{t.trichotomy}. %Hence, we will just give a sketch.

Fixed $1\leq m\leq d$, let  $f\in \mathcal{PH}_{\omega}(m)$. %( if $m=n$, we take $f$ in the open set $\mathcal{D}$).
Using Proposition \ref{identity}, we can find a diffeomorphism $g$, $C^1-$close to $f$, having a periodic point $p$  such that $Dg^{\tau(p,g)}|E^c(p)=Id$. %and by Remark \ref{rmkHT} we can also assume that $Dg^{\tau(p,g)}|E^{s}(p)$ has only simple eigenvalues.
Now, for any $\eps>0$, we can choose a symplectic basis $\{e_1, \ldots, e_{2m}\}$ in  $E^c(p)$,  such that $ang(e_1, e_{m+1})<\eps$. Hence, taking small constants $\eps_1\geq \eps_2\geq \ldots \geq \eps_m>0$, we define a symplectic linear map over $E^c(p)$ close to identity, induced by the symplectic matrix $A=(a_{ij})$ of order $2m\times 2m$, where $a_{ii}=1-\eps_i$, if $1\leq i\leq m$, $a_{ii}=(1-\eps_i)^{-1}$, if $m+1\leq i\leq 2m$, and $a_{ij}=0$ if $i\neq j$.

By repeating the process done in the proof of Theorem \ref{t.trichotomy},   we can find a difeomorphism $h$ $C^1-$close to $g$, such that $p$ still is a periodic point of $h$, $Dh^{\tau(p)}|E^c(p)=A$ and $Dh^{\tau(p)}|(E^c(p))^{\omega}=Id$. Therefore,  $p$ is a hyperbolic periodic point of $h$, and moreover by choice of $A$ and the symplectic basis, the $n-m+1-$strong stable and unstable manifolds are well defined, with $e_1\in E_{n-m+1}^{ss}(p)$ and $e_{m+1}\in E_{n-m+1}^{uu}(p)$, which implies that $ang(E_{n-m+1}^{ss}(p), E_{n-m+1}^{uu}(p))<\eps$.

\end{proof}



\begin{lemma}
\label{lemma2}
Let $f$ be a symplectic diffeomorphism with two hyperbolic periodic points
$p$ and $\tilde{p}$, having $k-$strong stable and unstable manifolds.
Given $\eps>0$, for any neighborhood $\mathcal{U}$
of $f$ and any neighborhood $V$ of $\tilde{p}$, there exists a
diffeomorphism $g\in \mathcal{U}$, such that the analytic continuation
$p(g)$ of $p$ for $g$, has a $k$-strong transversal homoclinic intersection
$q$ in $V$, that is,
$q\in W^{ss}_{k}(p(g))\cap W^{uu}_{k}(p(g))$, and
moreover $T_qW^{ss}_{k}(p(g))$ and $T_qW^{uu}_{k}(p(g))$ are
 $\eps$-close to $E_{k}^{ss}(\tilde{p}(g))$ and
$E_{k}^{uu}(\tilde{p}(g))$, respectively.
\end{lemma}



%\begin{lemma}
%\label{lemma2}
%Let $f\in \dw$ with two diagonalizable hyperbolic periodic points
%$p$ and $\tilde{p}$.
%Given $0<m\leq n$ and $\eps>0$, for any neighborhood $\mathcal{U}$
%of $f$ and any neighborhood $V$ of $\tilde{p}$, there exists a
%diffeomorphism $g\in \mathcal{U}$, such that the analytic continuation
%$p(g)$ of $p$ has a $(n-m+1)$-transversal strong homoclinic intersection
%$q$ in $V$, that is,
%$q\in W^{ss}_{n-m+1}(p(g))\cap W^{uu}_{n-m+1}(p(g))$, and
%moreover $T_qW^{ss}_{n-m+1}(p(g))$ and $T_qW^{uu}_{n-m+1}(p(g))$ are
%subbundles $\eps$-close to $E_{n-m+1}^{ss}(\tilde{p}(g))$ and
%$E_{n-m+1}^{uu}(\tilde{p}(g))$, respectively.
%\end{lemma}




\begin{proof}
%Passos: Toma V pequeno onde temo uma boa domina��o com respeito aos autoespa�os. Toma estas constantes, tipo, se ang(E,F)>\delta e E n�o tah contido num autoespaco, entao ele converge para E^uu, para um n grande fixo (delta depende de n).

First, we fix %$0<m\leq n$ and
$\eps>0$ small enough.
Without loss of generality we may assume $\tilde{p}$ is a fixed point
of $f$, by replacing $f$ by an iterated if necessary. Also, replacing $V$ by a smaller neighborhood of $\tilde{p}$, we can consider in this neighborhood a continuous splitting $TM=E^{ss}_{k}\oplus E^c_{k}\oplus E^{uu}_{k}$, not necessary invariant, such that  $T_{\tilde{p}}M=E^{ss}_{k}\oplus E^c_{k}\oplus E^{uu}_{k}(p)$ is an invariant partially hyperbolic splitting for $Df(\tilde{p})$. Hence, with respect to this decomposition we define the strong unstable cone fields $C_{\alpha}$ on $V$ as:
$$
C_{\alpha}(x)=\{w\in T_xM / \ w=w_{cs}+w_u, \ w_{cs}\in E^{ss}_k\oplus E^{c}_k,\  w_u\in E^{uu}_k \text{ and } \|w_{cs}\|\leq \alpha \|w_u\|\}
$$
Thus,  by domination of the fixed decomposition on $T_{\tilde{p}}M$, there exists $l>0$ such that for any $g$ $C^1-$close  to $f$, any $\alpha>0$, and any $x\in \displaystyle\bigcap_{0\leq i\leq l} g^{-i}(V)$:
$$
Dg^{l}(C_\alpha(x))\subset C_{\frac{1}{2}\alpha}(g^{l}(x)).
$$
Remember that these kind of cone fields allow us to find  a $g-$invariant dominated splitting on the maximal invariant set in $V$, which varies continuously in a neighborhood of $f$.  % that extends the $f-$dominated splitting on $T_{\tilde{p}}M$, in particular in the analytic continuation of $\tilde{p}$ to $g$, $\tilde{p}(g)$.


%$\R^{2n}=E_{\lambda_{1,\tilde{p}}}\oplus\ldots\oplus E_{\lambda_{2n,\tilde{p}}}$, where $E_{\lambda_{j,\tilde{p}}}$ are the eigenspaces of $Df(\tilde{p})$. To simplify notation we denote $E_{\lambda_{j,\tilde{p}}}$ by $E_{j,\tilde{p}}$, and in order to be more precise we denote by $E_{j,\tilde{p}}(.)$ the extension of the direction $E_{j,\tilde{p}}$ in the neighborhood $V$, and for any diffeomorphisms $C^1$ close to $f$,  identifying the base point.  %and for diffeomorphisms $C^1$ close to $f$ since this splitting is dominated.


%Hence, we remark that in $V$ it also makes sense the extension of the definition of $k-$strong directions $E^{ss (uu)}_{k}(.)$. %the $k-$strong manifolds of $\tilde{p}$.

From the existence of such cone fields we also have the following claim.



\vspace{0,2cm}
{\it Claim: Given $\delta>0$, there exists a positive integer $N_0$,  such that for every $N\geq N_0$ and any $g$, $C^1-$close to $f$,  if $x\in \cap_{0\leq j\leq N} f^{-j}(V)$
and $F(x)$ is a $k-$dimensional vector subspace in $T_xM$ such that $\ang( E^{ss}_k\oplus E^c_k(x), F(x))>\delta$, then $Dg^N(F(x))$ is $\eps/2-$close to $E^{uu}_{k}(g^{N}(x))$.
}
\vspace{0,2cm}

In fact, taking $0<\alpha \leq \beta$ and a positive integer $N_0-1=n_0l$ such that $F(x)\in C_{\alpha}(x)$, $Dg^j(C_{\alpha}(x))\subset C_{\beta}(g^j(x))$, for any $0\leq j< l$, and $\left(\displaystyle\frac{1}{2}\right)^{n_0}\beta<\displaystyle\frac{\eps}{2}$. Hence, for any $N\geq N_0$, we have $N=kl+j$ with $k\geq n_0$ and $0\leq j<l$, which implies
$$
Dg^{N}(C_{\alpha}(x))\subset Dg^{kl}(C_{\beta}(g^j(x)))\subset C_{(1/2)^{k}\beta}(g^{N}(x))\subset C_{\eps/2}(g^N(x))$$ for any $g$ $C^1-$close to $f$, and any $x\in \displaystyle\bigcap_{0\leq i\leq N} g^{-i}(V)$. %Hence, the claim follows since by hypothesis $F(x)\in C_{1/\delta}(x)$.





%In fact, we can choose $\alpha>0$ and a positive integer $N_0-1=n_0l$ such that $Dg^j(F(x))\in C_{\alpha}(g^j(x))$, for any $0\leq j< l$, and $\left(\displaystyle\frac{1}{2}\right)^{n_0}\alpha<\displaystyle\frac{\eps}{2}$. Hence, for any $N\geq N_0$, we have $N=kl+j$ with $k\geq n_0$ and $0\leq j<l$, which implies
%$$
%Dg^{N}(C_{\alpha}(x))=Dg^{kl}(C_{\alpha}(g^j(x)))\subset C_{(1/2)^{n_0}\alpha}(g^{N}(x))$$ for any $g$ $C^1-$close to $f$, any $N\geq N_0$, and any $x\in \displaystyle\bigcap_{0\leq i\leq N} g^{-i}(V)$. %Hence, the claim follows since by hypothesis $F(x)\in C_{1/\delta}(x)$.


\vspace{0,2cm}



Now, by the same method  as in the proof of Lemma \ref{conexao} we can perturb $f$ to create an intersection between the $k-$dimensional strong unstable manifold of $\tilde{p}$ and the $k-$dimensional strong stable manifold of  $p$.  See Figure a, in Figure \ref{fig4}. That is, there exists a diffeomorphism $h\in\mathcal{U}$, $C^1-$close to $f$, such that  there is  $x\in W^{ss}_{k}(p(h))\cap W^{uu}_{k}(\tilde{p}(h))$. Replacing $x$ by a past iterated we can suppose $x\in V$, and moreover by another perturbation, if necessary, we can also assume that $T_xW^{ss}_{k}(p(h))\cap E^{c}_k(x,h)\oplus E^{uu}_{k}(x,h)={0}$. Here, $E^{ss}\oplus E^c\oplus E^{uu}(x,h)$  means the extension of the partially hyperbolic splitting $E^{ss}\oplus E^c\oplus E^{uu}(\tilde{p}(h))$ on $V$, as we did before for $f$.


Hence, %by the partial hyperbolic splitting fixed on $V$ and
since $h^{-j}(x)\in V$ for every positive integer $j$, we have by domination that  $T_{h^{-j}(x)}(W^{ss}_{k}(p(h)))$ converges to  $E^{ss}_{k}(\tilde{p}(h))$, when $j$ goes to infinity. Therefore, we can choose a large positive integer $N_1$ such that $T_{h^{-N}(x)}(W^{ss}_{k}(p(h)))$ is $\eps/2$-close $E^{ss}_{k}(\tilde{p}(h))$, for every $N\geq N_1$. See Figure b, in Figure \ref{fig4}.



%the connected component of $W^{ss}_{k}(p,g)\cap V$ containing $g^{-N}(x)$ is $\eps/2$-close  to the connected component of $W^{ss}_{k}(\tilde{p}(g))\cap V$ containing $\tilde{p}$.

%the connected component of $W^{ss}_{n-m+1}(p(g))\cap V$ containing $g^{-j}(x)$ is $\eps/2-C^1$ close to the $n-m+1-$dimensional  local strong stable manifold of $\tilde{p}$, i.e., the connected component of $W^{ss}_{n-m+1}(\tilde{p}(g))\cap V$ containing $\tilde{p}$. %Note, since we are looking $V$ in local coordinates that this local strong stable manifold coincides with  $E^{ss}_{n-m+1}(\tilde{p}(g))\cap V$.


We consider now $\delta_0>0$ the small constant given by the Franks Lemma for the neighborhood $\mathcal{U}$ of $f$, and thus we take $N$ larger than $N_0$ (given by the Claim when we consider $\delta_0$) and $N_1$. % for a larger positive integer we can suppose that $\delta_N>0$ given by Claim  is smaller than $\delta_0$.

Let us fix a point $y$ in the connected component of  $W^{ss}_{k}(p(h))\cap V$ containing $h^{-2N}(x)$ such that $h^{-1}(y)\not\in cl(V)$ and $h^{j}(y)\in V$ for $0\leq j\leq 2N$.  Once again, similarly to the proof of Lemma \ref{conexao} we can find a diffeomorphism $g$ $C^1$-close to $h$ such that $y\in W^{ss}_{k}(p(g))\cap W^{uu}_{k}(p(g))$. It's important to observe that provided before we use the connecting lemma we need to take a generic transitive symplectomorphism, we can not guarantee that $y$ still belongs to the strong stable manifold of $p$, however by continuity of the strong manifolds, this connection could be found close to the point $y$. And thus, in order  to simplify the notation, we have continued denoting this intersection by $y$.


\begin{figure}[!htb]
\vspace{0,5cm}\centering
\includegraphics[scale=0.9]{fig4.eps}
\caption{\it{Figure a}: connecting $W^{ss}(p)$ and $W^{uu}(\tilde{p})$. \newline {\it Figure b}: Taking strong iterated disks of $p$ close to $\tilde{p}$.\newline  {\it Figure c}: Connecting $W^{uu}(p)$ and $W^{ss}(p)$ }\vspace{0,5cm}\label{fig4}\end{figure}




%As before, we can assume $g$ is topologically transitive by ARNAULD-BONATTI-CROVISIER, after a perturbation. Then, we can use once more connecting lemma in order to find a diffeomorphism $\phi$ $C^1$-close to $g$ such that $y\in W^{ss}_{n-m+1}(p(\phi))\cap W^{uu}_{n-m+1}(p(\phi))$.

Now, since $g^{-1}(y)\not\in cl(V)$, we can perform a local perturbation in a small neighborhood of $g^{-1}(y)$ by Franks Lemma, to obtain  $\ang(E^{ss}_{k}\oplus E^{c}_{k}(y,g),T_yW^{uu}_{k}(p(g)))> \delta_0$. In particular, $y$ is a $k-$strong transversal homoclinic point of $p(g)$.

Therefore,  by the Claim we conclude that $Dg^N(T_yW^{uu}_{k}(p(g)))$ is $\eps/2-$close to $E^{uu}_{k}(g^{N}(y))$, which is $\eps/2-$close to $E^{uu}_k(\tilde{p}(g))$ by continuity of the splitting on $V$ as defined. And thus we have $T_{g^N(y)}W^{uu}_{k}(p(g))$ is $\eps-$close to $E^{uu}_k(\tilde{p}(g))$. See Figure c, in Figure \ref{fig4}.

 % {\color{red} aqui est� tentando usar a mesma nota��o do ponto base, aquela ainda fixada pela f, mas isto n�o est� legal! O certo � fixar l� em cima esta extens�o como sendo extensao natural dependendo da continuacao do ponto p tilda}.
On the other hand, by the $C^1-$continuity of the k-strong stable manifold of $p(h)$ in compact parts, we can choose $g$ such that the connected component of $W^{ss}_{k}(p(g))\cap V$ and  $W^{ss}_{k}(p(h))\cap V$ containing $g^N(y)$ and $h^{N}(x)$, respectively, are $C^1-$close, which implies by choice of $N$ and again the $C^1-$continuity of  $k-$strong manifolds,  that $T_{g^N(y)}W^{ss}_{k}(p(g))$ is $\eps-$close to  $E_{k}^{ss}(\tilde{p}(g))$.


\end{proof}




%\begin{proof}
%Passos: Toma V pequeno onde temo uma boa domina��o com respeito aos autoespa�os. Toma estas constantes, tipo, se ang(E,F)>\delta e E n�o tah contido num autoespaco, entao ele converge para E^uu, para um n grande fixo (delta depende de n).

%First, we fix %$0<m\leq n$ and
%$\eps>0$ small enough.
%Without loss of generality we may assume $\tilde{p}$ is a fixed point
%of $f$, by replacing $f$ by an iterated. Also, replacing $V$ by a smaller neighborhood of $\tilde{p}$, we can  look at $V$ in local coordinates. Hence, we will consider in this neighborhood the decomposition which is a natural extension of the partially  hyperbolic dominated splitting in $T_{\tilde{p}}M$ given by $Df(\tilde{p})$. That is, we consider $\R^{2n}=E^{ss}_{k}\oplus E^c_{k}\oplus E^{uu}_{k}$, and
%to be more precise we denote by $E^{ss}_k(.)$, $E^{uu}_k(.)$  and $E^{c}_k(.)$ the extension of  directions $E^{ss}_{k}$, $E^{uu}_{k}$ and $E^{c}_{k}$, respectively, in the neighborhood $V$, and for any diffeomorphisms $C^1$ close to $f$,  identifying the base point.

%$\R^{2n}=E_{\lambda_{1,\tilde{p}}}\oplus\ldots\oplus E_{\lambda_{2n,\tilde{p}}}$, where $E_{\lambda_{j,\tilde{p}}}$ are the eigenspaces of $Df(\tilde{p})$. To simplify notation we denote $E_{\lambda_{j,\tilde{p}}}$ by $E_{j,\tilde{p}}$, and in order to be more precise we denote by $E_{j,\tilde{p}}(.)$ the extension of the direction $E_{j,\tilde{p}}$ in the neighborhood $V$, and for any diffeomorphisms $C^1$ close to $f$,  identifying the base point.  %and for diffeomorphisms $C^1$ close to $f$ since this splitting is dominated.


%Hence, we remark that in $V$ it also makes sense the extension of the definition of $k-$strong directions $E^{ss (uu)}_{k}(.)$. %the $k-$strong manifolds of $\tilde{p}$.


%The following claim is a directly consequence of dominated splitting properties:


%\vspace{0,2cm}
%{\it Claim: Given $\delta>0$, there exists a positive integer $N_0$,  such that for every $N\geq N_0$ and any $g$, $C^1-$close to $f$,  if $x\in \cap_{0\leq j\leq N} f^{-j}(V)$
%and $F(x)$ is a $k-$dimensional vector subspace in $T_xM$ such that $\ang( E^{ss}_k(x)\oplus E^c_k(x), F(x))>\delta$, then $Dg^N(F(x))$ is $\eps/2-$close to $E^{uu}_{k}(g^{N}(x))$. %for every $x\in \cap_{0\leq j\leq N} f^{-j}(V)$ %such that $g^j(x)\in V$ for all $0\leq j\leq N$
%and for every $g$ $C^1-$close to $f$.
%Moreover, $\delta_N$ could be taken arbitrarily close to zero if $N$ is arbitrarily large.
%}
%\vspace{0,2cm}







%\vspace{0,2cm}
%{\it Claim: Given a large positive integer $N$, there exists $\delta_N>0$ such that for every $x\in \cap_{0\leq j\leq N} f^{-j}(V)$ %such that $g^j(x)\in V$ for all $0\leq j\leq N$
%and any $g$, $C^1-$close to $f$, if $F(x)$ is a $k-$dimensional vector subspace in $T_xM$ such that $\ang( E^{ss}_k(x)\oplus E^c_k(x), F(x))>\delta_N$, then $Dg^N(F(x))$ is $\eps/2-$close to $E^{uu}_{n-m+1,\tilde{p}(g)}(g^{N}(x))$, %for every $x\in \cap_{0\leq j\leq N} f^{-j}(V)$ %such that $g^j(x)\in V$ for all $0\leq j\leq N$
%and for every $g$ $C^1-$close to $f$.
%Moreover, $\delta_N$ could be taken arbitrarily close to zero if $N$ is arbitrarily large.}
%\vspace{0,2cm}




%\vspace{0,2cm}
%{\it Claim: Given a large positive integer $N$, there exists $\delta_N>0$ such that if $\R^{2n}=  (E_{1,\tilde{p}(g)}(x)\oplus\ldots\oplus E_{n+m-1,\tilde{p}(g)}(x))\oplus F(x)$  and $\ang( E_{1,\tilde{p}(g)}(x)\oplus\ldots\oplus E_{n+m-1,\tilde{p}(g)}(x),$ $ F(x))>\delta_N$, then $Dg^N(F(x))$ is $\eps/2-$close to $E^{uu}_{n-m+1,\tilde{p}(g)}(g^{N}(x))$, for every $x\in \cap_{0\leq j\leq N} f^{-j}(V)$ %such that $g^j(x)\in V$ for all $0\leq j\leq N$
%and for every $g$ $C^1-$close to $f$. Moreover, $\delta_N$ could be taken arbitrary close to zero if $N$ is arbitrary large.}
%\vspace{0,2cm}



%Now, by the same arguments as in the proof of Lemma \ref{conexao} we can perturb $f$ to create an intersection between the $k-$dimensional strong unstable manifold of $\tilde{p}$ and the $k-$dimensional strong stable manifold of  $p$.  That is, there exists a diffeomorphism $g$ $C^1-$close to $f$ such that  there is  $x\in W^{ss}_{k}(p(g))\cap W^{uu}_{k}(\tilde{p}(g))$. Replacing $x$ by a past iterated we can suppose $x\in V$, and moreover by another perturbation, if necessary, we can also assume that $T_xW^{ss}_{k}(p(g))\cap E^{c}_k(x,g)\oplus E^{uu}_{k}(x,g)={0}$. Hence, %by the partial hyperbolic splitting fixed on $V$ and
%since $g^{-j}(x)\in V$ for every positive integer $j$, we have that  $T_{g^{-j}(x)}(W^{ss}_{k}(p,g))$ converges to  $E^{ss}_{k}(\tilde{p}(g))$, when $j$ goes to infinity. Therefore, we can choose a large positive integer $N_1$ such that $T_{g^{-N}(x)}(W^{ss}_{k}(p,g))$ is $\eps/2$-close $E^{ss}_{k}(\tilde{p}(g))$, for every $N\geq N_1$.



%the connected component of $W^{ss}_{k}(p,g)\cap V$ containing $g^{-N}(x)$ is $\eps/2$-close  to the connected component of $W^{ss}_{k}(\tilde{p}(g))\cap V$ containing $\tilde{p}$.

%the connected component of $W^{ss}_{n-m+1}(p(g))\cap V$ containing $g^{-j}(x)$ is $\eps/2-C^1$ close to the $n-m+1-$dimensional  local strong stable manifold of $\tilde{p}$, i.e., the connected component of $W^{ss}_{n-m+1}(\tilde{p}(g))\cap V$ containing $\tilde{p}$. %Note, since we are looking $V$ in local coordinates that this local strong stable manifold coincides with  $E^{ss}_{n-m+1}(\tilde{p}(g))\cap V$.


%We consider now $\delta_0>0$ the small constant given by the Franks Lemma for the neighborhood $\mathcal{U}$ of $f$, and thus we take $N$ larger than $N_0$ (given by Claim to $\delta_0$) and $N_1$. % for a larger positive integer we can suppose that $\delta_N>0$ given by Claim  is smaller than $\delta_0$.

%Let us fix a point $y$ in the connected component of  $W^{ss}_{k}(p,g)\cap V$ containing $g^{-2N}(x)$ such that $g^{-1}(y)\not\in cl(V)$ and $g^{j}(y)\in V$ for $0\leq j\leq 2N$.  Once again, using the methods contained in the proof of Lemma \ref{conexao} we can find a diffeomorphism $\phi$ $C^1$-close to $g$ such that $y\in W^{ss}_{k}(p(\phi))\cap W^{uu}_{k}(p(\phi))$.




%As before, we can assume $g$ is topologically transitive by ARNAULD-BONATTI-CROVISIER, after a perturbation. Then, we can use once more connecting lemma in order to find a diffeomorphism $\phi$ $C^1$-close to $g$ such that $y\in W^{ss}_{n-m+1}(p(\phi))\cap W^{uu}_{n-m+1}(p(\phi))$.

%By choice of $y$, we have that $\phi^{-1}(y)\not\in cl(V)$ and then, we can perform a local perturbation in a small neighborhood of $\phi^{-1}(y)$ by Franks Lemma, if necessary, in order to have that $\ang(E^{ss}_{k}(y,\phi)\oplus E^{c}_{k}(y,\phi),T_yW^{uu}_{k}(p(\phi)))> \delta_0$.

%Therefore,  by the Claim we conclude that $D\phi^N(T_yW^{uu}_{k}(p(\phi)))$ is $\eps/2-C^1$ close to $E^{uu}_{n-m+1}(\phi^{N}(y))$ {\color{red} aqui est� tentando usar a mesma nota��o do ponto base, aquela ainda fixada pela f, mas isto n�o est� legal! O certo � fixar l� em cima esta extens�o como sendo extensao natural dependendo da continuacao do ponto p tilda}. Moreover by continuity of the n-m+1-strong stable manifold of $p$ in compact parts, and the fact that $\phi^N(y)$ belongs to the connected component of $W^{ss}_{n-m+1}(p(\phi))\cap V$ containing $\phi^{N}(x)$, we could find $\phi$ such that the connected component of $W^{ss}_{n-m+1}(p(\phi))\cap V$ containing $g^{N}(y)$ is $2\eps/3$-close  to the connected component of $W^{ss}_{n-m+1}(\tilde{p}(\phi))\cap V$ containing $\tilde{p}$. And therefore, since $V$ is arbitrary small we have that $T_{\phi^{N}(y)}W^{ss}_{n-m+1}(p(\phi))$ and $T_{\phi^{N}(y)}W^{uu}_{n-m+1}(p(\phi))$ are sub-bundles $\eps-$close to $E_{n-m+1}^{ss}(\tilde{p}(\phi))$ and $E_{n-m+1}^{uu}(\tilde{p}(\phi))$, respectively, as we wanted.


%\end{proof}











\section{ Proof of proposition \ref{p.estimate}:}\label{section5}





The idea here is to use  Lemmas \ref{lemma1} and \ref{lemma2} to perturb a symplectic diffeomorphism $f\in \ph(m)$ in order to find a symplectic $(2d-2m)$-dimensional surface inside $M$, containing the hyperbolic periodic point $p$ and a segment of strong homoclinic points of $p$, such that we can use the arguments presented in \cite{CT}, to find a nice hyperbolic set by means of Newhouse's snake perturbations.


\vspace{0,2cm}

{\it Proof of Proposition \ref{p.estimate}}

%To simplify the notation we will first assume that $p$ is a diagonalizable hyperbolic fixed point, which implies $\tau(p,f)=1$.
%{\color{red} SEGUINTE, COMENTA AQUI A IDEIA DA PROVA, QUE SER� REDUZIR O PROBLEMA PARA USARMOS CT, ASSIM, FAREMOS NO CASO PONTO FIXO SENDO AN�LOGO O CASO PERI�DICO}

Let $f\in int(\ph^1(m))$ and $p$ be
a diagonalizable hyperbolic periodic point of $f$.  We will suppose first that $p$ is a fixed point.

Fixing an arbitrary $\eps>0$, %and choose $\delta>0$  satisfying Lemmas \ref{symplectic} and \ref{isotropic2} for $\eps/2$.
by Lemma \ref{lemma1}, after a perturbation, we can suppose there exists a hyperbolic periodic point $\tilde{p}$ of $f$ having $d-m+1-$strong stable and unstable manifolds such that $ang(E^{ss}_{d-m+1}(\tilde{p}),E^{uu}_{d-m+1}(\tilde{p}))<\eps/2$. Thus, provided we have defined $d-m+1-$strong manifolds to $p$, we can use Lemma \ref{lemma2}, to find a diffeomorphism $f_1$ $C^1-$close to $f$, such that $p(f_1)$ %of $f_1$,
has a $d-m+1-$strong transversal homoclinic point $q\in W^{ss}_{d-m+1}(p(f_1))\cap W^{uu}_{d-m+1}(p(f_1))$, such that
$$
ang(T_{q}W^{ss}_{d-m+1}(p(f_1)),T_{q}W^{uu}_{d-m+1}(p(f_1))<\eps.
$$
Note, since $f_1$ is arbitrary $C^1-$close to $f$, this perturbation could be found such that $p(f_1)$ still is a diagonalizable hyperbolic fixed point.


%{\color{red} Provided $E^{ss}_{d-m+1}\oplus E_{d-m+1}^{uu}(p(g))$ is a symplectic subspace, and $E_{d-m+1}^{ss \ (uu)}(p(g))$ is an isotropic subsapce,  % since Lemma \ref{lemma2} says that $T_{q}W^{ss}_{n-m+1}(p(f_1))\oplus T_{q}W^{uu}_{n-m+1}(p(f_1))$ is close to the symplectic subspace $E^{ss}_{n-m+1}(\tilde{p})\oplus E^{uu}_{n-m+1}(\tilde{p})$,
%by continuity of the symplectic form $\omega$
%there are unit vectors $v\in T_{q}W^{ss}_{d-m+1}(p(f_1))$ and $u\in T_{q}W^{uu}_{d-m+1}(p(f_1))$ such that
%$$
%ang(u,v)<\eps  \ \text{ and }\ \omega(u,v)\neq 0.
%$$}

%{\color{red} TALVEZ A EXISTENCIA DE U E V SEJA MELHOR APARECER L� EMBAIXO QUANDO J� FOR USAR


%VAMOS ASSUMIR, DEFINIR DEPOIS, QUE INTERSE��O HOMOCL�NICA FORTE TRANSVERSAL, IMPLICA O PONTO TAMB�M SER UM PONTO HOMOCL�NICO TRANSVERSAL. SENDO ASSIM TEM QUE COMENTAR ESTE PASSO NA DEMONSTRA��O DO LEMA 10}

Now, we can use a pasting lemma of Arbieto and Matheus \cite{AM} %lembre-se que em simpl�tico n�o precisa de regulariza��o
to linearize the diffeomorphism in a small enough neighborhood  $V$ of $p$. % after a $C^1$ local perturbation.
 More precisely, we can
 find $f_2$ $C^1-$close to $f_1$ such that $p(f_1)=p(f_2)$ and
$f_2=Df_2(p)$ in  $V$ (in local
coordinates). It's important to observe that after this perturbation, we could have no more a strong transversal intersection between $W^{ss}_{d-m+1}(p(f_2))$ and
$W^{uu}_{d-m+1}(p(f_2))$ near $q$. However, provided  these sub-manifolds varies continuously in compact parts for the diffeomorphism, this intersection could be recovered after a
local perturbation of $f_2$ in a neighborhood of $q$, having the same previous properties.  %using
%PROPOSI��O XIA (CITAR ALGUM DOS PAPERS QUE FAZEMOS ISTO) having the same previous properties. %Moreover, by the continuity of the symplectic form we still have the existence of vectors %


In the reminder of this proof by abuse of notation we will denote
by $p$ all of its continuations  with respect to nearby diffeomorphisms, and we will denote the $d-m+1-$strong directions and manifolds of $p$ only by $E^{ss\ (uu)}(p)$ and $W^{ss\ (uu)}(p)$. The same for the $m-1-$central direction, $E^c(p)$.  Also, we will look $V$ in local
coordinates, % and  by Darboux's Theorem we  will suppose in $V$ that $\omega$ is in the standard form, i.e.,  $\omega=\sum_{i=1}^n dx_i dx_{n+i}$.
and thus we consider the natural partially hyperbolic splitting induced by $Df_2(p)$ in local coordinates, i.e.,
$\R^{2d}=E^{ss}\oplus E^c \oplus E^{uu}(p)$.
%by the one dimensional eigenspaces of $Df_2(p)$.
As before, in order to be more precise, we  make some assumptions, we consider $E^{ss}(.)$ the extension of the direction $E^{ss}(p)$ in  $V$, identifying the base point. The same assumptions will be considered to  $E^c(.)$ and $E^{uu}(.)$.



We remark now that %provided $f_2=Df_2(p)$ in $V$,
the local strong stable and
unstable manifolds of $p$ coincide with their strong directions restrict to $V$, since $f_2$ is linear in this neighborhood. That is,
the local strong stable (unstable) manifold of $p$ is $E^{ss(uu)}(p)\cap V$.
Thus, since $q$ is a strong  homoclinic point, for any large positive integer $k$:
$$
f_2^{k}(q)\in E^{ss}(p)\cap V \text{ and } f_2^{-k}(q)\in
E^{uu}(p)\cap V.$$

The next lemma is the key point in this proof.

\begin{lemma}
There exists a symplectic diffeomorphism $f_3$ $C^1-$close to $f_2$, a po\-si\-ti\-ve integer $K$,  small neighborhoods $U_{-K},\ U_K\subset V$ of $f_3^{-K}(q)$ and $f_3^{K}(q)$, respectively, such that %$q$ still  is a strong  homoclinic point of $p$ and
replacing $V$ by a smaller neighborhood we have: %the following properties:
\begin{itemize}
\item $f_3=Df_3(p)$ is still linear on $V$ (in local coordinates),
\\
\item $f_3^{2K}\big((E^{ss}\oplus E^{uu}(f_3^{-K}(q)))\cap U_{-K}\big)\subset (E^{ss}\oplus E^{uu}(f_3^{K}(q)))\cap U_{K}$,  %for $m\geq m_0$,
%\item $f_3^{2m}\big((E_{\lambda_{n-m+1,p}}\oplus E_{\lambda_{2n-m-1,p}})\cap U_{-m}%\big)=(E_{\lambda_{n-m+1,p}}\oplus E_{\lambda_{2n-m-1,p}})\cap U_{m}$,  for $m\geq m_0$,
\\
\item there exists a segment of line $I$  in $W^{ss}(p)\cap W^{uu}(p)$ containing $q$. More precisely,
 $f_3^{2K}(E^{uu}(f_3^{-K}(q))\cap U_{-K})\cap E^{ss}(f_3^{K}(q))$ contains a segment of line. %That is, there is an interval of strong homoclinic points between $W^{ss}(p,f_3)$ and $W^{uu}(p,f_3)$ containing $q$.
%There is an interval $I\subset f_3^{2K}(E^{uu}_{n-m+1}(f_3^{-K}(q))\cap U_{-K})\cap (E^{ss}_{n-m+1}(f_3^{K}(q))\cap U_{K})$ containing $q$. That is, there exists an interval of strong homoclinic intersections.
%There is a homoclinic tangency between $W^{ss}_{n-m+1}(p,f_3)$ and $W^{uu}_{n-m+1}(p,f_3)$.
\end{itemize}
\label{keypoint}\end{lemma}



{\it Proof:}

First we observe that after a local perturbation, if necessary, %and using that $q$ is a strong transversal homoclinic point,
%%%%%%%%%%%%%%%%% como fazer isto %%%%%%%%%%%%%%%%%%%%%
%% Simples: por transversalidade um conjunto residual de matrizes simp�ticas pr�ximas da identidad
%% composta com a Df faria isto. Pronto, escolheria uma e usaria franks para fazer isto compondo com %% DF
we can suppose that for large positive integers k,

$$Df_2^{-k}(T_{q}W^{ss}(p))\cap (E^{c}\oplus E^{uu}(f_2^{-k}(q)))=0,$$
and
$$Df_2^{k}(T_{q}W^{uu}(p))\cap ( E^{ss}\oplus E^{c}(f_2^{k}(q)))=0.$$

%{\color{red}TEM QUE DEFINIR ESTE Ec L� EM CIMA}

% $q$ is in fact a transversal homoclinic point.  for any large positive integer $k$ we have $$T_{f^{-k}(q)}W^{ss}(p,f_2)\cap (E^{c}_{m-1}(f^{-k}(q))\oplus E^{uu}_{n-m+1}(f^{-k}(q)))=0.$$
Thus, by dominated splitting properties we have that $Df_2^{-k}(T_{q}W^{ss}(p))$ converges to $E^{ss}(p)$ when $k$ goes to infinity. Hence,  we can choose a large positive integer $k_0$ such that  $Df_2^{-k_0}(T_{q}W^{ss}(p))$ is close enough to  $E^{ss}(f_2^{-k_0}(q))$.






Provided for any  hyperbolic
periodic point of a symplectic map its stable ( unstable )  manifold is a Lagrangian submanifold (Remark \ref{dec.}), thus  $T_x W^{s}(p)$ ($T_x W^{u}(p)$) is a Lagrangian subspace for any $x\in W^{s}(p)$ ($W^{u}(p)$).  In particular $T_x W^{ss}$  ($T_x W^{uu}$) is an isotropic subspace (Remark \ref{decomp.}), for any $x\in W^{ss}(p)$ ($W^{uu}(p)$).

%Hence,  we can choose a large positive integer $k_0$ such that the isotropic subspace  $Df_2^{-k_0}(T_{q}W^{ss}(p,f_2))$ is close enough to the isotropic subspace $E^{ss}_{n-m+1}(f_2^{-k_0}(q))$,
Hence, we can use Lemma \ref{isotropic2} to find a symplectic linear map $\tilde{B}$ on $\R^{2d}$, $C^1-$close to $Id$, such that
$$\tilde{B}(Df_2^{-k_0}(T_{q}W^{ss}(p)))=E^{ss}(f_2^{-k_0}(q))  \text{ and } \tilde{B}| (E^c \oplus E^{uu}(f_2^{-k_0}(q)))=Id.$$ Thus, taking $B=Df_2\circ \tilde{B}^{-1} $, we can use Franks lemma to perform a local perturbation in a neighborhood of $f_2^{-k_0}(q)$ in order to obtain
$$
Df_2^{-k_0}(T_{q}W^{ss}(p))=E^{ss}(f_2^{-k_0}(q)).
$$
Moreover,  this perturbation does not change the action of $Df_2$ over $E^c\oplus E^{uu}(O(q))$.% keeps unchanged, %restrict to the subspace $E^c_{m-1}\oplus E^{uu}(n-m+1)$.
%{\color{red} which implies that $u\in T_q M$ still belongs to $T_q W^{uu}(p)$.}

On the other hand, also by dominated splitting properties
$Df_2^{k+k_0}(E^{uu}(f_2^{-k_0}(q)))$  converges to $E^{uu}(p)$ when $k$ goes to infinity. Hence, as before, for a large positive integer $k_1$ we can perform a local perturbation in a neighborhood of $f_2^{k_1-1}(q)$, in order to obtain
$$
Df_2^{k_1+k_0}(E^{uu}(f_2^{-k_0}(q)))=E^{uu}(f_2^{k_1}(q)),
$$
and keeps the action of $Df_2$ over $E^c\oplus E^{ss}(O(q))$.

%such that the action of $Df_2^{-1}$ over  $Df_2^{k_1+k_0}|E^c\oplus E^{ss}(q)$ does not change. %{\color{red}In particular, %$Df_2^{k_1+k_0}(E^{ss}_{n-m+1}(f_2^{-k_0}(q)))=E^{ss}_{n-m+1}(f_2^{k_1+k_0}(q))$, which implies
%$v\in T_q M$ still belongs to $T_q W^{ss}(p)$.}

We denote by $K$ the maximum positive number between $k_0$ and $k_1$, and we replace $V$ by a smaller neighborhood such that $f_2^{-K}(q), f_2^{K}(q)\in V$ and $f_2^{-K+1}(q),$ $ f_2^{K-1}(q)\not\in V$. Then, provided the perturbations we have performed were local, $f_2$ is still linear on $V$, which implies:
$$
Df_2^{2j}(E^{ss}\oplus E^{uu}(f_2^{-j}(q)))=E^{ss}\oplus E^{uu}(f_2^{j}(q)),$$
 for any $ j\geq K. $


\begin{figure}[!htb]
\vspace{0,5cm}\centering
\includegraphics[scale=1.3]{fig6.eps}
\caption{Creating a $(2d-2m)$-dimensional surface containing an interval of strong homoclinic intersections.}\vspace{0,5cm}\label{fig6}\end{figure}



Now, since $f_2$ is $C^1$ arbitrary close to $f_1$ and since the strong manifolds varies continuously, we still have that $ang(T_qW^{ss}(p), T_qW^{uu}(p))<\eps$. Moreover, provided $E^{ss}\oplus E^{uu}(p)$ is a symplectic subspace and moreover $E^{ss \ (uu)}(p)$ is an isotropic subsapce ( see \cite{BV}, section 4),  % since Lemma \ref{lemma2} says that $T_{q}W^{ss}_{n-m+1}(p(f_1))\oplus T_{q}W^{uu}_{n-m+1}(p(f_1))$ is close to the symplectic subspace $E^{ss}_{n-m+1}(\tilde{p})\oplus E^{uu}_{n-m+1}(\tilde{p})$,
%by continuity of the symplectic form $\omega$
there are unit vectors $v\in T_{q}W^{ss}(p)$ and $u\in T_{q}W^{uu}(p)$ such that
$$
ang(u,v)<\eps  \ \text{ and }\ \omega(u,v)\neq 0.
$$

Hence,  %$T_q W^{ss}(p)\oplus T_qW^{uu}(p)$ is a symplectic subspace, and thus
we  can find a symplectic basis of unit vectors of $T_q W^{ss}(p)\oplus T_qW^{uu}(p)\subset T_qM$ containing such vectors, $\{u,\ u_2\ldots,\  u_{d-m},$ $ \ v,\ v_2,\ldots,\ v_{d-m}\}$. Let $A$ be the symplectic linear map in $T_q W^{ss}(p)\oplus T_qW^{uu}(p)$ such that $A(u)=v$ and $A(u_i)=u_i$ for $2\leq i\leq d-m$. Note  $A$ is $\eps-C^1$ close to identity, since $ang(u,v)<\eps$. Now, using Lemma \ref{symplectic} we can find a symplectic linear map $L:T_qM\rightarrow T_qM$ such that $L|T_q W^{ss}(p)\oplus T_qW^{uu}(p)= A$ and $L| Df^{K}_2(E^c(f_2^{-K}(q)))=Id$. %$T_qM=T_q W^{ss}(p)\oplus T_qW^{uu}(p)\oplus E^c$ and
Then, we can use once more  Franks lemma to perform a local perturbation of $f_2$ in a neighborhood of $f_2^{-1}(q)$ and find a symplectic diffeomorphism $f_3$ close to $f_2$ such that $Df_3(f_3^{-1}(q))=L\circ Df_2(f_2^{-1}(q))$.  Which implies $T_q W^{ss}(p)\cap T_qW^{uu}(p)$ is non trivial. It's important to remark that the above local perturbation keeps the orbits of $p$ and $q$, and moreover does not change the action of the derivative over the direct sum of the strong sub bundles over the orbit of $q$. More precisely,  $q$ still is a strong homoclinic point of $p$, and %the sub-bundle $T_{o(q)}W^{ss}(p,f_2)\oplus T_{o(q)}W^{uu}(p,f_2)$ is $Df_3-$invariant. % for $Df_3$ the sub-bundle $T_{o(q)}W^{ss}(p,f_3)\oplus T_{o(q)}W^{uu}(p,f_3)$ in the orbit of $q$.
$$
Df_3^{2j}(E^{ss}\oplus E^{uu}(f_3^{-j}(q)))=E^{ss}\oplus E^{uu}(f_3^{j}(q)),$$
 for any $ j\geq K. $

Finally, we can use pasting Lemma again, \cite{AM}, to perform local perturbations, in order to linearizing the diffeomorphism $f_3^{2K}$ in a small neighborhood $U\subset V$ of $f_3^{-K}(q)$ such that we have:
$$
f_3^{2K}\big((E^{ss}\oplus E^{uu}(f_2^{-K}(q)))\cap U\big)\subset (E^{ss}\oplus E^{uu}(f_2^{K}(q)))\cap V.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%EXPLICA��O DO PORQUE PODEMOS USAR PASTING LEMMA AQUI
%%%           Note que o pasting lema de ARbieto e Mateus no caso simpl�tico, apesar de eu n�o concordar, eles %%%% fazem apenas para pontos peri�dicos. No entanto, acho que dah para fazer geral usando o fun��es %%%%%%% geradoras.
%%%%%% No entanto para n�o termos que colocar fun��es geradoras no paper, dah mesmo assim para %%%%%%%% argumentar usando o resultado deles. Mas note que isto tamb�m n�o vamos colocar, fica escondido. %%%%%%  Seguinte, seja g e um ponto x. Tome cartas \phi e \psi em vizinhan�as de x e f(x). Agora, vc tem o %%%%%%  difemorfismo local \psi^{-1}\circ f\circ \phi. Este difeomorfismo � simpl�tico, vc escolheu cartas %%%%%%  simpl�ticas. E e este difeomorfismo fixa o zero. Ent�o vc usa pasting lema e localmente este cara vira l%%%%%% inear. Agora, vc perturba f, usando esta perturba��o, que em coordenadas locais, por constru��o, � %%%%%%  linear. Note que quando vc muda as cartas, localmente n�o precisa ser linear, ent�o estas cartas \phi %%%%%%  e \psi devem ser tomadas como sendo restri��es das cartas de V, e a� tudo fica em casa e linear. %%%%%%  Pronto, ganhamos. acho :).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Therefore, provided $T_q W^{ss}(p)\cap T_qW^{uu}(p)$ was a non trivial subspace, we have now the existence of an interval of strong homoclinic points,  more precisely
$$
f_3^{2K}(E^{uu}(f_3^{-K}(q))\cap U)\cap E^{ss}(f_2^{K}(q)) \text{ contains a segment of line } I.
$$
We finish the proof of the lemma taking $U_{-K}=U$ and $U_K=f_3^{2K}(U)$. See figure \ref{fig6}.














%Now, let $U\subset V$ be a small neighborhood of $q$, and let $D\subset W^{uu}(p)\cap U$ a small open disk containing $f_3^{K}(q)\in V$. Hence, looking at $U$ in local coordinates, by similar methods to Lemma \ref{isotropic2}, we can find a symplectic map $\phi$, $C^1-$ close to $Id$, such that $\phi(D)\subset T_qW^{uu}(p)$, $\phi=Id$ in $U^c$(See Lemma 4.1 in CT for details). Finally, we can use this map to perform a local perturbation of $f_3$ in a neighborhood of $f_3^{K-1}(q)$, taking $f_4=\phi\circ f_3$.  perturbatioprovided this symplectic diffeomorphisms satisfies:

%\begin{itemize}
%\item $ f_3^{2K}\big((E^{ss}\oplus E^{uu})\cap U\big)\subset (E^{ss}\oplus E^{uu}(f_2^{K}(q)))\cap V,$

%\item $T_q W^{ss}(p)\cap T_qW^{uu}(p)$ contains an interval of homoclinic tangency.

%\end{itemize}
 %Which finishes the proof of the Lemma, since $f_2$ still is linear on $V$.

$\hfill\square$



%To prove the previous lemma we need the following  symplectic perturbative lemma.

%Before we prove the previous Lemma,  let us see how Proposition \ref{p.estimate} follows from it.

%Now, we continue the prove of Proposition \ref{propest}.

\vspace{0,2cm}
Now, let $f_3$ $C^1$ close to $f_2$ as
given by Lemma \ref{keypoint}, and thus  $I$ is the interval of strong homoclinic intersections containing $q$ inside $U_K$. We will look $U_K$ in local coordinates, where we consider the following decomposition $E^{ss}\oplus E^c\oplus E^{uu}$, which is the restriction to $U$ of the one considered in $V$.

Let $N$ be a large positive integer and $\delta>0$ an arbitrary small real number. Using pasting lemma, we can find a symplectic diffeomorphism $\Theta:M\rightarrow M$, $\delta-C^1$ near $Id$, $\Theta=Id$ in $U_K^c$ and
$$
\Theta(x_s,x_c,x_u)=\left(x_s,\ x_c,\,x_u+A\cos\frac{\pi x_s N}{2r}\right), \text{ for } (x_s,x_c,x_u)\in B(q,r)\subset U_K,
$$
for $A=\displaystyle\frac{2Rr\delta }{\pi N}$, $r>0$ small enough, and where $R$ is a constant depending only on the symplectic coordinate on $U_K$.
Hence,  the symplectic diffeomorphism $f_{4,N}=\Theta\circ f_3$ is $\delta-C^1$ close to $f_3$ and moreover:
\begin{itemize}
\item $f_{4,N}=f_3$ in the complement of $f_3^{-1}(U_K)$,

\item $f_{4,N}^{2K}\big((E^{ss}\oplus E^{uu}(f_{4,N}^{-K}(q))))\cap U_{-K}\big)\subset (E^{ss}\oplus E^{uu}(f_{4,N}^{K}(q)))\cap V,$

\item $f_{4,N}^{2K}\big(E^{uu}(f_{4,N}^{-K}(q))\cap U_{-K}\big)\cap E^{ss}(f_{4,N}^{K}(q))$ contains at least $N$ disjoint points.
\end{itemize}
This perturbation is a kind of Newhouse's snakes
perturbation  for higher dimensions, i.e., it destroys the interval of strong homoclinic intersections and creates $N$ transversal strong homoclinic points for $p$ inside $U_K$. See figure \ref{fig7}.


\begin{figure}[!htb]
\vspace{0,5cm}\centering
\includegraphics[scale=1.3]{fig7.eps}
\caption{Newhouse's snakes.}\vspace{0,5cm}\label{fig7}\end{figure}




After we have found these strong homoclinic points in $U_K$, the construction of a nice hyperbolic set that we wanted is exactly the same as done in \cite{CT}, step 3. Hence, we will give only a sketch of the remainder of the proof. %Also, for convenience, we suppose $p$ is a fixed point.

Following, we can choose a positive integer $t$, and a rectangle $D_t$ in $E^{ss}\oplus E^{uu}\cap U_K$ containing the strong transversal homoclinic points, such that $g^{t}(D_{t})\cap D_{t}$ has $N$ disjoint connected components, and $t$ is the smallest
possible one such that $D_t$ is $A/2-C^1$ close to $E^{ss}(f_{4,N}^{K}(q))$ and $f_{4,N}^t(D_t)$ is $A/2-C^1$ close to the connected component of $W^{uu}(p)\cap U_K$ containing the strong transversal homoclinic points. Note, $t$ goes to infinity when $N$ goes to infinity.

Therefore, the maximal invariant set in the orbit of $D_t$ is a hyperbolic set $\Lambda(p,N)$  conjugated to a product of shift maps, with topological entropy $h_{top}(f_{4,N}|\Lambda(p,N))=\displaystyle\frac{1}{t}\log N$.

Provided the dynamics of $f_{4,N}$ is linear on $V$, %for any $N$, and $D_t\subset E^{ss}\oplus E^{uu}$,
we have the following result:

\begin{lemma}[Lemma 4.2 in \cite{CT} ]
For $A$ and $t$ defined as before, there exists a positive integer $K_1$ independent of $A$, such that $$ A<K_1 \max \{ \|Df_{4,N}^{-t}|E^{uu}(p)\|,\,
\|Df_{4,N}^{t}|E^{ss}(p)\|\}.$$ \label{afirma}\end{lemma}


Finally, given a positive integer $k$, we can choose $N$ large enough such that  %as Proposition \ref{proposi��o princial} requires.
by choice of $A$ and Lemma \ref{afirma} %we can choose $N$ large enough such that
$$
\displaystyle\frac{1}{t}\log N> \min\left\{\frac{1}{t}\log\|Df_{4,N}^{-t}|E^{uu}(p)\|^{-1},\, \frac{1}{t}\log\|Df_{4,N}^{t}|E^{ss}(p)\|^{-1} \right\} -\frac{1}{2k}.
$$
%It is clear that $t$ goes to infinity when $N$ goes to infinity.
Therefore, provided $f_{4,N}=f_3$ in $V$ for every $N$, when $t$ goes to infinity the above minimum converges to the smallest  positive Lyapunov exponent of $p$ restrict to $E^{ss}\oplus E^{uu}(p)$. Hence, since $dim\ E^{ss}\oplus E^{uu}=d-m+1$, the smallest  positive Lyapunov exponent of $p$ restrict to this subspace  is exactly equal to
$\log\sigma(Df_{4,N}|E_m^c(p)$. Thus, taking $g=f_{4,N}$ for $N$ large enough, we have that
$$
h_{top}(g|\Lambda(p,N))> \log\sigma(Dg|E_m^c(p))-\frac{1}{n}.
$$



In the general case, where $p$ is not a fixed point, we also can create a strong homoclinic intersection between $W^{ss}(p)$ and $W^{uu}(p)$, as before, and thus repeating the above arguments for $\tilde{f}=f^{\tau(p,f)}$ we can find a nice hyperbolic set $\tilde{\Lambda}(p,N)$ of a symplectomorphism $\tilde{g}=g^{\tau(p,f)}$, such that $g$ is $C^1$ close to $f$, satisfying
$$
h_{top}(\tilde{g}|\tilde{\Lambda}(p,N))> \log\sigma(D\tilde{g}|E_m^c(p)-\frac{1}{n}.
$$

Therefore the proposition is proved, since the hyperbolic set $$\Lambda(p,N)=\bigcup_{0\leq i< \tau(p,f)} g^i(\tilde{\Lambda}(p,N))$$ of $g$ has topological entropy:
$$
h_{top}(g|\Lambda(p,N))=\frac{1}{\tau(p,f)} h_{top}(\tilde{g}|\tilde{\Lambda}(p,N)).
$$

$\hfill\square$



%Considering  now $f_{5,N}=f_{4,N}^{2K}$ %{\color{red} AQUI TEM QUE COLOCAR O PER�ODO DO PONTO PARA GARANTIR A INVARI�NCIA, OU MANT�M O PONTO FIXO TAMB�M}
%and $\mathcal{D}=V\cap (E^{uu}\oplus E^{ss}(p))$ a symplectic surface of dimension $2(d-m+1)$, we have that $\mathcal{D}$ contains the hyperbolic periodic point $p$ and the orbits of the $N$ transversal strong homoclinic points  of $p$ in $U_K$. %Therefore,  $N$ is a symplectic surface from now on the arguments are ``2(d-m+1)-dimensional''.

%to find a symplectic diffeomorphism $f_{5,j}$ $C^1-$close to $f_3$,  having a hyperbolic set $\Lambda(p,j)\subset N$ such that  $h_{top}(f_{5,j}|\Lambda(p,j))$ is lower bounded for the smallest positive Lyapunov exponent of $p$ restrict to $N$. That is, provided $p$ is a fix point, % And therefore, since $N=V\cap (E_{n-m+1}^{uu}(p)\oplus E_{n-m+1}^{ss}(p))$, and $f_{5,j}|N=f_4|N=Df_3^{2K}|N$,
%we have that
%$$
%h_{top}(f_{5,j}|\Lambda(p,j))\geq \log|\sigma(Df_{5,j}^{-1}|E_{d-m+1}^{ss}(p))|-\frac{1}{j}.
%$$

%In the general case, when $\tau(p,f)>1$, we can find analogously the symplectic ``invariant'' surface $N$, and then TC arguments provide to find the diffeomorphism $f_{5,j}$ such that
%$$
%h_{top}(f_{5,j}|\Lambda(p,j))\geq \frac{1}{\tau(p,f_{5,j})}\log|\sigma(Df_{5,j}^{-\tau(p,f)}|E_{d-m+1}^{ss}(p))|-\frac{1}{j}.
%$$

%Therefore, the proof o proposition is finished  if we observe  that $$|\sigma(Df_{5,j}^{-\tau(p,f)}|E_{d-m+1}^{ss})|=|\sigma(Df_{5,j}^{\tau(p,f)}|E_{m}^{c})|.$$





































%Now, let $f_3$ $C^1$ close to $f_2$ as given by Lemma \ref{keypoint}. Considering $f_4=f_3^{2K}$ %{\color{red} AQUI TEM QUE COLOCAR O PER�ODO DO PONTO PARA GARANTIR A INVARI�NCIA, OU MANT�M O PONTO FIXO TAMB�M}
%and $N=V\cap (E^{uu}\oplus E^{ss}(p))$ a symplectic surface of dimension $2(d-m+1)$, we have that $N$ contains the hyperbolic periodic point $p$ and an orbit of an interval of strong homoclinic points of $p$ for $f_4$. More precisely, there exists an interval of strong homoclinic points $I$ of $p$ such that $f_4^j(I)\subset N$ for every integer $j$. Therefore, since $N$ is a symplectic surface from now on the arguments are ``2(d-m+1)-dimensional'', that is, as in Lemma \ref{simplectic} we can perform local perturbations of $f_4$ keeping the ``invariance'' of $f_4|N$.
%More precisely, for any positive integer $j$ we can reproduce the arguments of TC ( Step 3 in the proof of Proposition 3.1), which extend the arguments of Newhouse to higher symplectic dimension, to make a Newhouse's snake perturbation to find a symplectic diffeomorphism $f_{5,j}$ $C^1-$close to $f_3$,  having a hyperbolic set $\Lambda(p,j)\subset N$ such that  $h_{top}(f_{5,j}|\Lambda(p,j))$ is lower bounded for the smallest positive Lyapunov exponent of $p$ restrict to $N$. That is, provided $p$ is a fix point, % And therefore, since $N=V\cap (E_{n-m+1}^{uu}(p)\oplus E_{n-m+1}^{ss}(p))$, and $f_{5,j}|N=f_4|N=Df_3^{2K}|N$,
%we have that
%$$
%h_{top}(f_{5,j}|\Lambda(p,j))\geq \log|\sigma(Df_{5,j}^{-1}|E_{d-m+1}^{ss}(p))|-\frac{1}{j}.
%$$

%In the general case, when $\tau(p,f)>1$, we can find analogously the symplectic ``invariant'' surface $N$, and then TC arguments provide to find the diffeomorphism $f_{5,j}$ such that
%$$
%h_{top}(f_{5,j}|\Lambda(p,j))\geq \frac{1}{\tau(p,f_{5,j})}\log|\sigma(Df_{5,j}^{-\tau(p,f)}|E_{d-m+1}^{ss}(p))|-\frac{1}{j}.
%$$

%Therefore, the proof o proposition is finished  if we observe  that $$|\sigma(Df_{5,j}^{-\tau(p,f)}|E_{d-m+1}^{ss})|=|\sigma(Df_{5,j}^{\tau(p,f)}|E_{m}^{c})|.$$


%\bibliographystyle{plain}
%\bibliography{bib}
\begin{thebibliography}{99}

%\bibitem[ABC]{abc} F. Abdenur, C. Bonatti and S. Crovisier,  Nonunifom Hyperbolicity for $C^1-$generic diffeomorphisms, {\it  Israel J. Math.} 183, 1-60, (2011).



\bibitem[AM]{AM} A. Arbieto  and  Matheus C. A pasting lemma and some applications for conservative systems. {\it Ergodic Theory and Dynamical Systems}, 27, no. 5, (2007), 1399-1417.


\bibitem[A]{A} M-C. Arnaud. The generic symplectic $C^1$-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, {\it Ergodic Theory and Dynamical Systems} 22 (2002) 1621-1639, 2002.




\bibitem[ABC]{ArBC} M-C. Arnaud, C. Bonatti, and S. Crovisier. Dynamiques symplectiques g�n�riques, {\it Ergodic Theory and Dynamical Systems} 25 (2005) 1401-1436, 2010.


\bibitem[ABW]{ABW} A. �vila, J. Bochi, and A. Wilkinson. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms.  {\it Ann. Scient. �c. Norm. Sup.} 42 (2009) 931-979.


%\bibitem[AACS]{AACS} A. Arbieto, A. Armijo, T. Catalan and L. Senos. Symbolic Extensions and dominated splittings for Generic $C^1$-Diffeomorphisms. {\it Journal
%Mathematische Zeitschrift.} DOI 10.1007/s00209-013-1180-7, 2013.

%\bibitem[AC]{AC}  A. Arbieto and T. Catalan. Hyperbolicity in the Volume Preserving Scenario. {\it Ergodic Theory and Dynamical Systems}, Available on CJO2012 doi:10.1017/etds.2012.111




%\bibitem[A]{A} M. Asaoka, Hyperbolic Sets Exhibiting $C^1-$Persistent Homoclinic Tangency for Higher Dimensions, {\it Proceedings of the American
%Mathematical Society}, Volume 136, n� 2, Pages 677-686, , Feb. 2008.


\bibitem[BV]{BV} J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, {\it Modern Dynamical Systems and Applications}, Cambridge Univ. Press, Cambridge, (2004), 271?297.


\bibitem[BC]{BC} C. Bonatti and S. Crovisier. Recurrence et generecite. Inv. Math. 158 (2004), 33-104


%\bibitem[BCDG]{BCDG} C. Bonatti, S. Crovisier,  L. Diaz. and N. Gourmelon. Internal perturbations of homoclinic non-domination, cycles, and self-replication. \newblock {\em preprint}, arXiv: 1011.2935v1.

%\bibitem[BD1]{BD} C. Bonatti and L. Diaz. Abundance of $C^1-$robust homoclinic tangencies,
%\newblock {\em Trans. Amer. Math. Soc.}, 364(10):5111-5148, 2012.

%\bibitem[BD2]{BD2} C. Bonatti and L. Diaz. Persistent non-hyperbolic transitive diffeomorphisms,
%{\em Annals of Math.}, 143 (2): 367-396, 1996


%\bibitem[BDF]{BDF} C. Bonatti, L. Diaz. and T. Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. {\it Discrete Contin. Dyn. Syst.} 20 (2008), no. 3, 589-604.


%\bibitem[BDG]{BDG} C. Bonatti, L. Diaz and N. Gourmelon. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication, {\em preprint} {\tt arXiv:1011.2935}, 2010.

\bibitem[BDP]{BDP} C. Bonatti, L. Diaz, and E. Pujals. A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,  {\em Annals of Mathematics}, 158 (2003), 355-418.


%\bibitem[BoyDo]{BoD} M. Boyle, and T. Downarowicz. The entropy theory of symbolic extensions.
%{\em Invent. Math.}, 156, 119-161, (2004).

%\bibitem[Bu]{B} D. Burguet.  $C^2$ surface diffeomorphisms have symbolic
%extensions, {\em Inventiones Mathematicae}, 186:191-236, 2010.

%\bibitem[BF]{BF} D. Burguet and T. Fisher. Symbolic extensions for partially hyperbolic dynamical systems with 2-dimensional center bundle, {\em preprint}, 2011.


%\bibitem[Cat]{C} T. Catalan. A generic condition for existence of symbolic extension of volume preserving diffeomorphisms,  {\it Nonlinearity} 25 (2012) 3505-3525.

\bibitem[CT]{CT} T. Catalan and A. Tahzibi. A lower bound for topological entropy of generic non Anosov
symplectic diffeomorphisms, {\em Ergodic Theory and Dynamical Systems} {\tt DOI: http://dx.doi.org/10.1017/etds.2013.12}, (2013).

%\bibitem[Cr]{Cro} S. Crovisier. Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems, {\em Annals of Math.}, 172:1641-1677, 2010.

%\bibitem[CP]{CP} S. Crovisier and E. Pujals.  Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms , {\em preprint.}, 2010.


%\bibitem[CSY]{C-S-Y} S. Crovisier, M. Sambarino, and D. Yang. Partial Hyperbolicity and Homoclinic Tangencies, {\em to appear in Journal of the European Mathematical Society.}  {\tt arXiv:1103.0869}, 2011.



%\bibitem[DF]{DF} L. Diaz, T. Fisher, Symbolic Extensions and Partially Hyperbloic, {\it Discrete and Continuous Dynamical Systems} Volume 29, n� 4,
 %pp. 1419-1441, April 2011.

%\bibitem[DFPV]{DFPV} L. Diaz, T. Fisher, M. Pac\'{i}fico, and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms,  {\it Discrete Contin. Dyn. Syst.} 32 (2012), no. 12.


\bibitem[DW]{DW} D. Dolgopyat, A. Wilkinson.
Stable accessibility is $C^1$ dense.
Geometric methods in dynamics. II.
{\it Ast�risque } No. 287 (2003), xvii, 33-60.

%\bibitem[DN]{D-N} T. Downarowicz and S. Newhouse. Symbolic extensions and smooth dynamical systems, {\em  Inventiones Mathematicae}, 160 (2005), 453-499.




%\bibitem[LVY]{L-V-Y}  L. Gang, M. Viana and J. Yang. The Entropy Conjecture for Diffeomorphisms away from Tangencies, {\it to appear Journal of the E.M.S.} {\tt arXiv:1012.0514}, 2011.


%\bibitem[G]{G} N. Gourmelon. Generation of Homoclinic tangencies by $C^1$-perturbations,
%{\em  Discrete and continuous Dynamical Systems: Series A}, 26:1-42, 2010.

\bibitem[F]{F} Franks, J. Necessary conditions for stability of diffeomorphisms. Trans. A.M.S. 158
(1971), 301-308.


\bibitem[H]{H} S. Hayashi. Connecting Invariant Manifolds and the Solution of the $C^1$ Stability and $C^1$-Stability Conjectures for Flows. The Annals of Mathematics, Second Series, Vol. 145, No. 1 (Jan., 1997), pp. 81-137 .


\bibitem[HPS]{HPS} M. Hirsch, C. Pugh, M. Shub, Invariant manifolds. Bull. AMS 76 (1970), 1015-1019.


\bibitem[HT]{HT} V. Horita and A. Tahzibi. Partial hyperbolicity for symplectic diffeomorphisms,
{\it Ann. I. H. Poincar�}- 23 (2006), 641-661.


%\bibitem[Ma]{M} R. Ma��, Contributions to the stability conjecture, {\it Topology} 17, 383-396, 1978.


%\bibitem[MMP]{MMP} C. Matheus, C. G. Moreira, e E. Pujals. Axioma A versus Newhouse Phenomena for Benedicks-Carleson Toy Models. {\it preprint}, {\it arxiv:1004.5453} 2013.




%\bibitem[Mo]{Mo} C.G.Moreira, There are no $C^1$-stable intersections of regular Cantor sets, {
%\em Acta Math.}, 206 (2011), 311-323.

%\bibitem[NP]{NP} M. Nassiri and E.  Pujals.
%Robust transitivity in Hamiltonian dynamics. {\it
%Ann. Sci. �c. Norm. Sup�r. }(4) 45, no. 2, 191?239. 2012.




%\bibitem[N1]{Newhouse4} S. E. Newhouse. Diffeomorfisms with infinitely many sinks,
%{\em Topology}, 12:9-18, 1973.

\bibitem[N1]{N3} S. E. Newhouse. Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces,
\emph{ Soci�t� Math�matique de France, Ast�risque}, 51 (1978 ), 323-334.


\bibitem[N2]{N2}  S. E. Newhouse.  Quasi-eliptic periodic points in conservative dynamical systems,
\emph{ American Journal of Mathematics}, 99(5) (1975), 1061-1087.


%\bibitem[PV]{P-V} M. J. Pacifico, J. Vieitez, Robust entropy expansiveness implies generic domination. {\it  Nonlinearity} 23 1971-1990 (2010).



%\bibitem[PT]{palis takens} J. Palis  and  F. Takens. {\em Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations}, (Cambridge Studies in Advanced Mathematics), Cambridge University Press, 1993.


%\bibitem[P]{P} R. Potrie. Generic Bi-Lyapunov stable homoclinic classes, \emph{ Preprint}, 2010.



%\bibitem[PS]{PS2} E. Pujals  and M. Sambarino.  Homoclinic tangencies and hyperbolicity
%for surface diffeomorphisms, {\em Annals of Mathematics}, 151:961-1023, 2000.


%\bibitem[S1]{S} K. Sigmund. On mixing measures for Axiom A diffeomorphisms. {\it Proceedings of the American
%Math. Society.} Volume 36. No. 2, December 1972.

%\bibitem[S2]{S2} K. Sigmund. Generic properties of invariant measures for Axiom A diffeomorphisms. {\it  Inventiones Mathematicae} 11, 99?109, 1970.

\bibitem[XW]{XW}  L. Wen and Z. Xia, $C^1$ connecting lemmas, {\it Trans. A.M.S.}, Vol. 352, No. 11 (2000) pp. 5213-5230.

\bibitem[X]{X}  Z. Xia, Homoclinic Points in Symplectic and Volume-Preserving Diffeomorphisms, {\it Com. in Math. Physics}, 177, (1996) pp. 435-449.

%\bibitem [W]{W} L. Wen. Homoclinic tangencies and dominated splittings.{\it Nonlinearity,} 15, 1445-1469 (2002).
\end{thebibliography}



\bigskip

\flushleft


{\bf Thiago Catalan} (tcatalan\@@famat.ufu.br)\\
Faculdade de Matem\'{a}tica, FAMAT/UFU \\
Av. Jo\~ao Naves de Avila, 2121\\
38.408-100, Uberl\^andia,MG, Brazil 

\bigskip

\flushleft


{\bf Vanderlei Horita} (vhorita\@@ibilce.unesp.br)\\
Departamento de Matem\'{a}tica, IBILCE/UNESP \\
Rua Crist\'{o}v\~{a}o Colombo 2265\\
15054-000 S. J. Rio Preto, SP, Brazil









\end{document}

1310.5162.extracted_bib
\begin{thebibliography}{99}

%\bibitem[ABC]{abc} F. Abdenur, C. Bonatti and S. Crovisier,  Nonunifom Hyperbolicity for $C^1-$generic diffeomorphisms, {\it  Israel J. Math.} 183, 1-60, (2011).



\bibitem[AM]{AM} A. Arbieto  and  Matheus C. A pasting lemma and some applications for conservative systems. {\it Ergodic Theory and Dynamical Systems}, 27, no. 5, (2007), 1399-1417.


\bibitem[A]{A} M-C. Arnaud. The generic symplectic $C^1$-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, {\it Ergodic Theory and Dynamical Systems} 22 (2002) 1621-1639, 2002.




\bibitem[ABC]{ArBC} M-C. Arnaud, C. Bonatti, and S. Crovisier. Dynamiques symplectiques g�n�riques, {\it Ergodic Theory and Dynamical Systems} 25 (2005) 1401-1436, 2010.


\bibitem[ABW]{ABW} A. �vila, J. Bochi, and A. Wilkinson. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms.  {\it Ann. Scient. �c. Norm. Sup.} 42 (2009) 931-979.


%\bibitem[AACS]{AACS} A. Arbieto, A. Armijo, T. Catalan and L. Senos. Symbolic Extensions and dominated splittings for Generic $C^1$-Diffeomorphisms. {\it Journal
%Mathematische Zeitschrift.} DOI 10.1007/s00209-013-1180-7, 2013.

%\bibitem[AC]{AC}  A. Arbieto and T. Catalan. Hyperbolicity in the Volume Preserving Scenario. {\it Ergodic Theory and Dynamical Systems}, Available on CJO2012 doi:10.1017/etds.2012.111




%\bibitem[A]{A} M. Asaoka, Hyperbolic Sets Exhibiting $C^1-$Persistent Homoclinic Tangency for Higher Dimensions, {\it Proceedings of the American
%Mathematical Society}, Volume 136, n� 2, Pages 677-686, , Feb. 2008.


\bibitem[BV]{BV} J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, {\it Modern Dynamical Systems and Applications}, Cambridge Univ. Press, Cambridge, (2004), 271?297.


\bibitem[BC]{BC} C. Bonatti and S. Crovisier. Recurrence et generecite. Inv. Math. 158 (2004), 33-104


%\bibitem[BCDG]{BCDG} C. Bonatti, S. Crovisier,  L. Diaz. and N. Gourmelon. Internal perturbations of homoclinic non-domination, cycles, and self-replication. \newblock {\em preprint}, arXiv: 1011.2935v1.

%\bibitem[BD1]{BD} C. Bonatti and L. Diaz. Abundance of $C^1-$robust homoclinic tangencies,
%\newblock {\em Trans. Amer. Math. Soc.}, 364(10):5111-5148, 2012.

%\bibitem[BD2]{BD2} C. Bonatti and L. Diaz. Persistent non-hyperbolic transitive diffeomorphisms,
%{\em Annals of Math.}, 143 (2): 367-396, 1996


%\bibitem[BDF]{BDF} C. Bonatti, L. Diaz. and T. Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. {\it Discrete Contin. Dyn. Syst.} 20 (2008), no. 3, 589-604.


%\bibitem[BDG]{BDG} C. Bonatti, L. Diaz and N. Gourmelon. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication, {\em preprint} {\tt arXiv:1011.2935}, 2010.

\bibitem[BDP]{BDP} C. Bonatti, L. Diaz, and E. Pujals. A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,  {\em Annals of Mathematics}, 158 (2003), 355-418.


%\bibitem[BoyDo]{BoD} M. Boyle, and T. Downarowicz. The entropy theory of symbolic extensions.
%{\em Invent. Math.}, 156, 119-161, (2004).

%\bibitem[Bu]{B} D. Burguet.  $C^2$ surface diffeomorphisms have symbolic
%extensions, {\em Inventiones Mathematicae}, 186:191-236, 2010.

%\bibitem[BF]{BF} D. Burguet and T. Fisher. Symbolic extensions for partially hyperbolic dynamical systems with 2-dimensional center bundle, {\em preprint}, 2011.


%\bibitem[Cat]{C} T. Catalan. A generic condition for existence of symbolic extension of volume preserving diffeomorphisms,  {\it Nonlinearity} 25 (2012) 3505-3525.

\bibitem[CT]{CT} T. Catalan and A. Tahzibi. A lower bound for topological entropy of generic non Anosov
symplectic diffeomorphisms, {\em Ergodic Theory and Dynamical Systems} {\tt DOI: http://dx.doi.org/10.1017/etds.2013.12}, (2013).

%\bibitem[Cr]{Cro} S. Crovisier. Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems, {\em Annals of Math.}, 172:1641-1677, 2010.

%\bibitem[CP]{CP} S. Crovisier and E. Pujals.  Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms , {\em preprint.}, 2010.


%\bibitem[CSY]{C-S-Y} S. Crovisier, M. Sambarino, and D. Yang. Partial Hyperbolicity and Homoclinic Tangencies, {\em to appear in Journal of the European Mathematical Society.}  {\tt arXiv:1103.0869}, 2011.



%\bibitem[DF]{DF} L. Diaz, T. Fisher, Symbolic Extensions and Partially Hyperbloic, {\it Discrete and Continuous Dynamical Systems} Volume 29, n� 4,
 %pp. 1419-1441, April 2011.

%\bibitem[DFPV]{DFPV} L. Diaz, T. Fisher, M. Pac\'{i}fico, and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms,  {\it Discrete Contin. Dyn. Syst.} 32 (2012), no. 12.


\bibitem[DW]{DW} D. Dolgopyat, A. Wilkinson.
Stable accessibility is $C^1$ dense.
Geometric methods in dynamics. II.
{\it Ast�risque } No. 287 (2003), xvii, 33-60.

%\bibitem[DN]{D-N} T. Downarowicz and S. Newhouse. Symbolic extensions and smooth dynamical systems, {\em  Inventiones Mathematicae}, 160 (2005), 453-499.




%\bibitem[LVY]{L-V-Y}  L. Gang, M. Viana and J. Yang. The Entropy Conjecture for Diffeomorphisms away from Tangencies, {\it to appear Journal of the E.M.S.} {\tt arXiv:1012.0514}, 2011.


%\bibitem[G]{G} N. Gourmelon. Generation of Homoclinic tangencies by $C^1$-perturbations,
%{\em  Discrete and continuous Dynamical Systems: Series A}, 26:1-42, 2010.

\bibitem[F]{F} Franks, J. Necessary conditions for stability of diffeomorphisms. Trans. A.M.S. 158
(1971), 301-308.


\bibitem[H]{H} S. Hayashi. Connecting Invariant Manifolds and the Solution of the $C^1$ Stability and $C^1$-Stability Conjectures for Flows. The Annals of Mathematics, Second Series, Vol. 145, No. 1 (Jan., 1997), pp. 81-137 .


\bibitem[HPS]{HPS} M. Hirsch, C. Pugh, M. Shub, Invariant manifolds. Bull. AMS 76 (1970), 1015-1019.


\bibitem[HT]{HT} V. Horita and A. Tahzibi. Partial hyperbolicity for symplectic diffeomorphisms,
{\it Ann. I. H. Poincar�}- 23 (2006), 641-661.


%\bibitem[Ma]{M} R. Ma��, Contributions to the stability conjecture, {\it Topology} 17, 383-396, 1978.


%\bibitem[MMP]{MMP} C. Matheus, C. G. Moreira, e E. Pujals. Axioma A versus Newhouse Phenomena for Benedicks-Carleson Toy Models. {\it preprint}, {\it arxiv:1004.5453} 2013.




%\bibitem[Mo]{Mo} C.G.Moreira, There are no $C^1$-stable intersections of regular Cantor sets, {
%\em Acta Math.}, 206 (2011), 311-323.

%\bibitem[NP]{NP} M. Nassiri and E.  Pujals.
%Robust transitivity in Hamiltonian dynamics. {\it
%Ann. Sci. �c. Norm. Sup�r. }(4) 45, no. 2, 191?239. 2012.




%\bibitem[N1]{Newhouse4} S. E. Newhouse. Diffeomorfisms with infinitely many sinks,
%{\em Topology}, 12:9-18, 1973.

\bibitem[N1]{N3} S. E. Newhouse. Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces,
\emph{ Soci�t� Math�matique de France, Ast�risque}, 51 (1978 ), 323-334.


\bibitem[N2]{N2}  S. E. Newhouse.  Quasi-eliptic periodic points in conservative dynamical systems,
\emph{ American Journal of Mathematics}, 99(5) (1975), 1061-1087.


%\bibitem[PV]{P-V} M. J. Pacifico, J. Vieitez, Robust entropy expansiveness implies generic domination. {\it  Nonlinearity} 23 1971-1990 (2010).



%\bibitem[PT]{palis takens} J. Palis  and  F. Takens. {\em Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations}, (Cambridge Studies in Advanced Mathematics), Cambridge University Press, 1993.


%\bibitem[P]{P} R. Potrie. Generic Bi-Lyapunov stable homoclinic classes, \emph{ Preprint}, 2010.



%\bibitem[PS]{PS2} E. Pujals  and M. Sambarino.  Homoclinic tangencies and hyperbolicity
%for surface diffeomorphisms, {\em Annals of Mathematics}, 151:961-1023, 2000.


%\bibitem[S1]{S} K. Sigmund. On mixing measures for Axiom A diffeomorphisms. {\it Proceedings of the American
%Math. Society.} Volume 36. No. 2, December 1972.

%\bibitem[S2]{S2} K. Sigmund. Generic properties of invariant measures for Axiom A diffeomorphisms. {\it  Inventiones Mathematicae} 11, 99?109, 1970.

\bibitem[XW]{XW}  L. Wen and Z. Xia, $C^1$ connecting lemmas, {\it Trans. A.M.S.}, Vol. 352, No. 11 (2000) pp. 5213-5230.

\bibitem[X]{X}  Z. Xia, Homoclinic Points in Symplectic and Volume-Preserving Diffeomorphisms, {\it Com. in Math. Physics}, 177, (1996) pp. 435-449.

%\bibitem [W]{W} L. Wen. Homoclinic tangencies and dominated splittings.{\it Nonlinearity,} 15, 1445-1469 (2002).
\end{thebibliography}