# A note on the density of periodic orbits of Anosov geodesic flow in
manifolds of finite volume

Nestor Nina Zarate and Sergio Romaña

###### Abstract

In this paper, we prove that manifolds of finite volume with Anosov geodesic
flow have dense periodic orbits. The same result works for conservative Anosov
flows in non-compact cases.

> Keywords: Geodesic Flow, Shadowing Lemma, Stable(Unstable) Manifolds.

Presentation: This article was organized by Sergio Romaña as a tribute to the
author Nestor Nina Zarate, a young Bolivian mathematician, with training in
Bolivia and Brazil, who left us too soon, but who left us with his valuable
contribution to mathematical sciences. This article is dedicated to his memory
and his loved ones, Bonifacia Zarate, Ernesto Nina, Rosemary Nina Zarate,
Rocio Nina Zarate, friends from Bolivia and Brazil, and especially friends
from UFRJ. Thank you Nestor, we will all always remember you.

##  1 Introduction

Let $N$ be a complete Riemannian manifold, a flow $\varphi^{t}\colon N\to N$
is said to be _Anosov_ if the tangent bundle of $N$, $TN$, has a splitting
$TN=E^{s}\oplus\langle\Phi\rangle\oplus E^{u}$ such that

$\displaystyle d\varphi^{t}_{\theta}(E^{s}(\theta))$ $\displaystyle=$
$\displaystyle E^{s}(\varphi^{t}(\theta)),$ $\displaystyle
d\varphi^{t}_{\theta}(E^{u}(\theta))$ $\displaystyle=$ $\displaystyle
E^{u}(\varphi^{t}(\theta)),$
$\displaystyle||d\varphi^{t}_{\theta}\big{|}_{E^{s}}||$ $\displaystyle\leq$
$\displaystyle C\lambda^{t},$
$\displaystyle||d\varphi^{-t}_{\theta}\big{|}_{E^{u}}||$ $\displaystyle\leq$
$\displaystyle C\lambda^{t},$

for all $t\geq 0$ with $C>0$ and $0<\lambda<1$, where $\Phi$ is the vector
field derivative of the flow.

If $M$ is a complete Riemannanian manifold, we denote $SM$ as its unit tangent
bundle. Given $\theta=(p,v)\in SM$, we define $\gamma_{{}_{\theta}}(t)$ as the
unique geodesic with initial conditions $\gamma_{{}_{\theta}}(0)=p$ and
$\gamma_{{}_{\theta}}^{\prime}(0)=v$. For $t\in\mathbb{R}$, let
$\phi^{t}:SM\to SM$ be the diffeomorphism given by
$\phi^{{t}}(\theta)=(\gamma_{{}_{\theta}}(t),\gamma_{{}_{\theta}}^{\prime}(t))$.
Recall that this family is a flow (called the geodesic flow) in the sense that
$\phi^{t+s}=\phi^{t}\circ\phi^{s}$ for all $t,s\in\mathbb{R}$.

Geodesic flows provide a wide range of examples of Anosov flows, for example,
the geodesic flow of manifolds with negative pinched curvature (see [An69] and
[Kn02]. Eberlein in [Eb73] found in manifolds without conjugate points,
necessary and sufficient geometrical conditions such that the geodesic flow is
Anosov. Some geometrical conditions to obtain an Anosov geodesic flow can be
also found in [MR20] and [CR23].

It is not difficult to prove that Anosov geodesic flows on compact manifolds
have a density of periodic orbits. The knowledge of the distribution and
density of periodic orbits of Anosov flows and more generally hyperbolic flows
is a very interesting problem because it gives us more information about the
dynamics of the system. A proof for the density of the periodic orbits of a
hyperbolic flow over one compact manifold is using the Spectral Decomposition
Theorem for flows (see [AK95] and [FH20]). Moreover, the information about the
density of periodic orbits of a hyperbolic flow over a compact manifold also
guarantees the topological transitivity of the flow, among other good
properties.

However, for Anosov geodesic flow on a manifold of finite volume, this result
is not easy to see, since some techniques used for the compact case maybe are
not valid in the non-compact case, for example, the uniformly size of local
stable and unstable manifolds. Everyone believes in this result, even already
used this result, but we do not find concrete and clear proof. Thus, the main
result of this work is to give a formal proof of this fact.

######  Theorem 1.1.

If $M$ has finite volume and $\phi^{t}:SM\to SM$ is an Anosov geodesic flow,
then the periodic orbits of $\phi^{t}$ are dense in $SM$.

With the same arguments, we can announce

######  Theorem 1.2.

If $\varphi^{t}:N\to N$ is an Anosov flow, which preserves a finite smooth
measure, then the periodic orbits of $\varphi^{t}$ are dense in $N$.

Those results imply the following corollary (valid also for conservative
Anosov flow).

######  Corollary 1.3.

If $M$ has finite volume such that $\phi^{t}:SM\to SM$ is an Anosov geodesic
flow, then

  * _(a)_

For all $x\in SM$, $W^{cu}(x)$ and $W^{cs}(x)$ are dense in $SM$.

  * _(b)_

$\phi^{t}$ is a transitive flow.

##  2 Preliminaries

###  2.1 Stable and Unstable Manifolds

Assume that $\varphi^{t}:N\to N$ is an Anosov flow. Given $x\in N$, we define
the stable, unstable, center stable and center unstable manifolds,
respectively, as

$\displaystyle W^{ss}(x)$ $\displaystyle=$ $\displaystyle\\{z\in
N:\lim_{t\to+\infty}d(\varphi^{t}(z),\varphi^{t}(x))=0\\},$ $\displaystyle
W^{uu}(x)$ $\displaystyle=$ $\displaystyle\\{z\in
N:\lim_{t\to+\infty}d(\varphi^{-t}(z),\varphi^{-t}(x))=0\\},$ $\displaystyle
W^{cs}(x)$ $\displaystyle=$
$\displaystyle\bigcup_{t\in\mathbb{R}}W^{ss}(\varphi^{t}(x)),$ $\displaystyle
W^{cu}(x)$ $\displaystyle=$
$\displaystyle\bigcup_{t\in\mathbb{R}}W^{uu}(\varphi^{t}(x)).$

Given $\eta>0$, we also consider the local stable and unstable manifolds
defined by

$\displaystyle W_{\eta}^{ss}(x)$ $\displaystyle=$ $\displaystyle\\{z\in
W^{ss}(x):d(\varphi^{t}(x),\varphi^{t}(z))\leq\eta,\,\,\,t\geq 0\\},$
$\displaystyle W_{\eta}^{uu}(x)$ $\displaystyle=$ $\displaystyle\\{z\in
N:d(\varphi^{-t}(x),\varphi^{-t}(z))\leq\eta,\,t\geq 0\\},$ $\displaystyle
W_{\eta}^{cu}(x)$ $\displaystyle=$
$\displaystyle\bigcup_{|t|<\eta}W^{uu}_{\eta}(\phi^{t}(x)).$ $\displaystyle
W_{\eta}^{cs}(x)$ $\displaystyle=$
$\displaystyle\bigcup_{|t|<\eta}W^{ss}_{\eta}(\phi^{t}(x)).$

When $\varphi^{t}\colon N\to N$ is a $C^{r}$-Anosov flow and $N$ is a compact
manifold, the stable and unstable manifolds theorem states that there exists
$\epsilon>0$ such that the local stable and unstable manifolds are
$C^{r}$-disk embedded tangent to the stable and unstable bundle, respectively.
However, the construction of the differentiability of the local stable-
unstable manifolds depends on the existence of some special charts depending
on the injectivity radius of manifold $M$ (see [AK95] for more details).

In the non-compact case, we can imitate the proof of the stable and unstable
manifold theorem to construct the differentiability of the local stable and
unstable manifolds on each point $x\in SM$. Thant can be guarantee since in
the Sasaki’s metric on $SM$ the injectivity radius is positive (cf. [El12,
page 41]). Thus we have a crucial difference with the compact case because in
the non-compact case, we have that the sizes of the local stable-unstable
manifolds depend on $x\in SM$. In other words, we have

######  Theorem 2.1.

_[Stable and Unstable Manifolds in the Non-Compact Case]_
Let $\phi^{t}:SM\to SM$ be a $C^{r}$\- Anosov geodesic flow, $r\geq 1$. Then,
for each $x\in SM$ there is a pair of embedded $C^{r}$-disks
$W_{\epsilon(x)}^{ss}$, $W_{\epsilon(x)}^{uu}\subset N$ called the local
strong-stable and local strong-unstable manifolds of $x\in N$ of size
$\epsilon(x)>0$, respectively such that:

  1. _(1)_

$T_{x}W_{\epsilon(x)}^{ss}(x)=E_{x}^{s}$,
$T_{x}W_{\epsilon(x)}^{uu}(x)=E_{x}^{u}$.

  2. _(2)_

For every compact subset $K\subset M$ there exists
$\epsilon_{K}:=\inf\\{\epsilon(x)>0:x\in K\\}>0$ such that for every $x\in K$
we have that $W_{\epsilon_{{}_{K}}}^{ss}(x)$ and
$W_{\epsilon_{{}_{K}}}^{uu}(x)$ are embedded.

  3. _(3)_

$W_{\epsilon(x)}^{ss}(x)\cap W_{\epsilon(x)}^{uu}(x)=\\{x\\}$.

###  2.2 Local Product Structure

If $\phi^{t}:N\to N$ is an Anosov flow and $N$ is compact, then there are
$\epsilon>0$ and $\delta>0$ such that $d(x,y)<\delta$

$W_{\epsilon}^{cu}(x)\cap W_{\epsilon}^{ss}(y):=\\{[x,y]\\},$

which is called the _Bowen-Bracket_ or also _Local Product Structure_.
The Bowen-Bracket can be defined in the non-compact case, but the size of the
local manifolds involved is not uniform. More specifically,

######  Theorem 2.2 (Local Product Structure ).

Let $\phi^{t}:N\to N$ be an Anosov flow and $\epsilon>0$ small enough. Then
for each $x\in N$ there is a $\delta>0$ and $\eta>0$ such that

$W_{\eta}^{cu}(y)\cap W_{\eta}^{ss}(z):=\\{[y,z]\\}\in B(x,\epsilon),$

for $y,z\in\overline{B(x,\delta)}$. Analogous result holds for
$W_{\eta}^{cs}(y)$ and $W_{\eta}^{uu}(z)$.

The proof of the last theorem is similar to the compact case, see [FH20] for
more details.

##  3 Shadowing Lemma

The orbit structure of a hyperbolic dynamical system has a distinctive and
iconic richness and complexity, and these features can be derived from what
thereby appears as a core feature of hyperbolic dynamics: the shadowing of
orbits. In this section, we remember the Shadowing Lemma for flows and the
expansivity in the compact case (See [FH20] and [AK95]). We also will give
some definitions of the shadowing and expansiveness that we need to prove the
density of periodic orbits when we have a geodesic flow on a finite volume
manifold.

Let $x\in N$ and $t_{0}>0$. We consider the segment of orbit
$\varphi^{[0,t_{0}]}(x):=\\{\phi^{t}(x):t\in[0,t_{0}]\\}$ of $x$ of size
$t_{0}$, the future orbit $O^{+}(x):=\\{\phi^{t}(x):t\geq 0\\}$ of $x$, and
the past orbit $O^{-}(x):=\\{\phi^{t}(x):t\leq 0\\}$.

######  Definition 3.1 (Shadowing forward and backward).

Let $x_{0},y\in SM$, $t_{0}>0$ and $\epsilon>0$.

  1. _1._

We said that $O^{+}(y)$ $\epsilon$-shades forward by piecewise the orbit
segment $\phi^{[0,t_{0}]}(x_{0})$ if

$\displaystyle d(\phi^{t}(y),\phi^{t}(x_{0}))\leq\epsilon,\hskip
8.5359pt\forall\hskip 5.69046ptt\in[0,t_{0}],$

and there exists a sequence $\\{s_{j}\\}_{j\geq 0}$ with $s_{j}\geq 0$
$($called _transition times_ $)$, such that for all $k\geq 1$ holds

$\displaystyle
d(\phi^{t}(\phi^{kt_{0}+\sum_{i=0}^{k-1}s_{i}}(y)),\phi^{t}(x_{0}))\leq\epsilon,\hskip
8.5359pt\forall\hskip 5.69046ptt\in[0,t_{0}].$

  2. _2._

Analogously, we said that $O^{-}(y)$ $\epsilon$-shades backward by piecewise
the orbit segment $\phi^{[0,t_{0}]}(x_{0})$ if

$\displaystyle d(\phi^{-t}(y),\phi^{-t}(x_{0}))\leq\epsilon,\hskip
8.5359pt\forall\hskip 5.69046ptt\in[0,t_{0}],$

and there exists a sequence $\\{r_{j}\\}_{j\geq 0}$ with $r_{j}\geq 0$
$($called _transition times_ $)$ such that for all $k\geq 1$ holds

$\displaystyle
d(\phi^{-t}(\phi^{-kt_{0}-\sum_{j=0}^{k-1}r_{j}}(y)),\phi^{-t}(x_{0}))\leq\epsilon,\hskip
8.5359pt\forall\hskip 5.69046ptt\in[0,t_{0}].$

We said that the orbit $O(y)=O^{+}(y)\cup O^{-}(y)$ $\epsilon$-shades by
piecewise the orbit segment $\phi^{[0,t_{0}]}(x_{0})$ if $O^{+}(y)$
$\epsilon$-shades forward by piecewise and $O^{-}(y)$ $\epsilon$-shades
backward by piecewise, respectively.

The above definition of shadowing forward and backward by piecewise is similar
to the specification property for flows (see [FH20]).

######  Definition 3.2.

Let $\phi^{t}:N\to N$ be a flow, a point $x\in N$ is a _recurrent point_ if
for all neighborhood $U$ of $x$ there is $t\in\mathbb{R}$ such that
$\phi^{t}(U)\cap U\neq\emptyset.$

Note that if $\phi^{t}$ is a conservative Anosov flow, then almost every point
on $N$ is recurrent. In particular, if $\phi^{t}$ a geodesic flow of a
manifold $M$ of finite volume, then almost every point of $SM$ is recurrent.
We denote the recurrent set of $\phi^{t}$ as $\text{Rec}(\phi^{t})$.

In the next proposition, we prove that segments of the orbit of a recurrent
point of $\phi^{t}$ can be $\epsilon$-shadowed by piecewise. For this sake, we
need the following general lemma.

Given $t_{0}>0$ and a sequence $\underline{c}:=\\{c_{j}\\}$, we consider the
family of real functions $P_{m,\underline{c}}\colon[0,t_{0}]\to\mathbb{R}$,
define by

$P_{m,\underline{c}}(t):=\lambda^{t}+\lambda^{t_{0}-t}+\sum_{j=1}^{m}\lambda^{t+jt_{0}+\sum_{i=1}^{j}c_{i}},$
(1)

where $\lambda\in(0,1)$ comes from the definition of Anosov flow.

######  Lemma 3.3.

Given $\eta>0$, $t_{0}>2\eta$, then the family of functions
$P_{m,\underline{c}}$, with $|c_{i}|<\eta$ is uniformly bounded.

###### Proof.

For all $m\in\mathbb{N}$, consider the sequence

$S_{m,\underline{c}}:=\sum_{j=1}^{m}\lambda^{jt_{0}+\sum_{i=1}^{j}c_{i}}.$

Since $|c_{i}|<\eta$, then for all $j\in\mathbb{N}$ we have

$t_{0}-\eta<t_{0}+\dfrac{1}{j}\sum_{i=1}^{j}c_{i}<t_{0}+\eta.$

Moreover, since $t_{0}>2\eta$, then for all $j\in\mathbb{N}$ we have

$0<\dfrac{t_{0}}{2}<t_{0}-\eta<t_{0}+\dfrac{1}{j}\sum_{i=1}^{j}c_{i}.$

Thus,

$\left(\lambda^{(t_{0}+\frac{1}{j}\sum_{i=1}^{j}c_{i})}\right)^{j}\leq({\lambda}^{t_{0}-\eta})^{j}\leq(\lambda^{\frac{t_{0}}{2}})^{j},$

which implies that, for all $m\in\mathbb{N}$.

$\displaystyle
S_{m,\underline{c}}=\sum_{j=1}^{m}\left(\lambda^{t_{0}+\frac{1}{j}\sum_{i=1}^{j}c_{i}}\right)^{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}(\lambda^{\frac{t_{0}}{2}})^{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}(\lambda^{\frac{t_{0}}{2}})^{j}$
$\displaystyle=$
$\displaystyle\dfrac{\lambda^{\frac{t_{0}}{2}}}{1-\lambda^{\frac{t_{0}}{2}}}=L.$

Note that
$P_{m,\underline{c}}(t)=\lambda^{t}+\lambda^{t_{0}-t}+\lambda^{t}S_{m,\underline{c}}$.
Since $0<\lambda<1$, then $0<\lambda^{t}\leq 1$ and $0<\lambda^{t_{0}-t}\leq
1$, for each $t\in[0,t_{0}]$, thus

$P_{m,\underline{c}}(t)\leq 2+L:=K,$

and the proof of the lemma is complete. ∎

######  Proposition 3.4.

Let $\phi^{t}:N\to N$ be an Anosov flow. Fix $x_{0}\in\emph{Rec}(\phi^{t})$.
For all $\epsilon>0$ there exists $t_{0}>0$ and $y\in SM$ such that $O(y)$
$\epsilon$-shades by piecewise the segment of orbit $\phi^{[0,t_{0}]}(x_{0}).$

###### Proof.

Given $\epsilon>0$, take $\delta>0$ and $\eta>0$ from Theorem 2.2. Since
$x_{0}\in\text{Rec}(\phi^{t})$ then consider $t_{0}\in\mathbb{R}$ such that
$\phi^{t_{0}}(x_{0})\in B(x_{0},\delta/3)$ and $t_{0}>2\eta$.

Consider the real function
$f(t)=\dfrac{\epsilon(1-\lambda^{\frac{t}{2}})}{2-\lambda^{\frac{t}{2}}}$.
Observe that $\displaystyle\lim_{t\to+\infty}f(t)=\frac{\epsilon}{2}$, then
given $\epsilon>0$ we can choose $\delta>0$, $\eta>0$ (as in Theorem 2.2,
which depends on $x_{0}$) and $t_{0}>2\eta$ such that
$x_{1}:=\phi^{t_{0}}(x_{0})\in B(x_{0},\frac{\delta}{3})$ and

$\displaystyle{C}{\eta}<\frac{\epsilon}{4}<f(t_{0})=\frac{\epsilon}{3K},\,\,\,\,C\lambda^{t_{0}}\eta<\delta/3,\,\,\
\text{and}\,\,\,\frac{\delta}{3}<\epsilon$ (2)

where $C$ is the constant on the definition of Anosov flow.

From the local product structure (see Theorem 2.2) there is $z_{1}\in
W_{\eta}^{cu}(x_{1})\pitchfork W_{\eta}^{ss}(x_{0})$, and consequently
$\theta_{1}\in W_{\eta}^{uu}(x_{1})$ such that

$\phi^{s_{1}}(\theta_{1})=z_{1},\,\,\text{for some}\,\,|s_{1}|<\eta$

and

$d(\phi^{-t}(\theta_{1}),\phi^{-t}(x_{1}))\leq C\lambda^{t}\eta,\,\,\ t\geq
0.$

Take $y_{1}=\phi^{-t_{0}}(\theta_{1})$, then for all $0\leq t\leq t_{0}$ we
have

$\displaystyle d(\phi^{t}(y_{1}),\phi^{t}(x_{0}))$ $\displaystyle=$
$\displaystyle
d(\phi^{t-t_{0}}(\theta_{1}),\phi^{t-t_{0}}(\phi^{t_{0}}(x_{0})))$ (3)
$\displaystyle=$ $\displaystyle
d(\phi^{t-t_{0}}(\theta_{1}),\phi^{t-t_{0}}(x_{1}))$ $\displaystyle\leq$
$\displaystyle C\lambda^{t_{0}-t}\eta.$

From (2) and (3), we have that $d(y_{1},x_{0})\leq
C\lambda^{t_{0}}\eta<\delta/3$, this implies $y_{1}\in B(x_{0},\delta/3)$,
since $x_{1}\in B(x_{0},\delta/3)$, then $d(x_{1},y_{1})<\delta$. Therefore,
by the local product structure (see Theorem 2.2) there is $z_{2}\in
W_{\eta}^{cu}(x_{1})\pitchfork W_{\eta}^{ss}(y_{1})$. So, we take
$\theta_{2}\in W_{\eta}^{uu}(x_{1})$ such that

$\phi^{s_{2}}(\theta_{2})=z_{2},\,\,\text{for some}\,\,|s_{2}|<\eta$

and

$d(\phi^{-t}(\theta_{2}),\phi^{-t}(x_{1}))\leq C\lambda^{t}\eta,\,\,\,\,\,\,\
d(\phi^{t}(z_{2}),\phi^{t}(y_{1}))\leq C\lambda^{t}\eta,\,\,\ t\geq 0.$ (4)

Consider $y_{2}:=\phi^{-t_{0}}(\theta_{2})$. Then,
$\phi^{t_{0}+s_{2}}(y_{2})=\phi^{s_{2}}(\theta_{2})=z_{2}$. Moreover, for all
$0\leq t\leq t_{0}$ we have

$d(\phi^{t}(y_{2}),\phi^{t}(x_{0}))=d(\phi^{t-t_{0}}(\theta_{2}),\phi^{t-t_{0}}(x_{1}))\leq
C\lambda^{t_{0}-t}\eta.$ (5)

From (3) and (4) we obtain

$\displaystyle d(\phi^{t}(z_{2}),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle
d(\phi^{t}(z_{2}),\phi^{t}(y_{1}))+d(\phi^{t}(y_{1}),\phi^{t}(x_{0}))$ (6)
$\displaystyle\leq$ $\displaystyle
C\lambda^{t}\eta+C\lambda^{t_{0}-t}\eta=C(\lambda^{t}+\lambda^{t_{0}-t})\eta.$

From (2) and (5) we have that $d(y_{2},x_{0})\leq
C\lambda^{t_{0}}\eta<\delta/3$. Thus, $y_{2},x_{1}\in B(x_{0},\delta/3)$ and
again by local product structure (see Theorem 2.2) there is $z_{3}\in
W_{\eta}^{cu}(x_{1})\pitchfork W_{\eta}^{ss}(y_{2})$. So, we take
$\theta_{3}\in W_{\eta}^{uu}(x_{1})$ such that

$\phi^{s_{3}}(\theta_{3})=z_{3},\,\,\text{for some}\,\,|s_{3}|<\eta$

and

$d(\phi^{-t}(\theta_{3}),\phi^{-t}(x_{1}))\leq C\lambda^{t}\eta,\,\,\,\,\,\,\
d(\phi^{t}(z_{3}),\phi^{t}(y_{2}))\leq C\lambda^{t}\eta,\,\,\ t\geq 0.$ (7)

Considering $y_{3}=\phi^{-t_{0}}(\theta_{3})$, for all $0\leq t\leq t_{0}$ we
have

$\displaystyle d(\phi^{t}(y_{3}),\phi^{t}(x_{0}))$ $\displaystyle=$
$\displaystyle d(\phi^{t-t_{0}}(\theta_{3}),\phi^{t-t_{0}}(x_{1}))$ (8)
$\displaystyle\leq$ $\displaystyle C\lambda^{t_{0}-t}\eta.$

From (5) and (11) we get

$\displaystyle d(\phi^{t}(z_{3}),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle
d(\phi^{t}(z_{3}),\phi^{t}(y_{2}))+d(\phi^{t}(y_{2}),\phi^{t}(x_{0}))$ (9)
$\displaystyle\leq$ $\displaystyle
C\lambda^{t}\eta+C\lambda^{t_{0}-t}\eta=C(\lambda^{t}+\lambda^{t_{0}-t})\eta.$

Since $\phi^{t_{0}+s_{2}}(y_{2})=z_{2}$, by (6) and (7) we get

$\displaystyle d(\phi^{t}(\phi^{t_{0}+s_{2}}(z_{3})),\phi^{t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
d(\phi^{t}(\phi^{t_{0}+s_{2}}(z_{3}))),\phi^{t}(\phi^{t_{0}+s_{2}}(y_{2})))+d(\phi^{t}(z_{2}),\phi^{t}(x_{0}))$
(10) $\displaystyle\leq$ $\displaystyle
C\lambda^{t+t_{0}+s_{2}}\eta+C(\lambda^{t}+\lambda^{t_{0}-t})\eta$
$\displaystyle=$ $\displaystyle
C(\lambda^{t+t_{0}+s_{2}}+\lambda^{t}+\lambda^{t_{0}-t})\eta$

One more time, from (2) and (8), $y_{3},x_{1}\in B(x_{0},\delta/3)$ and from
Theorem 2.2 there is $z_{4}\in W_{\eta}^{cu}(x_{1})\pitchfork
W_{\eta}^{ss}(y_{3})$. Thus, we take $\theta_{4}\in W_{\eta}^{uu}(x_{1})$ such
that

$\phi^{s_{4}}(\theta_{4})=z_{4},\,\,\text{for some}\,\,|s_{4}|<\eta$

and

$d(\phi^{-t}(\theta_{4}),\phi^{-t}(x_{1}))\leq C\lambda^{t}\eta,\,\,\,\,\,\,\
d(\phi^{t}(z_{4}),\phi^{t}(y_{3}))\leq C\lambda^{t}\eta,\,\,\ t\geq 0.$ (11)

Considering $y_{4}=\phi^{-t_{0}}(\theta_{4})$ and proceeding as we did in the
case of $y_{1}$, $y_{2}$ and $y_{3}$, we get that for all $0\leq t\leq t_{0}$
holds

$\displaystyle d(\phi^{t}(y_{4}),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle C\lambda^{t_{0}-t}\eta$ $\displaystyle
d(\phi^{t}(z_{4}),\phi^{t}(x_{0}))$ $\displaystyle\leq$ $\displaystyle
C(\lambda^{t}+\lambda^{t_{0}-t})\eta$ $\displaystyle
d(\phi^{t}(\phi^{t_{0}+s_{3}}(z_{4})),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle C(\lambda^{t+t_{0}+s_{3}}+\lambda^{t}+\lambda^{t_{0}-t})\eta$
$\displaystyle d(\phi^{t}(\phi^{2t_{0}+s_{2}+s_{3}}(z_{4})),\phi^{t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
C(\lambda^{t+2t_{0}+s_{2}+s_{3}}+\lambda^{t+t_{0}+s_{2}}+\lambda^{t_{0}-t}+\lambda^{t})\eta.$

Following this process inductively, we get four sequence $\\{y_{n}\\}_{n\geq
0}$, $\\{\theta_{n}\\}_{n\geq 0}$, $\\{z_{n}\\}_{n\geq 0}$ on $N$ and
$\\{s_{n}\\}$ on $\mathbb{R}$ such that

$z_{n}\in W_{\eta}^{cu}(x_{1})\pitchfork W_{\eta}^{ss}(y_{n-1})$
$y_{n}=\varphi^{-t_{0}}(\theta_{n})$, $\theta_{n}\in W_{\eta}^{uu}(x_{1})$,
and $\phi^{s_{n}}(\theta_{n})=z_{n}$ with $|s_{n}|<\eta$. Moreover, for $0\leq
t\leq t_{0}$

$\displaystyle d(\phi^{t}(y_{n}),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle C\lambda^{t_{0}-t}\eta$ $\displaystyle
d(\phi^{t}(\phi^{t_{0}+s_{n}}(y_{n})),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle C(\lambda^{t_{0}-t}+\lambda^{t})\eta$ (12) $\displaystyle
d(\phi^{t}(\phi^{2t_{0}+\sum_{i=0}^{1}s_{n-i}}(y_{n})),\phi^{t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
C(\lambda^{t_{0}-t}+\lambda^{t}+\sum_{j=1}^{1}\lambda^{t+jt_{0}+\sum_{i=0}^{j-1}s_{(n-1)+i}})\eta$
$\displaystyle
d(\phi^{t}(\phi^{3t_{0}+\sum_{i=0}^{2}s_{n-i}}(y_{n})),\phi^{t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
C(\lambda^{t_{0}-t}+\lambda^{t}+\sum_{j=1}^{2}\lambda^{t+jt_{0}+\sum_{i=0}^{j-1}s_{(n-2)+i}})\eta$
$\displaystyle\vdots$ $\displaystyle
d(\phi^{t}(\phi^{(n-1)t_{0}+\sum_{i=0}^{n-2}s_{n-i}}(y_{n})),\phi^{t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
C(\lambda^{t_{0}-t}+\lambda^{t}+\sum_{j=1}^{n-2}\lambda^{t+jt_{0}+\sum_{i=0}^{j-1}s_{2+i}})\eta.$
(13)

Considering the sequence $\underline{s}=\\{s_{j}\\}$, then the right side of
all last inequalities is exactly $C\cdot P_{k,\underline{s}}(t)\cdot\eta$, for
$k\in\\{1,\dots,n-2\\}$.

Thus, since $t_{0}>2\eta$ and $|s_{i}|<\eta$, then for $0\leq t\leq t_{0}$,
Lemma 3.3 and (2) provide

$\displaystyle d(\phi^{t}(y_{n}),\phi^{t}(x_{0}))\leq CK\eta<\epsilon/3,$ (14)

and

$\displaystyle
d(\phi^{t}(\phi^{jt_{0}+\sum_{i=0}^{j-1}s_{n-i}}(y_{n})),\phi^{t}(x_{0}))\leq
CK\eta<\epsilon/3.$ (15)

for each $j=1,2,...,n-1$.

Taking $t=0$ in the first inequality of (3) we have $d(y_{n},x_{0})\leq
C\lambda^{t_{0}}\eta<\delta/3$. This implies $y_{n}\in
B(x_{0},\delta/3)\subset B[x_{0},\delta/3]\subset B(x_{0},\epsilon)$, then
passing to a subsequence of $\\{y_{n}\\}_{n\geq 0}$ if necessary, there is
$y\in B(x_{0},\epsilon)$ such that $\displaystyle\lim_{n\to+\infty}y_{n}=y$.
Thus, from (14) and the continuity of the flow

$\displaystyle
d(\phi^{t}(y),\phi^{t}(x_{0}))=\lim_{n\to+\infty}d(\phi^{t}(y_{n}),\varphi^{t}(x_{0}))\leq\epsilon/3,\hskip
5.69046pt\forall\hskip 2.84544pt0\leq t\leq t_{0}.$

As $|s_{n}|<\eta$, for all $n\in\mathbb{N}$, then again passing to subsequence
if necessary, there is $s\in\mathbb{R}$ such that
$\displaystyle\lim_{n\to+\infty}s_{n}=s$ and $|s|\leq\eta$. Moreover, note
that for all $n\in\mathbb{N}$, $\phi^{t_{0}+s_{n}}(y_{n})=z_{n}$, then by
continuity of flow,
$\displaystyle\lim_{n\to+\infty}z_{n}=\lim_{n\to+\infty}\phi^{t_{0}+s_{n}}(y_{n})=\phi^{t_{0}+s}(y):=z$.
So, from (12),

$\displaystyle
d(\phi^{t}(\phi^{t_{0}+s}(y)),\phi^{t}(x_{0}))=d(\phi^{t}(z),\phi^{t}(x_{0}))=\lim_{n\to+\infty}d(\phi^{t}(z_{n}),\phi^{t}(x_{0}))\leq\epsilon/3,$

for all $0\leq t\leq t_{0}$.
Fixed $k\in\mathbb{N}$, then

$d(\phi^{t}(\phi^{k(t_{0}+s)}(y)),\phi^{t}(x_{0}))=\lim_{n\to+\infty}d(\phi^{t}(\phi^{k(t_{0}+s_{n})}(y_{n})),\phi^{t}(x_{0})).$
(16)

The right side of the last inequality is bounded by

$\displaystyle\lim_{n\to+\infty}\Big{(}d(\phi^{t}(\phi^{k(t_{0}+s_{n})}(y_{n})),\phi^{t}(\phi^{kt_{0}+\sum_{i=0}^{k-1}s_{n-i}}(y_{n})))+d(\phi^{t}(\phi^{kt_{0}+\sum_{i=0}^{k-1}s_{n-i}}(y_{n})),\phi^{t}(x_{0}))\Big{)}.$
(17)

Since $\displaystyle\lim_{n\to+\infty}s_{n}=s$, then

$\lim_{n\to+\infty}\Big{|}kt_{0}+\sum_{i=0}^{k-1}s_{n-i}-k(t_{0}+s_{n})\Big{|}=\lim_{n\to+\infty}\Big{|}\sum_{i=0}^{k-1}s_{n-i}-ks_{n}\Big{|}=0,$

consequently,

$\lim_{n\to+\infty}d(\phi^{t}(\phi^{k(t_{0}+s_{n})}(y_{n})),\phi^{t}(\phi^{kt_{0}+\sum_{i=0}^{k-1}s_{n-i}}(y_{n})))=0.$
(18)

From (15), (16), (17), and (18) we obtain

$d(\phi^{t}(\phi^{k(t_{0}+s)}(y)),\phi^{t}(x_{0}))\leq\epsilon/3,\,\,0\leq
t\leq t_{0}.$ (19)

In summary, $y\in B[x_{0},\delta/3]$ and from (19) we have that $O^{+}(y)$
$\epsilon$-shades forward by piecewise the orbit segment
$\phi^{[0,t_{0}]}(x_{0})$ with transition times of size $|s|\leq\eta$.
Analogously, we can find $z\in B[x_{0},\delta/3]$ such that $O^{-}(z)$
$\epsilon$-shades backward by piecewise the orbit segment
$\phi^{[0,t_{0}]}(x_{0})$ with transition times of size $|r|\leq\eta$, that
is, for each $k\in\mathbb{N}\cup\\{0\\}$

$d(\phi^{-t}(\phi^{-k(t_{0}+r)}(z)),\phi^{-t}(x_{0}))\leq\frac{\epsilon}{3},\,\,0\leq
t\leq t_{0}.$ (20)

To conclude our proof, note that
$d(z,y)<\frac{\delta}{3}+\frac{\delta}{3}=\frac{2\delta}{3}<\delta$, then for
the choosing of $\delta$ we have that

$w:=[z,y]=W_{\eta}^{cu}(z)\pitchfork W_{\eta}^{ss}(y)\in B(x_{0},\epsilon).$

Assume that $w=\phi^{r_{1}}(\tilde{z})$, with $\tilde{z}\in W_{\eta}^{uu}(z)$
and $|r_{1}|<\eta$. From (19) and (20) we have that

$\displaystyle d(\phi^{t+k(t_{0}+s)}(w),\phi^{t}(x_{0}))$ $\displaystyle\leq$
$\displaystyle
d(\phi^{t+k(t_{0}+s)}(w),\phi^{t+k(t_{0}+s)}(y))+d(\phi^{t+k(t_{0}+s)}(y),\phi^{t}(x_{0}))$
(21) $\displaystyle<$ $\displaystyle\eta+\frac{\epsilon}{3}<\epsilon,$

and

$\displaystyle d(\phi^{-t-k(t_{0}+r)-r_{1}}(w),\phi^{-t}(x_{0}))$
$\displaystyle\leq$ $\displaystyle
d(\phi^{-t-k(t_{0}+r)-r_{1}}(w),\phi^{-t-k(t_{0}+r)}(z))+d(\phi^{-t-k(t_{0}+r)}(z),\phi^{-t}(x_{0}))$
(22) $\displaystyle<$ $\displaystyle\eta+\frac{\epsilon}{3}<\epsilon.$

for all $t\in[0,t_{0}]$. Hence the orbit $O(w)$, $\epsilon$-shades by
piecewise the orbit segment $\phi^{[0,t_{0}]}(x_{0})$. ∎

###  3.1 Proof of the Theorem 1.1

Proposition 3.4 and the local product structure allow us to conclude that for
Anosov geodesic flow of a complete Riemannian manifold of finite volume, the
periodic orbits are dense.

######  _Proof of the Theorem 1.1_.

As $\text{vol}(M)<\infty$, then $SM=\overline{\text{Rec}(\phi^{t})}$. Then, it
is enough to prove that every recurrent point can be approximated by a
periodic orbit.

Let $x_{0}\in\text{Rec}(\phi^{t})$, then given $\epsilon>0$, as the proof of
Proposition 3.4 we obtain $\delta>0$ and $\eta>0$ satisfying

$\displaystyle\forall\hskip 2.84544ptx,y\in B(x_{0},\delta):\hskip
5.69046ptW_{\eta}^{cu}(x)\pitchfork W_{\eta}^{ss}(y)=\\{w\\}.$ (23)

Moreover, the numbers $\delta,\eta$ can be chosen to satisfy (2). Now taking
$l\in\mathbb{N}$ such that $\frac{\eta}{l}<\delta$ and such that for all
$z,w\in B(x_{0},\frac{\eta}{l})$, $|r|\leq 3\eta$

$d(\phi^{r}(z),\phi^{r}(w))\leq\eta,$ (24)

and also $\phi^{t_{0}}(x_{0})\in B(x_{0},\frac{\eta}{4l})$.
Thus, from the proof Proposition 3.4 there is $y\in
B(x_{0},\frac{\eta}{2l})\subset B(x_{0},\epsilon)$ such that orbit $O(y)$,
$\frac{\eta}{4l}$-shades piecewise the orbit arc $\phi^{[0,t_{0}]}(x_{0})$ for
some $t_{0}>0$, this means that for some $|s|\leq\eta$ (see (21)),
$|r_{1}|\leq\eta,|r_{2}|\leq\eta$ (see (22)) we have

$\displaystyle
d(\phi^{t}(\phi^{j(t_{0}+s)}(y)),\phi^{t}(x_{0}))\leq\frac{\eta}{4l},\,\,\,\,\forall\,j\in\mathbb{N}\cup\\{0\\},\,\,\forall\,0\leq
t\leq t_{0},$ (25)

and

$\displaystyle
d(\phi^{-t-j(t_{0}+r_{1})-r_{2}}(y),\phi^{-t}(x_{0}))\leq\frac{\eta}{4l},\,\,\,\,\forall\,j\in\mathbb{N}\cup\\{0\\},\,\,\,\,\,\forall\,0\leq
t\leq t_{0}.$ (26)

Claim: The point $y$ is unique.

######  _Proof of Claim_.

Assume that there is $\tilde{y}$ such that the orbit $O(\tilde{y})$,
$\frac{\eta}{4l}$-shades piecewise the arc orbit $\phi^{[0,t_{0}]}(x_{0})$,
_i.e._ , $O(\tilde{y})$ satisfy (25) and (26) for some
$|\tilde{s}|,|\tilde{r}_{1}|,|\tilde{r}_{2}|\leq\eta$.

Therefore, for all $j\in\mathbb{N}\cup\\{0\\}$ and $0\leq t\leq t_{0}$ hold

$\displaystyle
d(\phi^{t}(\phi^{j(t_{0}+s)}(y)),\phi^{t}(\phi^{j(t_{0}+\tilde{s})}(\tilde{y})))$
$\displaystyle\leq$ $\displaystyle
d(\phi^{t}(\phi^{j(t_{0}+s)}(y)),\phi^{t}(x_{0}))+d(\phi^{t}(\phi^{j(t_{0}+\tilde{s})}(\tilde{y})),\phi^{t}(x_{0}))$
(27) $\displaystyle\leq$
$\displaystyle\dfrac{\eta}{4l}+\dfrac{\eta}{4l}=\dfrac{\eta}{2l}.$

Analogously,

$d(\phi^{-t-j(t_{0}+r_{1})-r_{2}}(y),\phi^{-t-j(t_{0}+\tilde{r}_{1})-\tilde{r}_{2}}(\tilde{y}))\leq\frac{\eta}{2l}.$
(28)

In particular, for $t=t_{0}$ in (25), since $\phi^{t_{0}}(x_{0})\in
B(x_{0},\frac{\eta}{4l})$, then

$d(\phi^{(j+1)t_{0}+js}(y),x_{0})\leq\frac{\eta}{2l}\,\,\,\text{and}\,\,\,d(\phi^{(j+1)t_{0}+j\tilde{s}}(\tilde{y}),x_{0})\leq\frac{\eta}{2l}.$

From (24) we have

$d(\phi^{(j+1)t_{0}+js+r}(y),\phi^{(j+1)t_{0}+j\tilde{s}+r}(\tilde{y}))\leq\eta,\,\,\forall
j\in\mathbb{N}\cup\\{0\\},|r|\leq\eta.$ (29)

Finally, since $\max\\{|s|,|\tilde{s}|\\}\leq\eta$, then (27) and (29) give us

$\displaystyle d(\phi^{t}(\tilde{y}),\phi^{t}(y))\leq\eta,\hskip
5.69046pt\forall\hskip 2.84544ptt\geq 0.$ (30)

Consequently, $\tilde{y}\in W_{\eta}^{ss}(y)$.

If $t=0$ in (26)

$d(\phi^{-j(t_{0}+r_{1})-r_{2}}(y),x_{0})\leq\frac{\eta}{4l},\,\,\,\,\text{and}\,\,\,d(\phi^{-j(t_{0}+r_{1})-r_{2}}(\tilde{y}),x_{0})\leq\frac{\eta}{4l}.$

From (24) we have

$d(\phi^{-j(t_{0}+r_{1})-r_{2}+r}(y),\phi^{-j(t_{0}+r_{1})-r_{2}+r}(\tilde{y}))\leq\eta,\,\,\,\,\,\forall
j\in\mathbb{N}\cup\\{0\\},|r|\leq\eta.$ (31)

Joining (28) and (31) we have

$\displaystyle d(\phi^{-t}(\tilde{y}),\phi^{-t}(y))\leq\eta,\hskip
5.69046pt\forall\hskip 2.84544ptt\geq 0.$ (32)

Thus, $\tilde{y}\in W_{\eta}^{uu}(y)\subset W_{\eta}^{cu}(y)$, and
consequently, $\tilde{y}\in W_{\eta}^{ss}(y)\pitchfork W_{\eta}^{cu}(y)$.
However, $y\in B(x_{0},\frac{\eta}{l})\subset B(x_{0},\delta)$, then from (23)
it should be $\tilde{y}=y$. ∎

As an important observation, note that unicity in the last claim is also valid
even if $|r_{2}|<3\delta$. Thus, substituting $y$ by
$\tilde{y}=\phi^{t_{0}+s}(y)$ in (25) and (26), we have

$\displaystyle
d(\phi^{t}(\phi^{(j+1)(t_{0}+s)}(y)),\phi^{t}(x_{0}))\leq\frac{\eta}{4l},\,\,\,\,\forall\,j\in\mathbb{N}\cup\\{0\\},\,\,\forall\,0\leq
t\leq t_{0},$

and

$\displaystyle
d(\phi^{-t-(j-1)(t_{0}+r_{1})s-r_{1}-r_{2}}(y),\phi^{-t}(x_{0}))\leq\frac{\eta}{4l},\,\,\,\,\forall\,j\in\mathbb{N}\cup\\{0\\},\,\,\,\,\,\forall\,0\leq
t\leq t_{0}.$

Since $|s-r_{1}-r_{2}|\leq 3\eta$, then the last claim provides that
$y=\tilde{y}=\phi^{t_{0}+s}(y)$. Therefore, $O(y)$ is a periodic orbit such
that $y\in B(x_{0},\epsilon)$.

∎

##  4 Proof of Corollary 1.3

In this section, we use Theorem 1.1 to prove the transitive of Anosov geodesic
flow and the density of the central stable and central unstable manifold in
manifolds of finite volume, a similar result holds for conservative Anosov
flows. So, the proof of Corollary 1.3 will be given by Proposition 4.2 and
Theorem 4.3.
The main ingredients are the stable and unstable manifolds, which satisfy the
following properties (also valid for the stable case).

  1. (a)

For all $t\in\mathbb{R}$ holds $\phi^{t}(W^{uu}(x))=W^{uu}(\phi^{t}(x)).$

  2. (b)

For all $t\in\mathbb{R}$ holds $\phi^{t}(W^{cu}(x))=W^{cu}(x).$

  3. (c)

If $y\in W^{cu}(x)$, then $W^{cu}(y)=W^{cu}(x).$

As a consequence of these three properties, we can prove the following:

######  Lemma 4.1.

For any $x\in SM$ hold

  * •

$W^{cu}(x)=\bigcup_{y\in W^{cu}(x)}W^{cu}(y)$.

  * •

$\overline{W^{cu}(x)}=\bigcup_{y\in\overline{W^{cu}(x)}}W^{cu}(y)$.

###### Proof.

Since $x\in W^{cu}(x)$, then it is enough to prove that $\bigcup_{y\in
W^{cu}(x)}W^{cu}(y)\subset W^{cu}(x)$. In fact: let $z\in\bigcup_{y\in
W^{cu}(x)}W^{cu}(y)$, then there is $y\in W^{cu}(x)$ such that $z\in
W^{cu}(y)=W^{cu}(x)$ from property (c). This concludes the proof of the first
item. To proof the second part, note that as $y\in W^{cu}(y)$, then
$\overline{W^{cu}(x)}\subset\bigcup_{y\in\overline{W^{cu}(x)}}\\{y\\}\subset\bigcup_{y\in\overline{W^{cu}(x)}}W^{cu}(y)$.
Moreover, if $z\in\bigcup_{y\in\overline{W^{cu}(x)}}W^{cu}(y)$,
$z=\phi^{t_{1}}(\tilde{z})$ with $\tilde{z}\in W^{uu}(y)$,
$y\in\overline{W^{cu}(x)}$. In particular, there is $y_{k}\in W^{cu}(x)$ such
that $y_{k}\to y$. The continuity of the unstable manifold implies that there
is $\tilde{z}_{k}$ such that $\tilde{z}_{k}\to\tilde{z}$ and $\tilde{z}_{k}\in
W^{uu}(y_{k})$. Thus, $\phi^{t_{1}}(\tilde{z}_{k})\in W^{cu}(y_{k})$. From
property (c), $W^{cu}(y_{k})=W^{cu}(x)$, which allows us to conclude that
$z\in\overline{W^{cu}(x)}$.

∎

######  Proposition 4.2.

Let $\phi:SM\to SM$ be an Anosov geodesic flow with $M$ connected and
$\emph{vol}(M)<\infty$. Then for all $x\in SM$ we have $W^{cu}(x)$ and
$W^{cs}(x)$ are dense in $SM$.

###### Proof.

Consider $x\in SM$ and $W^{cu}(x)$ (the proof for $W^{cs}$ is analogous). We
know that $\overline{W^{cu}(x)}$ is a closed subset of $SM$. Since $SM$ is a
connected manifold, then we only need to prove that $\overline{W^{cu}(x)}$ is
an open set. For this sake, let $z\in\overline{W^{cu}(x)}$, then from Lemma
4.1, there is $y_{0}\in\overline{W^{cu}(x)}$ such that $z\in W^{cu}(y_{0})$.
Therefore, by property (c)

$W^{cu}(z)=W^{cu}(y_{0})\subset\bigcup_{v\in\overline{W^{cu}(x)}}W^{cu}(v)=\overline{W^{cu}(x)}.$

Let $U\subset SM$ be a neighborhood of $z$ and $p\in U\cap\text{Per}(\phi)$
(this last is possible because $\overline{\text{Per}(\phi)}=SM$). Taking $U$
sufficiently small such that the local structure product holds, then
$W^{ss}(p)\cap W^{cu}(z)\neq\emptyset$. Thus, let $y\in W^{ss}(p)\cap
W^{cu}(z)$. Then, since $W^{cu}(z)\subset\overline{W^{cu}(x)}$, the property
(b) implies that $O(y)\subset W^{cu}(z)\cap
W^{cs}(p)\subset\overline{W^{cu}(x)}\cap W^{cs}(p)$. Now, since $p$ is a
periodic point and $y\in W^{ss}(p)$, then $O(y)$ accumulates in the orbit
$O(p)$ and as $O(y)\subset\overline{W^{cu}(x)}$, we have
$O(p)\subset\overline{W^{cu}(x)}$. Hence $p\in\overline{W^{cu}(x)}$ and as
$\text{Per}(\phi)$ is dense in $U$ this implies
$U\subset\overline{W^{cu}(x)}$, as we wish.
∎

As a consequence of this proposition, we have the following theorem.

######  Theorem 4.3.

Let $\phi:SM\to SM$ be an Anosov geodesic flow with $\emph{vol}(M)<\infty$,
then $\phi$ is transitive.

###### Proof.

Let $U$ and $V$ be two open sets of $SM$. Fix $x\in V$ such that its
$\alpha$-limit, $\alpha(x)\neq\phi$. Take $y\in\alpha(x)$, then there is
$n_{k}\to+\infty$ such that
$\displaystyle\lim_{k\to+\infty}\phi^{-n_{k}}(x)=y$. From Proposition 4.2, we
have $\overline{W^{cs}(y)}=SM$, then there is $z\in W^{cs}(y)\cap U$. In
particular, there is $t_{1}\in\mathbb{R}$ such that $z\in
W^{ss}(\phi^{t_{1}}(y)\cap U$. By continuity of flow $\phi^{t_{1}}$ we obtain

$\lim_{k\to+\infty}\phi^{-n_{k}}(\phi^{t_{1}}(x))=\lim_{k\to+\infty}\phi^{t_{1}}(\phi^{-n_{k}}(x))=\phi^{t_{1}}(y).$

Now, note that $W^{ss}(\phi^{t_{1}}(y))\cap U\neq\emptyset$ and $U$ is an open
set, we consider a positive real number $b>0$ such that for all $k$ large
enough there is a disk $D_{k}\subset W^{ss}(\phi^{-n_{k}}(\phi^{t_{1}}(x)))$
centered at $\phi^{-n_{k}}(\phi^{t_{1}}(x))$ of radius at most $b$ such that
$D_{k}\cap U\neq\phi$.

Since $\phi^{-t_{1}}$ is continuous and $x\in V$, we can fix a neighborhood
$Q$ of $\phi^{t_{1}}(x)$ such that $\phi^{-t_{1}}(Q)\subset V$. Since $D_{k}$
has a radius at most $b$, then for $k$ large enough we have
$\phi^{n_{k}}(D_{k})\subset Q$ and then
$\phi^{-t_{1}}(\phi^{n_{k}}(D_{k}))\subset\phi^{-t_{1}}(Q)\subset V$.

Finally, for $k$ large enough we have that

$\phi^{-t_{1}}(\phi^{n_{k}}(D_{k}\cap
U))\subset\phi^{-t_{1}}(\phi^{n_{k}}(U))\cap\phi^{-t_{1}}(\phi^{n_{k}}(D_{k}))\subset\phi^{-t_{1}}(\phi^{n_{k}}(U))\cap
V.$

Thus, as $D_{k}\cap U\neq\emptyset$, then we obtain
$\phi^{-t_{1}}(\phi^{n_{k}}(U))\cap V\neq\emptyset$ as we wish. ∎

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