# Topological entropy of Lorenz-like flows

Jiagang Yang  Departamento de Geometria, Instituto de Matemática e
Estatística, Universidade Federal Fluminense, Niterói, Brazil<EMAIL_ADDRESS>
###### Abstract.

We use entropy theory as a new tool for studying Lorenz-like classes of flows
in any dimension. More precisely, we show that every Lorenz-like class is
entropy expansive, and has positive entropy which varies continuously with
vector fields. We deduce that every such class contains a transverse
homoclinic orbit and, generically, is an attractor.

J.Y. is partially supported by CNPq, FAPERJ, and PRONEX.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
  3. 3 Positive entropy
  4. 4 Entropy expansiveness
  5. 5 Lower semi-continuation of topological entropy
  6. A Proof of Theorem 5.1
  7. B Proof of Theorem 2.26

##  1\. Introduction

The study of Lorenz attractors, comes from the famous article of Lorenz [27],
where he did numerical investigation of the following system:

$\left\\{\begin{aligned} \dot{x}&=-\sigma x+\sigma
y\;\;\;\;\;\;\;\;\;\sigma=10\\\
\dot{y}&=r\\-y-xz\;\;\;\;\;\;\;\;\;\;\;\;r=28\\\
\dot{z}&=-bz+xy\;\;\;\;\;\;\;\;\;b=8/3\\\ \end{aligned}\right.$

Lorenz observed that, the solutions of the above simple equations, are
sensitive with respect to the initial conditions. Sometime later, a geometric
Lorenz model for this attractor was proposed in [17, 2, 18, 43], which
provides a classical non-uniformly hyperbolic, but robust transitive example
for time continuous systems. Finally it was proved much later by [39, 40],
with the help of computer that, the Lorenz attractor does exist and is exactly
a kind of the geometric model. The history of this story can be found in [41].

Inspired by the proposition of the Lorenz-like class, a theory for
3-dimensional robustly transitive strange attractor was built in [30, 31],
which showed that, any robust attractor of a 3-dimensional flow which contains
both singularity and regular orbits must be singular-hyperbolic, that is, it
admits a dominated splitting $E^{s}\oplus F^{cu}$ on the tangent bundle into a
1-dimensional uniformly contracting sub-bundle and a 2-dimensional volume-
expanding sub-bundle. The notion of Lorenz-like class, is a generalization of
the above constructions and results.

######  Definition 1.1.

A compact invariant set $\Lambda$ of a $\operatorname{C}^{1}$ flow $\phi_{t}$
is a Lorenz-like class if it contains both singularity and regular points, and
satisfies the following conditions:

  * (a)

$\Lambda$ is a chain recurrent class;

  * (b)

$\Lambda$ is Lyapunov stable, i.e., there is a sequence of compact
neighborhoods $\\{U_{i}\\}$ such that:

    * –

$U_{1}\supset U_{2}\supset\dots$ and $\cap_{i\geq 1}U_{i}=\Lambda$;

    * –

for each $i\geq 1$, $\phi_{t}(U_{i+1})\subset U_{i}$ for any $t\geq 0$.

  * (c)

$\Lambda$ is sectional-hyperbolic, i.e., $\Lambda$ admits a dominated
splitting $E^{s}\oplus F^{cu}$ such that the bundle $E^{s}$ is uniformly
contracting, and the bundle $F^{cu}$ is sectional-expanding, which means,
there are $K,\lambda>0$ such that for any subspace $V_{x}\subset F^{cu}(x)$
with $x\in\Lambda$ and $dim(V_{x})\geq 2$, we have:

$|\det(DX_{t}(x)|_{V_{x}})|\geq K\exp^{\lambda t}\;\;\text{for all}\;\;t>0.$

Note that we neither assume the singularities to be hyperbolic nor the class
to be isolated. An example for higher dimensional Lorenz-like class (which is,
in fact, an attractor), was constructed in [8] with $dim(F^{cu})>2$. More
recently, [35] proved that, for generic star flows, every non-trivial Lyapunov
stable chain recurrent class is Lorenz-like, where a $\operatorname{C}^{1}$
flow is a star flow if for any flow nearby, its critical elements are all
hyperbolic (see also [21], [47]).

The results on 3-dimensional Lorenz attractors are quite fruitful, e.g., see
[30, 31, 4, 3, 5]. In fact, as in the construction of geometric Lorenz model,
one can take 2-dimensional sections near to every singularity and analyze the
Poincaré return maps. The corresponding Poincaré return maps are partially
hyperbolic, the quotient along the stable lamination induces a family of
1-dimensional Lorenz maps (see [4]). By this argument, several important
properties were proved, for example: every 3-dimensional singular hyperbolic
attractor is a homoclinic class ( [3]) and is expansive ([4]). This argument
is powerful, but is hard to apply on general Lorenz-like classes. One reason
is that Lorenz-like classes may not be isolated, it will be difficult to
choose sections. Another reason is that the corresponding higher dimensional
Lorenz map is not well-studied. The non-hyperbolicity of the singularities
also brings difficulty on the analysis of the orbits close to the
singularities. In this paper, we will provide a new method to study Lorenz-
like classes, which is based on the new development of entropy theory ( [22]).

######  Theorem A.

Let $\Lambda$ be a Lorenz-like class of a $\operatorname{C}^{1}$ flow
$\phi_{t}$, then $\phi_{t}$ has positive topological entropy on $\Lambda$ and
is entropy expansive in a neighborhood of $\Lambda$.

The positive topological entropy of Lorenz-like class implies the existence of
non-trivial homoclinic class.

######  Corollary B.

Let $\Lambda$ be a Lorenz-like class of a $\operatorname{C}^{1}$ flow
$\phi_{t}$, then $\Lambda$ contains a non-trivial homoclinic class. Moreover,
if $\phi_{t}$ belongs to a $\operatorname{C}^{1}$ residual subset of flows,
then $\Lambda$ is an attractor.

Combining the result of [35], one can get the following description for star
flows:

######  Corollary C.

There is a $\operatorname{C}^{1}$ residual subset ${\mathcal{R}}$ of star
flows, such that for any star flow $\phi_{t}\in{\mathcal{R}}$, $\phi_{t}$ has
finitely many attractors, whose basins cover an open and dense subset of the
ambient manifold.

We also obtain regularity of the topological entropy for Lorenz like classes
which are quasi attractors, where an invariant compact set $\Lambda$ is quasi
attractor if for any neighborhood $V$ of $\Lambda$, there is compact set
$U\subset V$ such that $\phi_{t}(U)\subset int(U)$ for any $t>0$. The
continuity of topological entropy for 1-dimensional Lorenz maps was proved in
[28, 36].

######  Theorem D.

Let $\Lambda$ be a Lorenz-like class of a $\operatorname{C}^{1}$ flow
$\phi_{t}$ with all singularities hyperbolic. Suppose $\Lambda$ is a quasi
attractor, then there is an open neighborhood $U$ of $\Lambda$ and a
neighborhood ${\mathcal{U}}$ of $\phi_{t}$ such that the topological entropy
$h_{top}(.|_{U})$ varies continuously respect to the flows in ${\mathcal{U}}$.

######  Remark 1.2.

Let $U$ be an attracting compact set, i.e., $\phi_{t}(U)\subset int(U)$, and
$\Lambda$ the maximal invarint set of $\phi_{t}|_{U}$. Suppose $\Lambda$ is
sectional hyperbolic and not necessary to be chain recurrent. Then it is easy
to check that our proofs for Theorems A and D still work.

The first part of Theorem A, as well as Corollary B, exist an alternative
proof, which was announced at the International Conference on Dynamical
Systems held at IMPA, Rio de Janeiro in November 2013. A video of this lecture
is available at the conference website [45]. Similar methods and results to
that lecture appeared in a preprint circulated by A. Arbieto, C.A. Morales,
A.M. Lopez B in August 2014.

Now let us explain quickly how the entropy theory is used in the proof: We
study the time-one map of the flow in a neighborhood of Lorenz-like class. The
advantage to take the time-one diffeomorphism is because, we may make use of
‘fake foliations’, which comes from the dominated splitting of the time-one
map, and is in general not preserved by the flow. Making use of the fake
foliation and a new criterion for entropy expansiveness of [22], we prove that
the time-one map is entropy expansive in a small neighborhood of this Lorenz-
like class. In particular, the metric entropy is upper semi-continuous in this
neighborhood.

Then we show a lower bound of topological entropy for diffeomorphisms
admitting a dominated splitting, which enables us to prove that, the
topological entropy for flow in any small neighborhood of this Lorenz like
class is uniformly bounded from zero. Taking a sequence of neighborhoods which
converge to the Lorenz-like class, and combining the upper semi-continuity of
the entropy, we prove that the topological entropy on this Lorenz-like class
is always positive.

This paper is organized as follows. In Section 4 we prove Theorem A. And in
Section 5, we prove the lower semi-continuation of topological entropy and
finish the proof of Corollaries B, C and Theorem D.

##  2\. Preliminary

Throughout this article, $X$ denotes a $\operatorname{C}^{1}$ vector field on
a $d$-dimensional closed manifold $M^{d}$, ${\rm Sing}(X)$ the singularities
of $X$, $\phi_{t}$ the flow generated by $X$, and $f$ the corresponding time-
one map, i.e., $f=\phi_{1}$. We also denote the corresponding tangent flow by
$\Phi_{t}=d\phi_{t}:TM^{d}\rightarrow TM^{d}$.

###  2.1. Dominated splitting

Throughout this subsection, $g\in{\rm Diff}^{1}(M)$ is a diffeomorphism which
admits a dominated splitting $E\oplus F$ on the tangent space, i.e., there
exists $L\in\mathbb{N}$ such that for every $x\in M$, and every pair of non-
zero vectors $u\in E(x)$ and $v\in F(x)$, one has

$\frac{\|Dg_{x}^{L}(u)\|}{\|u\|}\leq\frac{1}{2}\frac{\|Dg^{L}_{x}(v)\|}{\|v\|}.$

For $a>0$ and $x\in M$, we define $(a,F)$-cone on the tangent space $T_{x}M$:

${\mathcal{C}}_{a}(F_{x})=\\{v;v=0\;\text{or}\;v=v_{E}+v_{F}\;\;\text{where}\;\;v_{E}\in
E;\;v_{F}\in F\;\;\text{and}\;\;\frac{\|v_{E}\|}{\|v_{F}\|}<a.\\}$

When $a$ is sufficiently small, the cone fields ${\mathcal{C}}_{a}(F_{x})$,
$x\in M$ is forward invariant, i.e., there is $\lambda<1$ such that for any
$x\in M$, $Dg_{x}({\mathcal{C}}_{a}(F_{x}))\subset{\mathcal{C}}_{\lambda
a}(g(x))$. In a similar way, we may define the $(a,E)$-cone
${\mathcal{C}}_{a}(E_{x})$, which is backward invariant. When no confusing is
caused, we call the two families of cones by $F$ cone and $E$ cone.

The images of the cones under the exponential map are also forward or backward
invariant. The invariance is induced from the invariance of the $E$ and $F$
cones on the tangent space. More precisely: Fix $\varepsilon_{0}>0$ small
enough, such that the exponential map is well defined on the $\varepsilon_{0}$
ball in the tangent space. The image of the exponential map restricted on the
set $B_{\varepsilon_{0}}(0)\cap{\mathcal{C}}_{a}(F_{x})\subset T_{x}M$ defines
a local $F$ cone in $B_{\varepsilon_{0}}(x)$, which we denote by
$\operatorname{C}_{a}(F_{x})$. Then for any $x\in M$, we have:

$g\big{(}\operatorname{C}_{a}(F_{x})\cap
B_{\frac{\varepsilon_{0}}{\|f\|_{\operatorname{C}^{1}}}}(x)\big{)}\subset\operatorname{C}_{\lambda
a}(F_{f(x)}).$

In the same way we define $\operatorname{C}_{a}(E_{x})$.

######  Definition 2.1.

Let $D$ be a $\operatorname{C}^{1}$ disk with dimension $\dim(E)$. We say $D$
is:

  * •

tangent to $F$ cone if for any $x\in D$,
$T_{x}D\subset{\mathcal{C}}_{a}(F_{x})$;

  * •

tangent to local $F$ cone at $x$ if $D\subset\operatorname{C}_{a}(F_{x})$;

  * •

tangent to local $F$ cone if for any $y\in D$ we have
$D\subset\operatorname{C}_{a}(F_{y})$.

$D$ is tangent to local $F$ cone implies that it is tangent to $F$ cone.
Conversely, if $D$ is tangent to $F$ cone, then it can be divided into
finitely many sub-disks, each sub-disk is tangent to local $F$ cone.

######  Remark 2.2.

Topologically, the local cones $\operatorname{C}_{a}(E_{x})$ and
$\operatorname{C}_{a}(F_{x})$ for $x\in M$ are transverse to each other, that
is, $\operatorname{C}_{a}(E_{x})\cap\operatorname{C}_{a}(F_{x})=\\{x\\}$.

######  Remark 2.3.

Suppose $D$ is a disk with dimension $\dim(F)$ and transverse to $E$ bundle,
then there is $n>0$ sufficiently large, such that $g^{n}(D)$ is tangent to $F$
cone. Hence, it can be divide into finitely connected pieces
$g^{n}(D)=\cup_{i=1}^{l}D_{i}$, such that each piece $D_{i}$ is tangent to
local $F$ cone.

######  Lemma 2.4.

There is $K_{0}>0$ such that for any $x\in M$ and any disk $D\subset
B_{\varepsilon_{0}}(x)$ which is tangent to the local $F$ cone,
$\operatorname{vol}(D)\leq K_{0}$.

###### Proof.

Take a smooth foliation ${\mathcal{F}}$ in $B_{\varepsilon_{0}}(x)$ such that
every leaf has dimension $dim(E)$ and is tangent to $E$ cone. Fix a disk
$D_{x}\ni x$ which is tangent to $F$ cone, for example, we may choose the
image of $exp_{x}(B_{\varepsilon_{0}}(x)\cap F_{x})$. By Remark 2.3, each leaf
of ${\mathcal{F}}$ intersects $D$ with at most one point. Hence the foliation
${\mathcal{F}}$ induces a holonomy map $H:D\rightarrow D_{x}$. Because both
$D$ and $D_{x}$ are tangent to $F$ cone, the Jacobian of $H$ is uniformly
bounded from above and below. Observe that the volume of $D_{x}$ is uniformly
bounded from above, the volume of $D_{x}$ is bounded by a uniform constant
$K_{0}>0$.

∎

By the invariance of local cones and induction, it is easy to prove the
following lemma:

######  Lemma 2.5.

For any $x\in M$,
$\varepsilon<\frac{\varepsilon_{0}}{\|g\|_{\operatorname{C}^{1}}}$ and $n>0$,
if $D\subset B_{\varepsilon}(x)$ is tangent to local $F$ cone (at $x$) and
$g^{i}(D)\subset B_{\varepsilon}(g^{i}(x))$ for every $0\leq i\leq n$, then
$g^{n}(D)$ is tangent to local $F$ cone (at $g^{n}(x)$).

######  Definition 2.6.

Let $D$ be a disk tangent to the $F$ cone, then the volume expansion of $D$,
$v_{F}(D)$, is defined by
$\limsup_{n}\frac{1}{n}\log(\operatorname{vol}(g^{n}(D)))$. The volume
expansion $v_{F}$ of bundle $F$, is defined by:

$v_{F}=\sup\\{v_{F}(D);\;\;D\;\;\text{is tangent to
the}\;\;F\;\;\text{cone}\\}.$

###  2.2. Entropy

Let $g:M\rightarrow M$ be a continuous map on a compact metric space $M$, $K$
be a subset of $M$ not necessary to be invariant. For each $\varepsilon>0$ and
$n\geq 1$, we consider the the dynamical ball of radius $\varepsilon>0$ and
length $n$ around $x\in M$:

$B_{n}(x,\varepsilon)=\\{y\in M;d(g^{j}(x),g^{j}(y))\leq\varepsilon\;\text{for
every}\;0\leq j<n\\}.$

A set $E\subset M$ is ($n,\varepsilon$)-spanning for $K$ if for any $x\in K$
there is $y\in E$ such that $d(g^{i}(x),g^{i}(y))\leq\varepsilon$ for all
$0\leq i<n$. In other words, the dynamical balls $B_{n}(y,\varepsilon),y\in E$
cover $K$. Let $r_{n}(K,\varepsilon)$ denote the smallest cardinality of any
($n,\varepsilon$)-spanning set, and

$r(K,\varepsilon)=\limsup_{n\rightarrow+\infty}\frac{1}{n}\log
r_{n}(K,\varepsilon).$

The topological entropy of $g$ on $K$ is defined by
$h_{top}(g,K)=\lim_{\varepsilon\rightarrow 0}r(K,\varepsilon)$. And the
topological entropy of $f$ is defined as $h_{top}(g)=h_{top}(g,M)$. The
topological entropy of a continuous flow equals to the topological entropy of
its time-one map.

Let $\mu$ be an invariant measure and ${\mathcal{A}}$ a finite partition, the
metric entropy of $\mu$ corresponding to the partition ${\mathcal{A}}$ is
defined as

$h_{\mu}({\mathcal{A}})=\lim_{n}-\sum\mu(B_{n,i})\log\mu(B_{n,i}),$

where $B_{n,i}$ are elements of partition

${\mathcal{A}}_{0}^{n-1}={\mathcal{A}}\vee g^{-1}{\mathcal{A}}\vee\dots\vee
g^{1-n}{\mathcal{A}}.$

The metric entropy of an invariant measure $\mu$ is defined as

$h_{\mu}=\sup_{{\mathcal{A}}\;\text{finite
partition}}\\{h_{\mu}({\mathcal{A}})\\}.$

By the variational principle,
$h_{top}(g)=\sup_{\mu\in{\mathcal{M}}_{inv}(g)}h_{\mu}$, where
${\mathcal{M}}_{inv}(g)$ denotes the space of invariant probability of $g$.
Metric entropy is not necessary to be upper semi-continuous depending on the
invariant measures, and the topological entropy may not be achieved by any
metric entropy, e.g., see [32, 14].

For each $x\in M$ and $\varepsilon>0$, let
$B_{\infty}(x,\varepsilon)=\\{y:d(g^{n}(x),g^{n}(y))\leq\varepsilon$ for
$n\in\mathbb{Z}\\}$. $g$ is ($\varepsilon$-)entropy expansive if

$\sup_{x\in M}h_{top}(g,B_{\infty}(x,\varepsilon))=0.$

For $\varepsilon>0$, we say $g$ is $\varepsilon$-almost entropy expansive if
for any invariant ergodic measure $\mu$ of $g$ and $\mu$ almost every point
$x\in M$, we have $h_{top}(g,B_{\infty}(x,\varepsilon))=0.$

A new criterion of entropy expansiveness was given in [22][Proposition 2.4]:

######  Lemma 2.7.

$g$ is $\varepsilon$-almost entropy expansive if and only if it is
$\varepsilon$-entropy expansive.

The positivity of topological entropy of Theorem A depends on the following
theorem, whose proof, as well as ones for the Corollaries 2.10, 2.11, will be
given in Section 3.

######  Theorem 2.8.

Suppose $g$ is a diffeomorphism which admits a dominated splitting $E\oplus
F$. Then $h_{top}(g)\geq v_{F}$.

Shub [37] has conjectured an estimation of lower bound of topological entropy:
the topological entropy is bounded from below by the spectral radius in
homology (see also Shub, Sullivan [38]).

Let $d=dimM$ and $g_{*,k}:H_{k}(M,\mathbb{R})\rightarrow
H_{k}(M,\mathbb{R}),0\leq k\leq d$ be the action induced by $g$ on the real
homology groups of $M$. Let

$\operatorname{sp}(g_{*})=\max_{0\leq k\leq d}\operatorname{sp}(g_{*,k}),$

where $\operatorname{sp}(g_{*,k})$ denotes the spectral radius of $g_{*,k}$.
The Shub entropy conjecture states that:

######  Conjecture 2.9.

For every $g\in{\rm Diff}^{1}(M)$,
$h_{top}(g)\geq\log\operatorname{sp}(g_{*})$.

This conjecture has been proved for $C^{\infty}$ maps in [46] and for
diffeomorphims away from tangencies in [22]. The history and more references
on the Shub entropy conjecture can be found in [22]. As an immediate corollary
of Theorem 2.8, we obtain a partial answer to the Shub entropy conjecture.

######  Corollary 2.10.

Suppose $g$ is a diffeomorphism which admits a dominated splitting $E\oplus
F$, then

(1) $h_{top}(g)\geq\log\operatorname{sp}(g_{*,dim(F)}).$

Theorem 2.8 and Corollary 2.10 were first stated in [33], but their proof only
works when the bundle $F$ is uniformly expanding (see the estimation of Bowen
ball in the proof of [33][Proposition 2]), we will give more explanation in
Remark 3.2.

It was shown in [7] that every $\operatorname{C}^{1}$ robustly transitive
diffeomorphism admits a dominated splitting, and the extreme bundles are
volume hyperbolic. As a corollary of Theorem 2.8, we prove that:

######  Corollary 2.11.

Suppose $g$ is a robust transitive diffeomorphism. Then its topological
entropy is positive.

New results on entropy expansiveness for partially hyperbolic diffeomorphisms
can also be found in [12] and [13].

###  2.3. Ergodic theory for flows

In this subsection, we state the results of ergodic theory for flows which
will be used later. Throughout this subsection, $\Lambda$ denotes a compact
invariant set of the flow $\phi_{t}$, and $\mu$ is a non-trivial invariant
measure of $\phi_{t}$, i.e., $\mu({\rm Sing}(X))=0$.

######  Definition 2.12.

We say $\Lambda$ admits a dominated splitting $E\oplus F$ if this splitting is
invariant for $\Phi_{t}$, and there exist $C>0$ and $\lambda<1$ such that for
every $x\in\Lambda$, and every pair of unit vectors $u\in E(x)$ and $v\in
F(x)$, one has

$\|(\Phi_{t})_{x}(u)\|\leq
C\lambda^{t}\|(\Phi_{t})_{x}(v)\|\;\;\text{for}\;\;t>0.$

We will see that in the above definition, the assumption of the invariance of
the splitting is not necessary.

######  Lemma 2.13.

$E\oplus F|_{\Lambda}$ is a dominated splitting for the flow
$\phi_{t}|_{\Lambda}$ if and only if it is a dominated splitting for the time-
one map $f|_{\Lambda}$. Moreover, if $\phi_{t}|_{\Lambda}$ is transitive, then
we have either $X|_{\Lambda\setminus{\rm Sing}(X)}\subset E$ or
$X|_{\Lambda\setminus{\rm Sing}(X)}\subset F$.

###### Proof.

The proof of the ‘only if’ part is trivial. Now we assume $E\oplus
F|_{\Lambda}$ is a dominated splitting for $f|_{\Lambda}$. Then, by the
commutative property between $f$ and $\phi_{t}$, it is easy to see that, the
splitting $\Phi_{t}(E)\oplus\Phi_{t}(F)|_{\Lambda}$ is also a dominated
splitting for $f|_{\Lambda}$.

Because the dominated splitting of fixed dimension is unique, we conclude that
this splitting $E\oplus F|_{\Lambda}$ is invariant for $\Phi_{t}$. Therefore,
$E\oplus F|_{\Lambda}$ is also a dominated splitting for
$\phi_{t}|_{\Lambda}$.

Now suppose $\phi_{t}|\Lambda$ is transitive. Take $x\in\Lambda\setminus{\rm
Sing}(X)$ such that ${\rm Orb}^{+}(x)$ is dense in $\Lambda$. If $X(x)\notin
E(x)\cup F(x)$, by the domination, for $t$ sufficient large, $X(\phi_{t}(x))$
is close to $F(\phi_{t}(x))$. We take $t_{0}$ large such that
$\phi_{t_{0}}(x)$ is close to $x$, then $X(\phi_{t_{0}}(x))$ is close to
$X(x)$ and $F(\phi_{t_{0}}(x))$ is close to $F(x)$, which implies in
particular that $X(x)$ is arbitrarily close to $F(x)$, a contradiction.

Because ${\rm Orb}^{+}(x)$ is dense, by the continuation of flow direction and
the subbundles $E$ and $F$, for $X(x)\in E(x)$ or $X(x)\in F(x)$, we may show
that $X|_{\Lambda\setminus{\rm Sing}(X_{t})}\subset E$ or
$X|_{\Lambda\setminus{\rm Sing}(X_{t})}\subset F$ respectively. The proof is
complete. ∎

######  Definition 2.14.

The topological entropy (resp. metric entropy) of a continuous flow equals to
the topological entropy (resp. metric entropy) of its time-one map. A flow is
$\varepsilon$-entropy expansive if its time-one map is $\varepsilon$-entropy
expansive.

######  Lemma 2.15.

Let $\mu$ be an ergodic invariant measure of $\phi_{t}$, and $\tilde{\mu}$ be
an ergodic component of $\mu$ for the diffeomorphism $f$. Then
$h_{\mu}(\phi_{t})=h_{\tilde{\mu}}(f)$.

###### Proof.

Observe that $\tilde{\mu}_{t}=(\phi_{t})_{*}\tilde{\mu}$ is also an
$f$-invariant measure and

$\mu=\int_{[0,1]}(\phi_{t})_{*}\tilde{\mu}_{t}dt.$

It is easy to show $h_{\tilde{\mu}_{t}}(f)=h_{\tilde{\mu}}(f)$ by the
following observation: for any partition
${\mathcal{A}}=\\{A_{1},\dots,A_{k}\\}$, write
${\mathcal{A}}_{t}=\\{\phi_{t}(A_{1}),\dots,\phi_{t}(A_{k})\\}$, then
$\tilde{\mu}(A_{i})=\tilde{\mu}_{t}(\phi_{t}(A_{i}))$.

Because metric entropy is an affine function respect to the invariant
measures,

$h_{\mu}(f)=\int_{[0,1]}h_{\tilde{\mu}}(f)dt=h_{\tilde{\mu}}(f)=h_{\tilde{\mu}}(\phi_{t}).$

∎

As a corollary of the above lemma, we state the following two results in the
version for flows.

######  Lemma 2.16.

([9]) Suppose the flow $\phi_{t}$ is entropy expansive, then the metric
entropy function is upper semi-continuous. In particular, there exists a
maximal measure.

######  Lemma 2.17.

([22][Lemma2.3]) Let ${\mathcal{U}}$ be a $\operatorname{C}^{1}$ open set of
flows which are $\delta$-entropy expansive for some $\delta>0$. Then the
topological entropy varies in an upper semi-continuous manner for flows in
${\mathcal{U}}$.

The linear Poincaré flow $\psi_{t}$ is defined as following. Denote the normal
bundle of $\phi_{t}$ over $\Lambda$ by

${\mathcal{N}}_{\Lambda}=\cup_{x\in\Lambda\setminus{\rm
Sing}(X)}{\mathcal{N}}_{x},$

where ${\mathcal{N}}_{x}$ is the orthogonal complement of the flow direction
$X(x)$, i.e.,

${\mathcal{N}}_{x}=\\{v\in T_{x}M^{d}:v\bot X(x)\\}.$

Denote the orthogonal projection of $T_{x}M$ to ${\mathcal{N}}_{x}$ by
$\pi_{x}$. Given $v\in{\mathcal{N}}_{x},x\in M^{d}\setminus{\rm Sing}(X)$,
$\psi_{t}(v)$ is the orthogonal projection of $\Phi_{t}(v)$ on
${\mathcal{N}}_{\phi_{t}(x)}$ along the flow direction, i.e.,

$\psi_{t}(v)=\pi_{\phi_{t}(x)}(\Phi_{t}(v))=\Phi_{t}(v)-\frac{<\Phi_{t}(v),X(\phi_{t}(x))>}{\|X(\phi_{t}(x))\|^{2}}X(\phi_{t}(x)),$

where $<.,.>$ is the inner product on $T_{x}M$ given by the Riemannian metric.
The following is the flow version of Oseledets theorem.

######  Proposition 2.18.

For $\mu$ almost every $x$, there exist $k=k(x)\in\mathbb{N}$, real number
$\hat{\lambda}_{1}(x)>\dots>\hat{\lambda}_{k}(x)$ and a measurable splitting

${\mathcal{N}}_{x}=\hat{E^{1}_{x}}\oplus\dots\oplus\hat{E^{k}_{x}}$

such that this splitting is invariant under $\psi_{t}$, and

$\underset{t\to\pm\infty}{\lim}\frac{1}{t}\log\|\psi_{t}(v_{i})\|=\hat{\lambda_{i}}(x)\;\;\text{for
every non-zero}\;\;v_{i}\in\hat{E}^{i}(x).$

Now we state the relation between Lyapunov exponents and the Oseledets
splitting for $\phi_{t}$ and for $f$:

######  Theorem 2.19.

For $\mu$ almost every $x$, denote $\lambda_{1}(x)>\dots>\lambda_{k}(x)$ the
Lyapunov exponents and the Oseledets splitting
$T_{x}M=E^{1}_{x}\oplus\dots\oplus E^{k}_{x}$ of $\mu$ for $f$. Then
${\mathcal{N}}_{x}=\pi_{x}(E^{1}_{x})\oplus\dots\oplus\pi_{x}(E^{k}_{x})$ is
the Oseledets splitting of $\mu$ for $\phi_{t}$. And the Lyapunov exponents of
$\mu$ counted with multiplicity for the flow $\phi_{t}$ is a subset of the
exponents of $f$ by removing one of the zero exponent which comes from the
flow direction.

######  Definition 2.20.

$\mu$ is called a hyperbolic measure for flow $\phi_{t}$ if it is an ergodic
measure of $\phi_{t}$ and all the exponents are non-vanishing. We call the
number of the negative exponents of $\mu$, counted with multiplicity, its
index.

###  2.4. Lorenz-like classes

We will prove that Lorenz-like class is always entropy expansive. This is a
generalization of the result in [4], where expansiveness has been shown for
every 3-dimensional Lorenz attractor.

######  Proposition 2.21.

Let $\Lambda$ be a Lorenz-like class. Then there are $\delta>0$ and a
neighborhood $U$ of $\Lambda$, such that $\phi_{t}|_{U}$ is $\delta$-entropy
expansive. Moreover, if $\Lambda$ is a quasi attractor and
$Sing(X)\cap\Lambda$ consists of only hyperbolic singularities, then there is
a $\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of $X$, such that the
flows generated by vector fields in ${\mathcal{U}}$ restricted on $U$ are all
$\delta$-entropy expansive.

The proof of Proposition 2.21 will be given in Section 4. By Lemmas 2.16 and
2.17, entropy expansiveness implies the metric entropy and topological entropy
vary in an upper semi-continuous manner. This fact is important for us to
prove that the topological entropy of any Lorenz-like class is positive. The
lower semi-continuation of the topological entropy comes from a similar
argument as Katok [20] and a shadowing lemma of Liao [26], which permits the
pseudo orbit to pass near singularities.

######  Theorem 2.22.

Let $\Lambda$ be a compact invariant set of flow $\phi_{t}$ and admit a
dominated splitting $E\oplus F$. Suppose $\mu$ is a non-trivial hyperbolic
ergodic measure of $\phi_{t}$ with positive entropy and index $dim(E)$. Then
for any $n>0$, there is a hyperbolic set $\Lambda_{n}$ arbitrarily close to
$\operatorname{supp}(\mu)$ such that
$h_{top}(\phi_{t}|_{\Lambda_{n}})>h_{top}(\phi_{t}|_{\operatorname{supp}(\mu)})-\frac{1}{n}.$

Theorem 2.22 and the Corollary 2.24 are proved in Section 5.

######  Remark 2.23.

We need to note that, the original argument of [20] cannot be applied directly
on flows, even for the flows which are uniformly hyperbolic. The obstruction
comes from the difference of the shadowing lemmas for diffeomorphisms and for
flows. In the latter case, the period of periodic pseudo orbit and the period
of the shadowing periodic orbit are in general different. The key point of our
argument is that, we need an uniform estimation on the difference of the
times– (d) of Theorem 5.1. As far as the author knows, this estimation is new.

Immediately, we have the following corollary, which is similar to [35][Theorem
5.6]. The difference is that here we permit the singularities to be non-
hyperbolic.

######  Corollary 2.24.

Under the assumptions of Theorem 2.22, the chain recurrent class of $\Lambda$
contains a non-trivial homoclinic class.

We need the following theorem from [35]:

######  Theorem 2.25 ( [35] Theorem C).

There is a residual subset ${\mathcal{R}}_{1}$ of star flows, such that for
every flow contained in ${\mathcal{R}}_{1}$, its non-trivial Lyapunov stable
chain recurrent classes are all Lorenz-like.

In Appendix B, we will also show that:

######  Theorem 2.26.

There is residual subset ${\mathcal{R}}_{2}$ of $\operatorname{C}^{1}$ flows,
such that for every flow contained in ${\mathcal{R}}_{2}$, if $\Lambda$ is a
Lorenz-like class which contains a periodic orbit, then it is an attractor.

##  3\. Positive entropy

In this section, we give the proof of Theorem 2.8, Corollaries 2.10 and 2.11.
We assume $g$ is a diffeomorphism which admits a dominated splitting $E\oplus
F$, $a$ and $\varepsilon_{0}$ are defined as in Subsection 2.1. $K_{0}$ is
given by Lemma 2.4.

######  Lemma 3.1.

Suppose $D$ is a disk with dimension $dim(F)$ and tangent to local $F$ cone.
Then for any $x\in D$ and
$\varepsilon<\frac{\varepsilon_{0}}{\|g\|_{\operatorname{C}^{1}}}$, one has
$\operatorname{vol}(g^{n}(D\cap B_{n}(x,\varepsilon)))\leq K_{0}$.

Without the assumption on dominated splitting, the previous lemma is false
even for analytic diffeomorphism, e.g., see [10].

###### Proof.

By the definition of dynamical ball $B_{n}(x,\varepsilon)$ and Lemma 2.5,
$D_{n}=f^{n}(D\cap B_{n}(x,\varepsilon))$ is tangent to local $F$ cone and is
contained in $B_{\varepsilon}(x)$. By Lemma 2.4, the volume of $D_{n}$ is
bounded by $K_{0}$.

∎

######  Remark 3.2.

As we mentioned in the introduction, the proofs of [33] about Theorem 2.8 and
Corollary 2.10 do not work. A main reason is because in their proof of
[33][Proposition 2], they need $f^{n}(D\cap B_{n}(x,\varepsilon))$ to be
almost a ball, which occurs only when the bundle $F$ is uniformly expanding.
In general, this set is even not connected.

###### Proof of Theorem 2.8 :.

For any $\delta>0$, choose $D$ a disk which is tangent to the $F$ cone and
satisfies $v_{F}(D)\geq v_{F}(f)-\delta$, that is:

$\limsup_{n}\frac{1}{n}\log(\operatorname{vol}(f^{n}(D)))\geq v_{F}-\delta.$

Take $\varepsilon>0$ and
$\Gamma_{n}=\\{x_{1},\dots,x_{r_{n}(D,\varepsilon)}\\}$ an
($n,\varepsilon$)-spanning set of $D$. By Lemma 3.1,
$\operatorname{vol}(f^{n}(D\cap B_{n}(x,\varepsilon)))\leq K_{0}$. Observe
that $\\{B_{n}(x_{i},\varepsilon)\\}_{i=1}^{r_{n}(D,\varepsilon)}$ is a cover
of $D$, we have

$\operatorname{vol}(f^{n}(D))\leq\sum_{i}\operatorname{vol}(f^{n}(D\cap
B_{n}(x_{i},\varepsilon)))\leq K_{0}r_{n}(D,\varepsilon).$

Hence,

$\begin{split}&\limsup_{n}\frac{1}{n}\log(r_{n}(D,\varepsilon))\geq\limsup\frac{1}{n}\log(\operatorname{vol}(f^{n}(D))/K_{0})\\\
&=\limsup\frac{1}{n}\log(\operatorname{vol}f^{n}(D))\geq v_{F}-\delta.\\\
\end{split}$

This implies that $h_{top}(g,D)\geq v_{F}-\delta$. Since $\delta$ can be
chosen arbitrarily small, we conclude the proof of Theorem 2.8.

∎

Now let us begin the proof of Corollaries 2.10 and 2.11:

###### Proof of Corollary 2.10:.

It is well known that the supremum of volume expansion of disks with dimension
$dim(F)$ is larger than or equal to $\operatorname{sp}(f_{*,dim(F)})$ (for
example, see p.287 [46]). Now we will show that the same proof also provides
that

$v_{F}(f)\geq\operatorname{sp}(f_{*,dim(F)}).$

As explained in [46]: the norm of the homology class is bounded by integrals
of some fixed differential forms on this class; these integrals, in turn, are
bounded by the volume of the chain, representing this class. We may choose a
chain such that every simplex is transverse to the $E$ bundle. Because of the
dominated splitting, under the iteration of $f$, the image of every simplex is
finally tangent to the $F$ cone. Hence, their volume expansion speed will be
bounded by $v_{F}(f)$. Then the previous argument still works here.

As a corollary of Theorem 2.8, we show that
$h_{top}(f)\geq\operatorname{sp}(f_{*,dim(F)})$.

∎

###### Proof of Corollary 2.11:.

Because $f$ is robustly transitive, by [7], $f$ admits a dominated splitting
$E_{1}\oplus\dots\oplus E_{k}$ where $E_{1}$ is volume contracting and $E_{k}$
is volume expanding. More precisely, there is $\lambda<1$ and $C>0$ such that
for any $x\in M$

$\det(Df^{n}|_{E_{1}(x)})<C\lambda^{n}\;\;\;\text{and}\;\;\;\det(Df^{n}|_{E_{k}(x)})>C\lambda^{-n}.$

Now we claim that $v_{E_{k}}(f)\geq-\log\lambda$. Then by Theorem 2.8, we
finish the proof.

It remains to prove this claim. Take $D$ any disk tangent to $F$ cone. By the
invariance of $F$ cone, $f^{n}(D)$ is still tangent to $F$ cone for any $n>0$.
Then the determine of the tangent map of $f^{n}|D$ is close to
$\det(Df^{n}|_{F})$, which means that

$\operatorname{vol}(f^{n}(D))\geq C\lambda^{-n}.$

Hence
$v_{F}(D)=\limsup\frac{1}{n}\log\operatorname{vol}(f^{n}(F))\geq-\log\lambda$.

∎

##  4\. Entropy expansiveness

Throughout this section, let $\Lambda$ be a Lorenz-like class which admits a
sectional-hyperbolic splitting $E^{s}\oplus F^{cu}$.

The structure of this section is the following. In Subsection 4.1, we use
Proposition 2.21 to prove Theorem A. The proof of Proposition 2.21 on the
entropy expansiveness of Lorenz-like class will be given in Subsection 4.2.

###  4.1. Proof of Theorem A

First let us recall the proof of Theorem 2.8, where we only use the forward
invariance of the $F$ cone. In a small attracting neighborhood of $\Lambda$,
the $F$ cone is well defined and forward invariant, hence, Theorem 2.8 still
works.

###### Proof of Theorem A:.

By the definition of Lorenz-like class, $\Lambda$ is a compact invariant set
of the time-one map $f$ and admits a dominated splitting $E^{s}\oplus F^{cu}$,
where $F^{cu}$ is volume expanding. We take a forward invariant $F^{cu}$ cone
on $\Lambda$.

Let $\\{U_{i}\\}_{i=1}^{\infty}$ be a sequence of neighborhoods of $\Lambda$
as in Definition 1.1, i.e., they satisfy:

  * •

$U_{1}\supset U_{2}\dots$ and $\cap U_{i}=\Lambda$;

  * •

$f^{n}(U_{i+1})\subset int(U_{i})$ for any $i,n\in\mathbb{N}$.

Assume $U_{1}$ is sufficiently small, we may extend the $F^{cu}$ cone to
$U_{1}$ which is still forward invariant. Then for any disk $D\subset U_{1}$
which is tangent to $F$ cone, $f^{n}(D)$ is still tangent to $F$ cone.
Moreover, because $F^{cu}|_{\Lambda}$ is volume expanding, there exist $C>0$
and $0<\lambda<1$ such that:

$\operatorname{vol}(f^{n}(D))\geq C\lambda^{-n}.$

This implies $v_{F}(D)\geq-\log\lambda$.

For each $i\in\mathbb{N}$, applying Theorem 2.8 on $f|_{U_{i}}$, we deduce
that $h_{top}(f|_{U_{i}})\geq-\log\lambda$. By the variation principle, there
is an $f$ ergodic invariant measure $\mu_{i}$ supported on $U_{i}$ with
$h_{\mu_{i}}(f)\geq-\log\lambda-\frac{1}{i}$. Replacing by a subsequence, we
suppose that $\mu_{i}\to\mu_{0}$ in the weak-* topology, where $\mu_{0}$ is an
invariant measure supported on $\Lambda$. As shown in Lemma 2.16, entropy
expansiveness implies the metric entropy function is upper semi-continuous,
this implies that $h_{\mu_{0}}(f)\geq-\log\lambda$. Therefore, by the
definition of topological entropy for flows, we conclude that

$h_{top}(\phi_{t}|_{\Lambda})=h_{top}(f|_{\Lambda})\geq
h_{\mu_{0}}(f)\geq-\log\lambda,$

the proof is complete. ∎

###  4.2. Entropy expansiveness

Let $U$ be a small neighborhood of $\Lambda$, such that the maximal invariant
set of $\phi_{t}|_{U}$, denoted by $\tilde{\Lambda}$, is sectional-hyperbolic.
This means, $\phi_{t}|_{\tilde{\Lambda}}$ admits a dominated splitting
$E^{s}\oplus F^{cu}$ and there is $0<\lambda_{0}<1$ such that, for any
2-dimensional subspace $\Sigma_{x}\subset F^{cu}(x)$ ($x\in\tilde{\Lambda}$),
we know that $\det(Tf_{x}|_{\Sigma_{x}})>\frac{1}{\lambda_{0}}$. We may assume
the above inequality also holds for any 2-dimensional subspace contained in
$2a_{0}$ $F^{cu}$-cone for $a_{0}$ sufficiently small. In order to simplify
the notation, we write briefly $x_{t}=\phi_{t}(x)$.

The proof of Proposition 2.21 occupies the rest of this subsection. We need
notice that although this result holds for flow, in the proof, we will only
consider the time-one diffeomorphism. The main reason is because we need use
the fake foliations, which are locally invariant for the diffeomorphism, but
not for the flow! Now let us sketch the proof.

Although the singularities can be non-isolated–by non-hyperbolicity, in
Subsubsection 4.2.1 we will show that typical points of $\tilde{\Lambda}$ can
only approach finitely many singularities in ${\rm Sing}(X)$. Let us explain
this fact with more details. It will be shown that $f|_{{\rm
Sing}(X)\cap\tilde{\Lambda}}$ admits a partially hyperbolic splitting with a
1-dimensional center direction, and the singularities are either isolated or
contained in finitely many center segments. Moreover, typical points of
$\tilde{\Lambda}$ cannot approach the interior of each center segment, and
when a singularity is approached by a typical point from one direction along
the center, then the corresponding branch of center leaf is (topologically)
contracting and with uniform size.

As a standard argument in [22], the infinite Bowen ball is contained in a fake
$cu$ leaf. In Subsubsection 4.2.2, we will see that the distance between
different flow orbits contained in the same infinity Bowen ball is expanding.
There are two kinds of analysis: the orbits are away from singularities or
close to singularities. We need notice the reader that, in the latter case, we
cannot use linearization for the dynamics close to a singularity, because
which can not be hyperbolic. In fact, a new argument is used to analyze these
orbits which are close to singularities. More precisely, we use two different
kinds of fake foliations– the fake foliations defined in a neighborhood of
singularities which come from the partially hyperbolic splitting on ${\rm
Sing}(X)$; and the fake foliations come from the dominated splitting on
$\tilde{\Lambda}$–to build a local product structure in $cu$ leaves.

In Subsubsection 4.2.3, we give the proof of Proposition 2.21.

####  4.2.1. Singularities and center segments

######  Lemma 4.1.

$f|_{{\rm Sing}(X)\cap\tilde{\Lambda}}$ admits a partially hyperbolic
splitting $E^{s}\oplus E^{c}\oplus E^{u}$, where $E^{c}$ is a 1-dimensional
bundle corresponding to an eigenvalue with norm less than or equal to one.

###### Proof.

We will only prove that the singularities contained in $\Lambda$ admit such a
partially hyperbolic splitting. This is because when we choose $U$
sufficiently small, ${\rm Sing}(X)\cap\tilde{\Lambda}$ is contained in a small
neighborhood of ${\rm Sing}(X)\cap\Lambda$, and then induced the same
partially hyperbolic splitting.

Recall that $\Lambda$ admits a dominated splitting $E^{s}\oplus F^{cu}$ where
$F^{cu}$ is sectional-expanding. It suffices to prove that for any singularity
$\sigma\in\Lambda$, $Tf|_{F^{cu}(\sigma)}$ has one eigenvalue with norm less
than or equal to one. Suppose this fact and denote $E^{c}$ the direction of
the eigenvector. Then the splitting $F^{cu}|_{{\rm
Sing}(X)\cap\Lambda}=E^{c}\oplus E^{u}$ is continuous, invariant, and the
latter subbundle is uniformly expanding. Therefore this splitting is a
dominated splitting, and $E^{s}\oplus E^{c}\oplus E^{u}$ is the corresponding
partially hyperbolic splitting on ${\rm Sing}(X)\cap\Lambda$.

It remains to show that for any singularity $\sigma\in\Lambda$,
$Tf|_{F^{cu}(\sigma)}$ has a eigenvalue with norm less than or equal to one.
Suppose by contradiction that there is a singularity $\sigma\in\Lambda$ such
that $Tf|_{F^{cu}(\sigma)}$ is uniformly expanding. Then $\sigma$ is an
isolated hyperbolic saddle of $f$. We claim that
$\Lambda\cap(W^{s}(\sigma)\setminus\sigma)\neq\emptyset$.

The proof of this claim is standard. By the definition of Lorenz-like class,
$\Lambda$ contains a regular point $x$. Then for any $n\in\mathbb{N}$, there
is a $\frac{1}{n}$ pseudo orbit $\\{\phi_{t}(x);0\leq t\leq
t_{1}\\};\dots;\\{\phi_{t}(x^{k_{n}});0\leq t\leq t_{k_{n}}\\}$ contained in
$\Lambda$ with $t_{1},\dots,t_{k_{n}}\geq 1$ such that
$d(\phi_{t_{k_{n}}}(x^{k_{n}}),\sigma)\leq\frac{1}{n}$. Let $y^{n}$ be the
last time this pseudo orbit enters $B_{\varepsilon}(\sigma)$ for some small
$\varepsilon$. By taking a subsequence, we may assume that $\lim y^{n}=y^{0}$.
Then $y^{0}\in\Lambda\setminus\sigma$ and $\phi_{t}(y^{0})\in
B_{\varepsilon}(\sigma)$ for any $t>0$. This implies that $y^{0}\in
W^{s}(\sigma)\setminus\sigma$, which proves the claim.

By the invariance of the stable manifold of $\sigma$, $\phi_{t}(y^{0})\subset
W^{s}(\sigma)$ for $t\in\mathbb{R}$. This implies in particular that
$X(y^{0})$ is contained in the $E^{s}$ cone. Take a local orbit
$l=(\phi_{t}(y^{0}))_{t\in[-\delta,\delta]}$, which is tangent to $E^{s}$
cone. By the expansion of vectors in ${\mathcal{C}}(E^{s})$ under the
iteration of $f^{-1}$, $\operatorname{length}(f^{-n}(l))\rightarrow\infty$,
which contradicts the fact that

(2)
$\operatorname{length}(f^{-n}(l))\leq\frac{\max\\{\|X(x)\|;x\in\Lambda\\}}{\min\\{\|X(y)\|;y\in
l\\}}\operatorname{length}(l)$

is uniformly bounded. ∎

The bundles in a dominated splitting are in general not integrable. The
following lemma was borrowed from [22][Lemma 3.3] (see also [11][Proposition
3.1]) which shows that one can always construct local fake foliations.
Moreover, these fake foliations have local product structure, and this
structure is preserved as soon as they stay in a neighborhood.

######  Lemma 4.2.

[[22][Lemma 3.3]] Let $K$ be a compact invariant set of $f$. Suppose $K$
admits a dominated splitting $T_{K}M=E^{1}\oplus E^{2}\oplus E^{3}$. Then
there are $\rho>r_{0}>0$ such that the neighborhood $B_{\rho}(x)$ of every
$x\in K$ admits foliations
${\mathcal{F}}^{1}_{x}$,${\mathcal{F}}^{2}_{x}$,${\mathcal{F}}^{3}_{x}$,${\mathcal{F}}^{12}_{x}$
and ${\mathcal{F}}^{23}_{x}$ such that for every $y\in B_{r_{0}}(x)$ and
$*\in\\{1,2,3,12,23\\}$:

  * (i)

${\mathcal{F}}_{x}^{*}(y)$ is $\operatorname{C}^{1}$ and tangent to the
respective cone.

  * (ii)

$f({\mathcal{F}}^{*}_{x}(y,r_{0}))\subset{\mathcal{F}}^{*}_{f(x)}(f(y))$ and
$f^{-1}({\mathcal{F}}^{*}_{x}(y,r_{0}))\subset{\mathcal{F}}^{*}_{f^{-1}(x)}(f^{-1}(y))$.

  * (iii)

${\mathcal{F}}^{1}_{x}$ and ${\mathcal{F}}^{2}_{x}$ subfoliate
${\mathcal{F}}^{12}_{x}$ and ${\mathcal{F}}^{2}_{x}$ and
${\mathcal{F}}^{3}_{x}$ subfoliate ${\mathcal{F}}^{23}_{x}$.

######  Remark 4.3.

In the following proof, we will consider two kinds of fake foliations:

  * •

The partially hyperbolic splitting $E^{s}\oplus E^{c}\oplus E^{u}$ on the
singularities ${\rm Sing}(X)\cap\tilde{\Lambda}$ induces fake foliations
$\hat{{\mathcal{F}}}^{i}$ $i=s,c,u,sc,cu$, where $\hat{{\mathcal{F}}}^{s}$ and
$\hat{{\mathcal{F}}}^{u}$ have contracting and expanding property.

  * •

The dominated splitting $E^{s}\oplus F^{cu}$ on $\tilde{\Lambda}$ induces fake
foliations $\tilde{{\mathcal{F}}}^{s}$ and $\tilde{{\mathcal{F}}}^{cu}$.

We note that the proposition (ii) above may not hold for flows, i.e., the fake
foliation does not preserved by $\phi_{t}$ when $t\notin\mathbb{Z}$. Moreover,
the flow orbit is not even locally saturated by $\tilde{{\mathcal{F}}}^{cu}$
leaf. In fact, the fake foliations depends on extension of the dynamics in the
tangent bundle (see [11][Proposition 3.1]), which is in general not preserved
by the flow.

The following lemma provides an important observation on infinity Bowen ball,
a proof can be found in the proof of [22][Theorem 3.1], which is based on the
local product structure and hyperbolicity of $\tilde{{\mathcal{F}}}^{s}$:

######  Lemma 4.4.

For any $x\in\tilde{\Lambda}$,
$B_{\infty}(x,r_{0})\subset\tilde{{\mathcal{F}}}_{x}^{cu}(x)$.

As an immediately corollary of Lemma 4.4, we obtain a kind of weak saturated
property for $cu$ fake leaf $\tilde{{\mathcal{F}}}^{cu}(x)$:

######  Corollary 4.5.

There is $r_{1}>0$ such that for any $y\in B_{\infty}(x,r_{0}/2)$,
$y_{t}\in\tilde{{\mathcal{F}}}_{x}^{cu}(x)$ for $|t|\leq r_{1}$.

###### Proof.

Denote $D_{0}=\max_{x\in M}{\|X(x)\|}$ and $r_{1}=\frac{r_{0}}{2D_{0}}$. Then
for any $|t|\leq r_{1}$, the segment of flow orbit between $f^{n}(y_{t})$ and
$f^{n}(y)$ has length bounded by $r_{1}D_{0}\leq r_{0}/2$. Hence,

$d(f^{n}(x),f^{n}(y_{t}))\leq
d(f^{n}(x),f^{n}(y))+d(f^{n}(y),f^{n}(y_{t}))\leq r_{0}.$

Which implies that $y_{t}\in
B_{\infty}(x,r_{0})\subset\hat{{\mathcal{F}}}^{cu}(x)$ for $|t|\leq r_{1}$.

∎

The discussion of the infinity Bowen ball of a singularity is the same as that
in [22][Theorem 3.1]:

######  Lemma 4.6.

There is $L>0$ such that for any singularity $\sigma\in\tilde{\Lambda}$,
$B_{\infty}(\sigma,r_{0}/2)$ is a single point or a 1-dimensional center
segment with length bounded by $L$. In the latter case, the center segment
consists singularities and saddle connections.

######  Remark 4.7.

By Lemma 4.6, ${\rm
Sing}(X)\cap\tilde{\Lambda}\subset\\{\tilde{\sigma}_{1},\dots,\tilde{\sigma}_{m}\\}\cup
I_{1}\cup\dots\cup I_{t}$ where
$\\{\tilde{\sigma}_{1},\dots,\tilde{\sigma}_{m}\\}$ are finite isolated
singularities, and $\\{I_{1},\dots,I_{t}\\}$ are center segments tangent to
the center direction. Moreover, each center segment $I_{i}$ is fixed by $f$
and contained in $B_{\infty}(\sigma_{i},r_{0}/2)$ for some singularity
$\sigma_{i}\in I_{i}$. We assume that different center segments $I_{i}\neq
I_{j}$ only intersect on the boundary points, which are singularities. By a
double covering, we may always suppose $E^{c}$ is orientable and call the
boundary points of each center segment $I_{i}$ by the left extreme point
$\sigma_{i}^{-}$ and the right extreme point $\sigma_{i}^{+}$.

Now we give more description for the dynamics close to the center segments.
Consider the fake leaf $\hat{{\mathcal{F}}}_{\sigma_{i}}^{cu}(\sigma_{i})$
which contains $I_{i}$. Define:

$W^{uu}_{\delta}(I_{i})=\cup_{x\in
I_{i}}\hat{{\mathcal{F}}}^{u}_{\sigma_{i}}(x,\delta)\subset\hat{{\mathcal{F}}}_{\sigma_{i}}^{cu}(\sigma_{i})\;\;\text{and}\;\;{\mathcal{N}}_{i}(\delta)=\cup_{y\in
W^{uu}_{\delta}(I_{i})}\hat{{\mathcal{F}}}^{s}_{\sigma_{i}}(y,\delta).$

######  Lemma 4.8.

There are $\delta,r_{2}>0$ and center segments $I_{i,j}\subset I_{i}$ for
$1\leq j\leq k_{i}$ such that $\operatorname{length}(I_{i,j})<r_{2}$, ${\rm
Sing}(X)\cap I_{i}={\rm Sing}(X)\cap(\cup_{j}I_{i,j})$ and for any non-trivial
ergodic measure $\mu$ of $\phi_{t}$, one has
$\mu({\mathcal{N}}_{i,j}(\delta))=0$ for each $I_{i,j}$.

###### Proof.

There is $\varepsilon>0$ such that for any
$z\in{\mathcal{F}}^{s}_{r_{0}}(\sigma)\setminus{\mathcal{F}}^{s}_{r_{0}/2}(\sigma)$
where $\sigma$ belongs to ${\rm Sing}(X)\cap\tilde{\Lambda}$, and for any
$z^{{}^{\prime}}\in B_{\varepsilon}(z)$, $X(z^{{}^{\prime}})$ is tangent to
the $E^{s}$ cone.

Take $r_{2}>0$ such that for any $x,y\in I_{i}$ with $d(x,y)<r_{2}$ and for
any
$z\in\hat{{\mathcal{F}}^{s}}_{\sigma}(x,r_{0})\setminus\hat{{\mathcal{F}}}^{s}_{\sigma}(x,r_{0}/2)$,
there is
$z^{{}^{\prime}}\in\hat{{\mathcal{F}}^{s}}_{\sigma}(y,r_{0})\setminus\hat{{\mathcal{F}}}^{s}_{\sigma}(y,r_{0}/2)$
satisfying $d(z,z^{{}^{\prime}})<\varepsilon$.

Recall that $I_{i}\subset B_{\infty}(\sigma_{i},r_{0}/2)$ is invariant and
$f|I_{i}$ is partially hyperbolic. By the stable manifold theorem ([19]) for
normally hyperbolic submanifold $I_{i}$, for any $x\in I_{i}$,
$\hat{{\mathcal{F}}}_{\sigma}^{s}(x)$ is the local strong stable manifold of
$x$. Similar property holds for the local strong unstable manifold. Moreover,

$W^{s}_{loc}(I_{i})=\cup_{x\in
I_{i}}\hat{{\mathcal{F}}}_{\sigma}^{s}(x,r_{0})\;\;\;\text{and}\;\;\;W^{u}_{loc}(I_{i})=\cup_{x\in
I_{i}}\hat{{\mathcal{F}}}_{\sigma}^{u}(x,r_{0}).$

Choose finite sub-segments $(I_{i,j})_{j=1}^{k_{i}}$ such that

  * •

$\operatorname{length}(I_{i,j})<r_{2}$;

  * •

the boundary points of each $I_{i,j}$ are singularities, and different
subsegments only intersect on the boundary points;

  * •

${\rm Sing}(X)\cap I_{i}={\rm Sing}(X)\cap(\cup_{j}I_{i,j})$.

Suppose there are $\delta_{n}\rightarrow 0$ such that
$\mu({\mathcal{N}}_{i,j}(\delta_{n}))>0$. Since $\mu$ is a non-trivial ergodic
measure, $\mu(W^{s}_{loc}(I_{i,j}))=\mu(W^{u}_{loc}(I_{i,j}))=0$. There exists

$x^{n}\in{\mathcal{N}}_{i,j}(\delta_{n})\setminus(W^{s}_{loc}(I_{i,j})\cup
W^{u}_{loc}(I_{i,j})).$

By [19], the negative iteration of $x^{n}$ must leave
${\mathcal{N}}_{i,j}(r_{0})$. Take $t_{n}<0$ the last time such that
$f^{m}(x^{n})\in{\mathcal{N}}_{i,j}(r_{0})$ for any $t_{n}\leq m\leq 0$. Then
$t_{n}\rightarrow-\infty$ and we may suppose

$f^{-t_{n}}(x^{n})\rightarrow x^{*}\in W^{s}_{loc}(I_{i,j})\setminus I_{i,j}.$

As discussed before, for $n$ is sufficiently large, $X(f^{t_{n}}(x^{n}))$ is
tangent to $E^{s}$ cone. Denote by
$l_{\delta}=(\phi_{t}(f^{t_{n}}(x_{n})))_{|t|\leq\delta}$. Then for $\delta$
sufficiently small, $l_{\delta}$ is tangent to the $E^{s}$ cone. It follows
easily that
$\lim_{m\to\infty}\operatorname{length}f^{t_{n}-m}(l_{\delta})\to\infty$,
which contradicts the fact that the length of any image of a flow orbit with
bounded length is always bounded, as explained in the last paragraph of the
proof of Lemma 4.1. ∎

From now on, we treat every isolated singularity as a trivial center segment.
Re-order the center segments $\\{I_{i,j}\\}$ by $\\{I_{i}\\}_{1\leq i\leq t}$
and choose a singularity $\sigma_{i}$ inside each $I_{i}$. From the
construction, $\operatorname{length}(I_{i})<r_{2}$ for every $I_{i}$. Lemma
4.8 shows that, for any non-trivial ergodic measure supported in $U$, its
generic point can only approach the extreme singularities of every center
segment from their half neighborhoods which is defined as following. Let
$\sigma_{i}^{+}$ be a right extreme point of $I_{i}$,
$\cup_{y\in\hat{{\mathcal{F}}}^{u}_{\sigma_{i}}(\sigma_{i}^{+},r_{0})}\hat{{\mathcal{F}}}^{s}_{\sigma_{i}}(y,r_{0})$
is a topological codimension-one disk, which separates
$B_{r_{0}}(\sigma^{+}_{i})$ into two connect components, the right
neighborhood of $\sigma_{i}^{+}$, denoted by $B^{r}_{r_{0}}(\sigma_{i}^{+})$,
is just the the right component. In a similar way, we may define the left
neighborhood $B^{l}_{r_{0}}(\sigma_{i}^{-})$ for the left extreme point
$\sigma_{i}^{-}$.

In the following proof, we only consider the right extreme points, similar
results always hold for the left ones.

######  Lemma 4.9.

There is $\delta>0$ such that for every non-trivial ergodic measure $\mu$ of
$\phi_{t}$ and for every center segment $I_{i}$, if
$\mu(B^{r}_{\delta}(\sigma^{+}_{i}))>0$, then the right branch of the center
leaf $\hat{{\mathcal{F}}}^{c}_{\sigma_{i}}(\sigma^{+}_{i},r_{2})$, denoted by
$\hat{{\mathcal{F}}}^{c,+}_{\sigma_{i}}(\sigma^{+}_{i},r_{2})$, is
topologically contracting.

###### Proof.

Since there are only finitely many extreme points, it suffices to prove only
for one right extreme point $\sigma_{i}^{+}$.

First observe that there are no singularities contained in
$\hat{{\mathcal{F}}}^{c,+}_{\sigma_{i}}(\sigma^{+}_{i},r_{2})$. Suppose such a
singularity does exist, there is a center segment connecting this singularity
and $\sigma_{i}^{+}$ from the right. Then the proof of Lemma 4.8 implies that
generic point of $\mu$ cannot approach $\sigma_{i}^{+}$ from the right, a
contradiction.

Because $\hat{{\mathcal{F}}}^{c,+}_{\sigma_{i}}(\sigma^{+}_{i},r_{2})$ is
invariant and contains no singularities, it is topologically expanding or
contracting. Assume it is topologically expanding, then
$\hat{{\mathcal{F}}}^{cu,+}_{\sigma_{i}}(\sigma_{i}^{+},r_{2})$ belongs to the
unstable set of $\sigma_{i}^{+}$. We claim that there is
$\delta^{{}^{\prime}}>0$ such that
$\mu(B^{r}_{\delta^{{}^{\prime}}}(\sigma_{i}^{+}))=0$. Then we may take
$\delta<\delta^{{}^{\prime}}$ and conclude the proof.

It remains to prove this claim. Suppose there are positive numbers
$\lim\delta_{n}^{{}^{\prime}}\to 0$ such that
$\mu(B^{r}_{\delta^{{}^{\prime}}_{n}}(\sigma_{i}^{+}))>0$. Because $\mu$ is
non-trivial, we may assume there are $\mu$ generic points

$x^{n}\in
B^{r}_{\delta^{{}^{\prime}}_{n}}(\sigma_{i}^{+})\setminus(W^{u}(\sigma^{+}_{i})\cup{\mathcal{F}}^{s}(\sigma_{i}^{+})).$

Consider $t_{n}<0$ the last time such that $f^{m}(x^{n})\in
B^{r}_{r_{0}}(\sigma^{+}_{i})$ for any $t_{n}\leq m\leq 0$. It is easy to see
that $t_{n}\rightarrow\infty$. By the topological expansion of
$\hat{{\mathcal{F}}}^{cu,+}_{\sigma_{i}}(\sigma_{i}^{+},r_{2})$, we may
suppose $f^{-t_{n}}(x^{n})\rightarrow
x^{*}\in{\mathcal{F}}^{s}(\sigma^{+}_{i})$.

The rest proof is the same as the last paragraph of the proof of Lemma 4.1:
for $n$ sufficiently large, $f^{t_{n}}(x^{n})$ is close to $x^{*}$ and
$X(f^{-t_{n}}(x^{n}))$ is tangent to $E^{s}$ cone. This is a contradiction,
since the length of a small segment of the flow orbit which contains
$f^{-t_{n}}(x^{n})$ grows exponentially fast. The proof is complete. ∎

Lemmas 4.8 and 4.9 explain to us that the generic point of any non-trivial
ergodic measure can only approach finitely many half neighborhood of
singularities which are topologically ‘hyperbolic’.

Take $\delta_{0}<r_{2}$ satisfying Lemmas 4.8 and 4.9. From now on, we rename
the half neighborhoods of the extreme points by
$B_{\delta_{0}}^{h}(\sigma_{1}),\dots,B_{\delta_{0}}^{h}(\sigma_{k})$. Note
that, in this notation, when we are talking about the two half neighborhoods
of the same isolated singularity, we give different name for this singularity.

####  4.2.2. Expanding in a $cu$ fake leaf

In this subsubsection, we will build the expansion of the distance between
different flow orbits contained in the same infinity Bowen ball. The
discussion can be divided into two cases: the orbits are away from
singularities (Lemma 4.13), and are close to singularities (Lemma 4.17). We
need the following notation:

######  Definition 4.10.

For $x\in U\setminus{\rm Sing}(X)$, write
$B_{\delta}^{\bot}(x)=\exp_{x}({\mathcal{N}}_{x}(\delta))$. And for $y\in
B_{\delta}^{\bot}(x)\cap\tilde{{\mathcal{F}}}^{cu}(x)$, denote by
$d_{x}^{*}(x,y)$ the distance between $x$ and $y$ in the submanifold
$B_{\delta}^{\bot}(x)\cap\tilde{{\mathcal{F}}}^{cu}(x)$.

######  Lemma 4.11.

For any $\varepsilon>0$ and $L>0$, there is $\delta>0$ such that for every
$x\in\tilde{\Lambda}\setminus B_{\varepsilon}({\rm Sing}(X))$ and
$y\in\tilde{{\mathcal{F}}}_{x}^{cu}(x)\cap B^{\bot}_{\delta}(x)$, we have

$d(x,y)<d^{*}_{x}(x,y)<Ld(x,y).$

###### Proof.

This comes from the continuity of $\tilde{{\mathcal{F}}}^{cu}_{x}(x)$ in the
$\operatorname{C}^{1}$ topology.

∎

######  Definition 4.12.

For any regular point $x\in U$, denote $P_{x}$ the projection along the flow:

$P_{x}:B_{\delta}(x)\rightarrow B_{r_{0}}^{\bot}(x)$

which is defined in a neighborhood of $x$. For any point $y$ in this
neighborhood, define $t_{x}(y)$ the time which satisfies
$y_{t_{x}(y)}=P_{x}(y)$.

######  Lemma 4.13.

For $\varepsilon>0$ and $1<b_{0}<\frac{1}{\lambda_{0}}$, there is $\delta>0$
such that for any $x$ satisfying
$(x_{t})_{t\in[0,1]}\subset\tilde{\Lambda}\setminus B_{\varepsilon}({\rm
Sing}(X))$, and for any $y\in B_{\infty}(x,\delta)$:

$d_{x_{1}}^{*}(x_{1},P_{x_{1}}(y_{1}))>b_{0}\frac{\|X(x)\|}{\|X(x_{1})\|}d^{*}_{x}(x,P_{x}(y)).$

###### Proof.

Write $C=(\frac{1}{\lambda_{0}b_{0}})^{1/6}$ and take
$0<t_{0}<\min\\{\frac{r_{0}}{2},\frac{r_{1}}{2}\\}$ such that for any
2-dimensional subspace $\Sigma$ in the tangent space,
$1/C<\operatorname{Jac}(\Phi_{t}|_{\Sigma})<C$ for any $|t|<t_{0}$. For two
vectors $u,v\in T_{x}M$, denote by ${\mathcal{P}}[u,v]$ the parallelogram
defined by these two vectors and ${\mathcal{A}}(u,v)$ its area.

When $\delta>0$ is small, for $y\in B_{\infty}(x,\delta)$,
$|t_{x}(y)|,|t_{x_{1}}(y_{1})|<t_{0}$. Then by Corollary 4.5,
$P_{x}(y)\in\tilde{{\mathcal{F}}}^{cu}_{x}(x)\cap B^{\bot}_{r_{0}}(x)$ and
$P_{x_{1}}(y_{1})\in\tilde{{\mathcal{F}}}^{cu}_{x}(x)\cap
B^{\bot}_{r_{0}}(x)$.

There is $0<\varepsilon_{1}\ll r_{0}/2$ which depends on $\varepsilon$ and
$b_{0}$, such that for any $z\in B^{\bot}_{\varepsilon_{1}}(x_{t})$, the
following conditions are satisfied:

  * (a)

$\frac{1}{C}<\frac{\|X(z)\|}{\|X(x_{t})\|}<C$.

  * (b)

For $v\in T_{z}B^{\bot}_{\varepsilon_{1}}(x_{t})$, we have
$\frac{1}{C}\|v\|\|X(z)\|\leq{\mathcal{A}}(v,X(z))\leq C\|v\|\|X(z)\|$.

We may further suppose that $\delta$ is sufficiently small, such that
$d^{*}_{x}(x,P_{x}(y))<\varepsilon_{1}$,
$d^{*}_{x_{1}}(x_{1},P_{x_{1}}(y_{1}))<\varepsilon_{1}$ and by Lemma 4.11:

(3) $\begin{split}d(x,P_{x}(y))&<d^{*}_{x}(x,P_{x}(y))<Cd(x,P_{x}(y)),\\\
d(x_{1},P_{x_{1}}(y_{1}))&<d^{*}_{x_{1}}(x_{1},P_{x_{1}}(y_{1}))<Cd(x_{1},P_{x_{1}}(y_{1})).\\\
\end{split}$

Moreover, take a curve $\tilde{l}_{1}\subset
B^{\bot}_{\varepsilon_{1}}(x_{1})\cap\hat{{\mathcal{F}}}^{cu}_{x_{1}}(x_{1})$
which links $x_{1}$ and $P_{x_{1}}(y_{1})$ with
$\operatorname{length}(\tilde{l}_{1})=d_{x_{1}}^{*}(x_{1},P_{x_{1}}(y_{1})))$,
and write $\tilde{l}_{0}=f^{-1}(\tilde{l}_{1})$ and $l=P_{x}(\tilde{l}_{0})$,
we suppose $|t_{x}|_{\tilde{l}_{0}}|\leq t_{0}$ and $l\subset
B^{\bot}_{\varepsilon_{1}}(x)$. Then $l$ is a curve contained in
$B_{\varepsilon_{1}}^{\bot}(x)$ which connects $x$ and $P_{x}(y)$. We note
that although by the local invariance,
$\tilde{l}_{0}\subset\tilde{{\mathcal{F}}}^{cu}_{x}(x)$, $l$ is not necessary
contained in $\tilde{F}^{cu}_{x}(x)$. Denote $H$ the smooth holonomy map
between $\tilde{l}_{1}$ and $l$ which is induced by flow, then
$H(P_{x_{1}}(y_{1}))=P_{x}(y)$. We claim that

$\|dH\|\leq\lambda_{0}C^{5}\frac{\|X(x_{1})\|}{\|X(x)\|}.$

Which implies that

$d(x,P_{x}(y))\leq\operatorname{length}(l)\leq
C^{5}\lambda_{0}\frac{\|X(x_{1})\|}{\|X(x)\|}d_{x_{1}}^{*}(x_{1},P_{x_{1}}(y_{1})).$

Suppose this claim for a while, as a corollary of (3), we conclude the proof
of this lemma:

$\begin{split}d^{*}_{x}(x,P_{x}(y))&<Cd(x,P_{x}(y))\\\
&<C^{6}\lambda_{0}\frac{\|X(x_{1})\|}{\|X(x)\|}d_{x_{1}}^{*}(x_{1},P_{x_{1}}(y_{1}))\\\
&=\frac{1}{b_{0}}\frac{\|X(x_{1})\|}{\|X(x)\|}d_{x_{1}}^{*}(x_{1},P_{x_{1}}(y_{1})).\\\
\end{split}$

It remains to prove this claim.

For any $\tilde{z}_{1}\in\tilde{l}_{1}$, denote $\tilde{v}$ a tangent vector
of $\tilde{l}_{1}$, write $\tilde{z}_{0}=\phi_{-1}(\tilde{z}_{1})$,
$z=H(\tilde{z})\in l$ and $v=dH^{-1}(\tilde{v})$.

Then by sectional-hyperbolic,

${\mathcal{A}}(\Phi_{-1}(\tilde{v}),X(\tilde{z}_{0}))\leq\lambda_{0}{\mathcal{A}}(\tilde{v},X(\tilde{z}_{1})).$

Because $|t_{x}|_{\tilde{l}_{0}}|<t_{0}$, by the assumption on $t_{0}$,

(4)
${\mathcal{A}}(\Phi_{t_{x}(\tilde{z}_{0})}{\mathcal{P}}[\Phi_{-1}(\tilde{v}),X(\tilde{z}_{0})])\leq
C\lambda_{0}{\mathcal{A}}(\tilde{v},X(\tilde{z}_{1})).$

Since
$\Phi_{t_{x}(\tilde{z}_{0})}(\Phi_{-1}(\tilde{v}))=\Phi_{-1+t_{x}(\tilde{z}_{0})}(\tilde{v})$
and $\Phi_{t_{x}(\tilde{z}_{0})}(X(\tilde{z}_{0}))=X(z)$, we have

(5)
${\mathcal{A}}(\Phi_{t_{x}(\tilde{z}_{0})}{\mathcal{P}}[\Phi_{-1}(\tilde{v}),X(\tilde{z}_{0})])={\mathcal{A}}(\Phi_{-1+t_{x}(\tilde{z}_{0})}(\tilde{v}),X(z)).$

Note that $dH(\tilde{v})$ is the projection of
$\Phi_{-1+t_{x}(\tilde{z}_{0})}(\tilde{v})$ along $X(z)$ on
$T_{z}B^{\bot}(x)$, combine equations (4), (5), we have that

(6)
${\mathcal{A}}(v,X(z))={\mathcal{A}}(\Phi_{-1+t_{x}(\tilde{z}_{0})}(\tilde{v}),X(z))\leq
C\lambda_{0}{\mathcal{A}}(\tilde{v},X(\tilde{z}_{1})).$

By the assumptions (a) and (b) above, $C^{-2}\|v\|\|X(x)\|\leq
C^{3}\lambda_{0}\|\tilde{v}\|\|X(x_{1})\|$, which implies:

$\|dH_{\tilde{z}}\|=\frac{\|v\|}{\|\tilde{v}\|}\leq
C^{5}\lambda_{0}\frac{\|X(x_{1})\|}{\|X(x)\|}.$

∎

######  Lemma 4.14.

For every $\varepsilon>0$, there are $\delta>0$ and $K>0$ such that for any
non-trivial ergodic measure $\mu$ which satisfies
$\mu(B_{\varepsilon}^{h}(\sigma_{i}))=0$ for every $i=1,\dots,k$, and for any
$x\in\operatorname{supp}(\mu)$, $B_{\infty}(x,\delta)$ is a flow segment with
length bounded by $K>0$.

###### Proof.

By Lemma 4.8 and the assumption, $\operatorname{supp}(\mu)$ is away from
singularities. Apply Lemma 4.13 on $\\{x_{i};i\in\mathbb{N}\\}$, there is
$\delta>0$ such that for any $y\in B_{\infty}(x,\delta)$:

$d_{x_{n}}^{*}(x_{n},P_{x_{n}}(y_{n}))>b^{n}_{0}\frac{|X(x_{n})|}{|X(x)|}d^{*}_{x}(x,P_{x}(y)).$

Note that $\operatorname{supp}(\mu)\cap{\rm Sing}(X)=\emptyset$,
$X|_{\operatorname{supp}(\mu)}$ is uniformly bounded from above and below.
Because $d_{x_{n}}^{*}(x_{n},P_{x_{n}}(y_{n}))$ is bounded from above, we
conclude that $d^{*}_{x}(x,P_{x}(y))=0$, i.e., $y$ belongs to the local orbit
of $x$, and there is $K>0$ such that each connected component of ${\rm
Orb}(x)\cap B_{\delta}(x)$ has length bounded by $K$. We complete the proof. ∎

Now let us deal with the case $\operatorname{supp}(\mu)$ contains
singularities. We will show that there exists $\delta>0$ such that for every
$y\in B_{\infty}(x,\delta)$, $d_{x_{n}}^{*}(x_{n},P_{x_{n}}(y_{n}))$ is
expanding when $x$ and $x_{n}$ both are away from singularities.

######  Definition 4.15.

An extreme singularity $\sigma_{i}$ ($1\leq i\leq k$) is accumulated if there
is a non-trivial ergodic measure $\mu$ of $\phi_{t}$, such that
$\mu(B^{h}_{\delta_{0}}(\sigma_{i}))>0$.

######  Remark 4.16.

By Lemma 4.9, for each accumulated singularity $\sigma_{i}$,
$B^{h}_{\delta_{0}}(\sigma_{i})$ is topologically hyperbolic.

For $1\leq i\leq k$ and $0<\delta\ll\delta_{0}$, we consider the set

$J^{h}_{\delta}(\sigma_{i})=\\{y:y\in
B_{\delta_{0}}^{h}(\sigma_{i})\;\;\;\text{with}\;\;\;\|X(y)\|\leq\delta\\}.$

There is $\delta_{1}>0$ small enough, such that
$\partial(J_{\delta_{1}}^{h}(\sigma_{i}))\cap\partial(B^{h}_{\delta_{0}}(\sigma_{i}))=\emptyset$
where $\partial(.)$ denotes the boundary of a set.

######  Lemma 4.17.

For every $L>1$, there are $0<\varepsilon\ll\delta_{1}$ and $\delta>0$ such
that for each accumulated singularity $\sigma$ and any $x\in
B^{h}_{\delta_{0}}(\sigma)\cap\tilde{\Lambda}$ which satisfies

  * •

$(x_{t})_{t\in[0,T]}\subset B^{h}_{\delta_{0}}(\sigma)$ and
$(x_{t})_{t\in[0,T]}\cap B^{h}_{\varepsilon}(\sigma)\neq\emptyset$.

  * •

$x,x_{T}\notin J_{\delta_{1}}(\sigma)$,
$(x_{t})_{t\in[0,1]}\cap\partial{J^{h}_{\delta_{1}}}(\sigma)\neq\emptyset$ and
$(x_{t})_{t\in[T-1,T]}\cap\partial{J^{h}_{\delta_{1}}}(\sigma)\neq\emptyset$.

then for any $y\in B_{\infty}(x,\delta)$, we have

$d^{*}_{x_{T}}(x_{T},P_{x_{T}}(y_{T}))>Ld^{*}_{x}(x,P_{x}(y)).$

In order to simplify the proof, we may assume $T\in\mathbb{N}$. The following
proof depends on a local product structure in
$\mathcal{{\mathcal{F}}}^{cu}_{x_{n}}(x_{n})$ ($n=0,\dots,T$), which blends
two different families of fake foliations (see Remark 4.3):

  * •

$\hat{{\mathcal{F}}}_{\sigma}^{i}$ ($i=s,c,u,cs,cu$) which comes from the
partially hyperbolic splitting $E^{s}\oplus E^{c}\oplus E^{u}$ of the
singularities;

  * •

$\tilde{{\mathcal{F}}}^{i}_{x}$ ($i=s,cu$) which comes from the sectional-
hyperbolic splitting $E^{s}\oplus E^{cu}$ defined on $\tilde{\Lambda}$.

###### Proof of Lemma 4.17:.

First we define a new 1-dimensional center fake foliation
$\overline{{\mathcal{F}}}^{c}_{x_{t}}$ in
$\tilde{{\mathcal{F}}}^{cu}_{x_{t}}(x_{t})$, $0\leq t\leq T$ which is induced
by $\hat{{\mathcal{F}}}^{cs}_{x_{t}}$:

$\overline{{\mathcal{F}}}^{c}_{x_{t}}(z)=\hat{{\mathcal{F}}}^{cs}_{x_{t}}(z)\cap\tilde{{\mathcal{F}}}^{cu}_{x_{t}}(x_{t}),\;\;\text{for
every}\;\;z\in{\mathcal{F}}^{cu}_{x_{t}}(x_{t}).$

By the local invariance of the fake foliations
$\tilde{{\mathcal{F}}}^{cu}_{x_{t}}$ and $\hat{{\mathcal{F}}}^{cs}_{x_{t}}$,
the new center fake foliation is also locally invariant, i.e.

$f(\overline{{\mathcal{F}}}^{c}_{x_{t},r_{0}}(z))\subset\overline{{\mathcal{F}}}^{c}_{x_{t+1}}(z_{1})\;\;\;\text{for
any}\;\;\;z\in\tilde{{\mathcal{F}}}^{cu}_{x,r_{0}}\;\;\text{and}\;\;t\in[0,T-1].$

Let $\overline{{\mathcal{D}}}^{u}_{x}\ni
x\subset\tilde{{\mathcal{F}}}^{cu}_{x}(x)$ be a leaf with dimension
$dim(E^{u}_{\sigma})$ and tangent to the $a_{0}$-$E^{u}$ cone (which is
defined near to $\sigma$). By the forward invariance of the $E^{u}$ cone, this
leaf is expanding for $f^{n}$ $0\leq n\leq T$ and the connected component of
$f^{n}(\overline{{\mathcal{D}}}^{u}_{x})\cap\tilde{{\mathcal{F}}}^{cu}_{x_{n}}(x_{n})$
which contains $x_{n}$, denoted by $\overline{{\mathcal{D}}}^{u}_{x_{n}}$, is
still tangent to the $E^{u}$ cone. Hence, there is a unique transverse
intersection between $\overline{{\mathcal{D}}}^{u}_{x_{n}}$ and
$\overline{{\mathcal{F}}}^{c}_{x_{n}}(z)$ for any
$z\in\tilde{{\mathcal{F}}}^{cu}_{x_{n}}(x_{n},r_{0})$, we denote this
intersection by

$[z,x_{n}]=\overline{{\mathcal{F}}}^{c}_{x_{n}}(z)\pitchfork\overline{{\mathcal{D}}}^{u}_{x_{n}}.$

This local product structure is preserved by the iteration of $f$. In fact, by
the local invariance of fake foliation
$\overline{{\mathcal{F}}}^{c}_{x_{n}}(.)$ and the expanding property of
$\overline{{\mathcal{D}}}^{u}_{x_{n}}$, for every $y\in B_{\infty}(x,r_{0})$,
$f^{n}([y,x])=[y_{n},x_{n}]$ for $0\leq n\leq T$.

Suppose $\delta$ is sufficiently small, there is $r_{\delta}>0$ such that
$P_{z}|_{B_{\delta}(z)}$ is well defined and $|t_{z}|_{B_{\delta}(z)}|\ll
r_{\delta}\ll r_{0}$ for every $z\in B_{\delta_{0}}^{h}(\sigma)\setminus
J^{h}_{\delta_{1}}(\sigma)$. The choice of $\delta$ depends on $\varepsilon$
and will be given later. Now consider $y\in B_{\infty}(x,\delta)$ and write
$\tilde{y}_{T}=P_{x_{T}}(y_{T})$. Then $\tilde{y}=f^{-T}(\tilde{y}_{T})=y_{t}$
for some $|t|<r_{\delta}$. In particular, by Corollary 4.5,
$\tilde{y}\in\tilde{{\mathcal{F}}}^{cu}_{x}(x)\cap
B_{\infty}(x,\delta+r_{\delta}D)$, recall that $D=\max\|X(.)\|$.

Denote $\tilde{z}=[\tilde{y},x]$, by the invariance of the product structure,
$\tilde{z}_{n}=[\tilde{y}_{n},x_{n}]$ for any $0\leq n\leq T$. Take
$\tilde{l}_{T}=\tilde{l}_{T}^{c}\cup\tilde{l}_{T}^{u}$ a piecewise smooth
curve connecting $\tilde{y}_{T}=P_{x_{T}}(y_{T})$ and $x_{T}$ induced from the
local product structure:
$\tilde{l}_{T}=\tilde{l}_{T}^{c}\cup\tilde{l}_{T}^{u}$ where
$\tilde{l}_{T}^{c}\subset\overline{{\mathcal{F}}}^{c}_{x_{T}}(\tilde{y}_{T})$
connects $\tilde{y}_{T}$ and $\tilde{z}_{T}$; and $\tilde{l}_{T}^{u}$ is a
shortest segment contained in $\overline{{\mathcal{F}}}^{u}_{x_{T}}(x_{T})$
which connects $x_{T}$ and $\tilde{z}_{T}$. For $0\leq n\leq T$, write
$\tilde{l}_{n}=\tilde{l}^{c}_{n}\cup\tilde{l}^{u}_{n}$ a piecewise smooth
curve connects $x_{n}$ and $\tilde{y}_{n}$ where
$\tilde{l}_{n}^{c}=f^{n-T}(\tilde{l}_{T}^{c})$ and
$\tilde{l}_{n}^{u}=f^{n-T}(\tilde{l}_{T}^{u})$. Then $\tilde{l}_{n}^{c}$ is a
fake center curve contained in
$\overline{{\mathcal{F}}}^{c}_{x_{n}}(\tilde{y}_{n})$ which connects
$\tilde{y}_{n}$ and $\tilde{z}_{n}$. Recall that
$\overline{{\mathcal{D}}}^{u}_{x_{T}}$ is exponentially contracting under the
map $f^{n-T}$, $\tilde{l}^{n}_{u}$ is contained in the fake unstable leaf
$\overline{{\mathcal{D}}}^{u}_{x_{n}}$. By the uniform transversality between
these two fake leaves, there are $C_{1}>0$ and $0<\lambda_{0}<1$ such that

(7)
$\begin{split}&\max\\{\operatorname{length}(\tilde{l}^{u}_{(T)}),\operatorname{length}(\tilde{l}^{c}_{(T)})\\}\leq
C_{1}d(x_{T},\tilde{y}_{T}),\\\
&\operatorname{length}(\tilde{l}^{c}_{(n)})\leq
C_{1}(\delta+r_{\delta}D)\;\;\text{and}\;\;\operatorname{length}(\tilde{l}^{u}_{n})\leq\lambda_{0}^{n-T}\operatorname{length}(\tilde{l}^{u}_{T}).\\\
\end{split}$

Hence $\operatorname{length}(\tilde{l}_{(n)})\leq 2C_{1}(\delta+r_{\delta}D)$.
Now we give the condition $\delta$ need to satisfy. Let $\delta$ be
sufficiently small such that:

  * •

$P_{x}(\tilde{l}_{0})=l$ is well defined and $|t_{x}|_{\tilde{l}_{0}}|\leq
t_{0}\ll r_{0}/4$, where $t_{0}$ satisfies that for any 2-dimensional subspace
$\Sigma$ contained in the tangent space and for any $|t|\leq t_{0}$,
$|\operatorname{Jac}(\Phi_{t}|_{\Sigma})|\leq C_{1}$.

  * •

by Lemma 4.11:

$d(x,P_{x}(y))\leq d^{*}_{x}(x,P_{x}(y))\leq C_{1}d(x,P_{x}(y))$
$d(x_{T},\tilde{y}_{T})\leq d^{*}_{x_{T}}(x_{T},\tilde{y}_{T})\leq
C_{1}d(x_{T},\tilde{y}_{T}).$

We claim that there is $C_{2}$ such that
$\frac{1}{C_{2}}<\frac{\operatorname{length}(\tilde{l}^{u}_{0})}{\operatorname{length}(l^{u})}<C_{2}$,
which follows from the observation that the half neighborhood
$B^{h}_{\delta_{0}}(\sigma)$ is topologically hyperbolic, i.e., if we denote
$\hat{{\mathcal{F}}}^{cs,h}_{\sigma}(\sigma,r_{2})$ the corresponding branch
of center stable fake leaf located in the same side as the half neighborhood,
then by Lemma 4.9, $\hat{{\mathcal{F}}}^{cs,h}_{\sigma}(\sigma,r_{2})$ is
contained in the stable set of $\sigma$. When $\varepsilon$ is sufficiently
small, $x$ is close to $\hat{{\mathcal{F}}}^{cs,h}_{\sigma}(\sigma,r_{2})$,
therefore, the flow vector $X(x)$ is tangent to the $E^{s}\oplus E^{c}$ cone,
and is transverse to the $E^{u}$ cone. Note that $\tilde{l}^{u}_{0}$ is
tangent to the $E^{u}$ cone, it has a uniform angle with the flow direction.
Then one may apply the idea of the proof in Lemma 4.13. More precisely,
consider the holonomy map between $\tilde{l}^{u}_{0}$ and $l^{u}$ induced by
the flow, we are going to compare the areas of parallelograms generated by the
flow directions and the tangent vectors. Because the Jacobian restricted on
the parallelograms are bounded, and which are equivalent to the product of
length of vector of flow and the tangent vector, by the reason of the uniform
angle between them. We conclude the proof of this claim.

By the claim above, we deduce that

(8) $\operatorname{length}(l^{u})\leq
C_{2}\lambda_{0}^{T}\operatorname{length}(\tilde{l}^{u}_{T}).$

The estimation of $\operatorname{length}(l^{c})$ is quite different, we use a
similar argument as in the proof of Lemma 4.13. Let us give more explanation.
Consider the tangent map of the holonomy map $H:\tilde{l}^{c}_{T}\rightarrow
l^{c}$ induced by the flow, which contracts exponentially the area of the
parallelogram formed by the tangent vector of $\tilde{l}^{c}_{T}$ and the flow
vector. For this parallelogram and its image, both areas are equivalent to the
product of the length of the tangent vector and the length of the flow vector,
we conclude the exponentially contraction of the map $H$. There is $C_{3}>0$
such that that

(9)
$\operatorname{length}(l^{c})\leq\operatorname{length}(\tilde{l}^{c}_{T})\lambda_{0}^{T}C_{3}^{3}\frac{\|X(x_{T})\|}{\|X(x)\|}.$

Let $\hat{C}=\max\\{C_{1},C_{2},C_{3}\\}$. We now give the condition what
$\varepsilon$ need to satisfy. It is easy to see that $\lim_{\varepsilon\to
0}T\to\infty.$ We assume $\varepsilon$ is sufficiently small such that
$\lambda_{0}^{T}\hat{C}^{3}(\hat{C}^{2}D/\delta_{1}+1)\leq L^{-1}$, that is,
$T>-\frac{\operatorname{ln}L\hat{C}^{3}(\hat{C}^{2}D/\delta_{1}+1)}{\operatorname{ln}\lambda_{0}}$.

Because $x\notin J_{\delta_{1}}^{h}(\sigma)$, $\|X(x)\|\geq\delta_{1}$. By
(7), (8) and (9), one has that

$\begin{split}d^{*}_{x}(x,P_{x}(y))&\leq\hat{C}d(x,P_{x}(y))\leq\hat{C}(\operatorname{length}(l^{c})+\operatorname{length}(l^{u}))\\\
&\leq\hat{C}[C_{2}\lambda_{0}^{T}\operatorname{length}(\tilde{l}^{u}_{T})+\lambda_{0}^{T}C_{3}^{3}\frac{\|X(x_{T})\|}{\|X(x)\|}\operatorname{length}(\tilde{l}^{c}_{T})]\\\
&\leq\lambda_{0}^{T}\hat{C}^{3}(\hat{C}^{2}\frac{D}{\delta_{1}}+1)d_{x_{T}}^{*}(x_{T},P_{x_{T}}(y_{T}))\\\
&\leq\frac{1}{L}d_{x_{T}}^{*}(x_{T},P_{x_{T}}(y_{T})).\\\ \end{split}$

The proof is finished. ∎

####  4.2.3. Proof of Proposition 2.21

Now we are preparing to finish the proof of Proposition 2.21.

###### Proof of Proposition 2.21:.

We first prove that $\phi_{t}$ is entropy expansive restricted on a small
attracting neighborhood of $\Lambda$.

By Lemma 2.7, we need show that there is $\delta>0$ such that for any
invariant ergodic measure $\mu$ of $f$ and for $\mu$ almost every $x\in M$, we
have $h_{top}(f,B_{\infty}(x,\delta))=0.$

Fix $1<b_{0}<\frac{1}{\lambda_{0}}$. Let

$D^{{}^{\prime}}=\min\\{\frac{\|X(x)\|}{\|X(x^{{}^{\prime}})\|};\text{there
are singularities}\;\;\sigma,\sigma^{{}^{\prime}}\;\text{such that}$
$x\in(\phi_{t}(\partial(J^{h}_{\delta_{1}}(\sigma))))_{t\in[0,1]}\;\;\text{and}\;\;x^{{}^{\prime}}\in(\phi_{t}(\partial(J^{h}_{\delta_{1}}(\sigma^{{}^{\prime}}))))_{t\in[-1,0]}\\}.$

For $L=b_{0}/D^{{}^{\prime}}$, apply Lemma 4.17, we obtain $\varepsilon$ and
$\delta$. By taking a smaller value, we assume that $\delta<r_{0}/2$ and
satisfies Lemma 4.13 for the $\varepsilon$ above.

If $\mu$ is trivial, that means, it is an atomic measure supported on a
singularity $\sigma$. Then by Lemma 4.6, $B_{\infty}(x,\delta)$ is a
1-dimensional center segment with length bounded by $L_{0}>0$, hence has
vanishing topological entropy.

From now on, we suppose $\mu$ is non-trivial, and $x$ is any generic point of
$\mu$. By Lemma 4.6, we assume that $x\notin B_{r_{0}}({\rm Sing}(X))$.

Define a sequence of integers $0\leq
T_{1}<T_{1}^{{}^{\prime}}<T_{2}<T_{2}^{{}^{\prime}}\dots$ such that for each
$n\geq 1$, the following conditions are satisfied:

  * •

$(x_{t})_{[T^{{}^{\prime}}_{n},T_{n+1}]}\cap B_{\varepsilon}({\rm
Sing}(X))=\emptyset$;

  * •

there is a singularity $\sigma$ such that
$(x_{t})_{[T_{n},T_{n}^{{}^{\prime}}]}\subset B^{h}_{r_{0}}(\sigma)$ and
$(x_{t})_{[T_{n},T_{n}^{{}^{\prime}}]}\cap
B^{h}_{\varepsilon}(\sigma)\neq\emptyset$;

  * •

$x_{T_{n}},x_{T_{n}^{{}^{\prime}}}\notin J^{h}_{\delta_{1}}(\sigma)$,
$(x_{t})_{[T_{n},T_{n}+1]}\cap\partial{J^{h}_{\delta_{1}}(\sigma)}\neq\emptyset$
and
$(x_{t})_{[T_{n}^{{}^{\prime}}-1.T_{n}^{{}^{\prime}}]}\cap\partial{J^{h}_{\delta_{1}}(\sigma)}\neq\emptyset$.

If the above sequence is bounded, this implies that
$\mu(B_{\varepsilon}(\sigma))=0$. Then by Lemma 4.14, $B_{\infty}(x,\delta)$
is a 1-dimensional segment with bounded length, therefore has topological
entropy zero.

Hence we assume the sequence is defined with infinite length. For $n\geq 1$,
by Lemma 4.13, we have

$\begin{split}d^{*}_{x_{T_{n+1}}}(x_{T_{n+1}},P_{x_{T_{n+1}}}(y_{T_{n+1}}))&>b_{0}^{T_{n+1}-T_{n}^{{}^{\prime}}}\frac{\|X(x_{T_{n+1}})\|}{\|X(x_{T_{n}^{{}^{\prime}}})\|}d^{*}_{x_{T_{n}^{{}^{\prime}}}}(x_{T_{n}^{{}^{\prime}}},P_{x_{T_{n}^{{}^{\prime}}}}(y_{T_{n}^{{}^{\prime}}}))\\\
&\geq
b_{0}^{T_{n+1}-T_{n}^{{}^{\prime}}}D^{{}^{\prime}}d^{*}_{x_{T_{n}^{{}^{\prime}}}}(x_{T_{n}^{{}^{\prime}}},P_{x_{T_{n}^{{}^{\prime}}}}(y_{T_{n}^{{}^{\prime}}})).\\\
\end{split}$

And by Lemma 4.17,

$d^{*}_{x_{T_{n}^{{}^{\prime}}}}(x_{T_{n}^{{}^{\prime}}},P_{x_{T_{n}^{{}^{\prime}}}}(y_{T_{n}^{{}^{\prime}}}))>\frac{b_{0}}{D^{{}^{\prime}}}d^{*}_{x_{T_{n}}}(x_{T_{n}},P_{x_{T_{n}}}(y_{T_{n}})).$

As a conclusion, we conclude that

$d^{*}_{x_{T_{n}^{{}^{\prime}}}}(x_{T_{n}^{{}^{\prime}}},P_{x_{T_{n}^{{}^{\prime}}}}(y_{T_{n}^{{}^{\prime}}}))>b_{0}^{n}d^{*}_{x_{T_{1}}}(x_{T_{1}},P_{x_{T_{1}}}(y_{T_{1}})).$

This implies that $d^{*}_{x_{T_{1}}}(x_{T_{1}},P_{x_{T_{1}}}(y_{T_{1}}))=0$.
Therefore, $y$ belongs to the local orbit of $x$. In particular,
$B_{\infty}(x,\delta_{1})$ is a segment of the orbit of $x$, and has
topological entropy zero.

It remains to prove that when $\Lambda$ is a quasi attractor and all the
singularities are hyperbolic, this entropy expansiveness also holds for nearby
flows. Take $U$ be a sufficiently small contratcing neighborhood. Because the
singularities are all isolated, each singularity can be treated as extreme
points, and $\Lambda$ contains only finitely many singularities. By a similar
argument as Lemma 4.9, one can show that each singularity has negative center
exponent. One may check that Lemmas 4.13, 4.14 and 4.17 all hold robustly for
nearby flows restricted in $U$, so does the entropy expansiveness.

The proof is complete. ∎

##  5\. Lower semi-continuation of topological entropy

In this section, we will obtain the lower semi-continuation of the topological
entropy. In Subsubsection 5.1, we assume Theorem 5.1, a $\operatorname{C}^{1}$
version of Pesin theory for flows, to finish the proof of Theorem 2.22 and
Corollary 2.24. The proof of Theorem 5.1 is postponed to Appendix A. The proof
of Corollary B, Corollary C and Theorem D are given in Subsubsection5.2.

###  5.1. Proof of Theorem 2.22 and Corollary 2.24

First let us state a $\operatorname{C}^{1}$ version of Pesin theory for flows,
where we replace the regularity of maps by domination on the tangent bundle.
Similar statements for diffeomorphisms can be found at [1, 44].

######  Theorem 5.1.

Let $\Lambda$ be a compact invariant set of flow $\phi_{t}$ and admit a
dominated splitting $E\oplus F$. Suppose $\mu$ is a non-trivial hyperbolic
ergodic measure supported on $\Lambda$ with index $dim(E)$ and $\tilde{\mu}$
an ergodic decomposition of $\mu$ respect to $f$. Then there is a compact,
positive $\tilde{\mu}$ measure subset $\Lambda_{0}\subset\Lambda\setminus{\rm
Sing}(X)$ which satisfies the following properties: for any $\varepsilon>0$,
there are $L,L^{{}^{\prime}},\delta>0$ such that for any $x$ and
$f^{n}(x)\in\Lambda_{0}$ with $n>L^{{}^{\prime}}$ and $d(x,f^{n}(x))<\delta$,
there exists a point $p\in M^{d}$ and a $\operatorname{C}^{1}$ strictly
increasing function $\theta:[0,T]\rightarrow\mathbb{R}$ such that:

  * (a)

$\theta(0)=0$ and $1-\varepsilon<\theta^{{}^{\prime}}(t)<1+\varepsilon$;

  * (b)

$p$ is a periodic point of $\phi_{t}$ with $\phi_{\theta(T)}(p)=p$;

  * (c)

$d(\phi_{t}(x),\phi_{\theta(t)}(p))\leq\varepsilon\|X(\phi_{t}(x))\|,t\in[0,T]$;

  * (d)

$d(\phi_{t}(x),\phi_{t}(p))\leq Ld(x,\phi_{T}(x))$

  * (e)

$p$ has uniform size of stable manifold and unstable manifold.

Now we are ready to give the proof of Theorem 2.22, the argument is similar to
[20].

###### Proof of Theorem 2.22:.

By Theorem 5.1, take $\tilde{\mu}$ an ergodic decomposition of $\mu$ respect
to $f$ and $\Lambda_{0}$ the compact set with positive $\tilde{\mu}$ measure,
and for $\varepsilon>0$, take $L,L^{{}^{\prime}},\delta>0$. Replacing by a
subset, we may assume the diameter of $\Lambda_{0}$ is small enough such that
any two distinct periodic orbits obtained in Theorem 5.1 are homoclinic
related to each other. By Lemma 2.15, $h_{\mu}(f)=h_{\tilde{\mu}}(f)>0$.

The following is the same as the proof of [20][Theorem 4.3]. For $\eta,l>0$
and $n\in\mathbb{N}$, there is a finite set $K_{n}=K(\eta,l)$ which satisfies
the following properties (for some $n$ the set $K_{n}(\eta,l)$ may be empty).

  * •

$K_{n}\subset\Lambda_{0}$;

  * •

if $x,y\in K_{n}$, then $d^{f}_{n}(x,y)>\frac{1}{l}$;

  * •

For every $x\in K_{n}$, there exists a number $m(x)$: $n\leq
m(x)\leq(1+\eta)n$ such that $f^{m(x)}(x)\in\Lambda_{0}$ and
$d(x,f^{m}(x))<\frac{1}{4Ll}$;

  * •

For every $\eta>0$,
$\lim_{l\rightarrow\infty}\lim_{n\rightarrow\infty}\frac{\operatorname{ln}\operatorname{Card}K_{n}(\eta,l)}{n}\geq
h_{\tilde{\mu}}(f)-\eta$.

Let $n,l$ be sufficiently large such that $n>L^{{}^{\prime}}$,
$\frac{1}{4l}<\delta$ and
$\operatorname{Card}K_{n}(\eta,l)>\exp^{n(h_{\tilde{\mu}}-\eta)}$.

Then for every $x\in K_{n}$, there is a pseudo orbit
$(\phi_{t}(x))_{[0,m(x)]}$, which is shadowed by a periodic point $p_{x}$ with
period at most $n(1+\varepsilon)(1+\eta)$ and
$d^{f}_{n}(x,p_{x})\leq\frac{1}{4l}$. In particular, for $x\neq y\in K_{n}$,

$d^{f}_{n}(p_{x},p_{y})\geq
d^{f}_{n}(x,y)-d^{f}_{n}(x,p_{x})-d^{f}_{n}(y,p_{y})\geq\frac{1}{2l}.$

When $x$ and $y$ are both shadowed by the same periodic orbit ${\mathcal{O}}$,
write $p_{x}=\phi_{t_{y,x}}(p_{y})$, then $|t_{y,x}|>\frac{1}{2lD}$ where
$D=\max\\{\|X(.)\|\\}$.

As a corollary of the above argument, there are at most
$[n(1+\varepsilon)(1+\eta)2lD]$ points of $K_{n}$ are shadowed by the same
periodic orbit. Hence, there are at least
$\frac{e^{n(h_{\tilde{\mu}}-\eta)}}{[n(1+\varepsilon)(1+\eta)2lD]}$ many
different periodic points with period bounded by $n(1+\eta)$. Moreover, these
periodic orbits are homoclinic related to each other near to $\Lambda_{0}$,
and form a Horse-shoe with topological entropy larger than or equal to

$\operatorname{ln}\frac{\exp^{n(h_{\tilde{\mu}}-\eta)}}{[n(1+\varepsilon)(1+\eta)2lD]}/n(1+\eta),$

which converges to $h_{\tilde{\mu}}(f)=h_{\mu}(f)$ when $\varepsilon,\eta\to
0$.

The proof is complete. ∎

###### Proof of Corollary 2.24:.

By Theorem 5.1, take $\tilde{\mu}$ an ergodic decomposition of $\mu$ respect
to $f$ and $\Lambda_{0}$ the compact set with positive $\tilde{\mu}$ measure,
and for $\varepsilon>0$, take $L,L^{{}^{\prime}},\delta>0$. Replacing by a
subset, we may assume the diameter of $\Lambda_{0}$ is small enough such that
for any two distinct periodic orbits obtained in Theorem 5.1 are homoclinic
related to each other.

Take $x\in\Lambda_{0}$ a generic point of $\tilde{\mu}$. By the Birkhoff
ergodic theorem, there are $0<n_{1}<\cdots$ such that
$f^{n_{i}}(x)\in\Lambda_{0}$ for each $i\in\mathbb{N}$ and
$\lim_{i}d(x,f^{n_{i}}(x))\to 0$. By Theorem 5.1, each pseudo orbit
$(x_{t})_{[0,n_{i}]}$ is shadowed by a periodic point $p_{i}$. Moreover, these
periodic orbits are homoclinic related to each other and $\lim_{i}p_{i}\to x$.
This implies that $x$, and the closure of its forward orbit which coincides to
$\operatorname{supp}(\tilde{\mu})$, are contained in the homoclinic class.
This implies immediately that the chain recurrent class of $\Lambda$ contains
a non-trivial homoclinic class.

∎

###  5.2. Proof of Corollaries B, C and Theorem D

###### Proof of Corollary B.

By Theorem A and the variation principle, there is an $f$-invariant measure
$\mu$ supported on $\Lambda$ with positive metric entropy. Then Corollary B
follows from Corollary 2.24 and Theorem 2.26. ∎

###### Proof of Corollary C:.

Let ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$ be the residual subsets
defined in Theorem 2.25 and Theorem 2.26 respectively. Then for any star flow
$\phi_{t}\in{\mathcal{R}}_{1}\cap{\mathcal{R}}_{2}$, every non-trivial
Lyapunov stable chain recurrent class of $\phi_{t}$ is Lorenz-like, and is in
fact an attractor. In particular, $\phi_{t}$ has only finitely many
attractors.

By the main result of [29], there is a $\operatorname{C}^{1}$ residual subset
of flows ${\mathcal{R}}_{3}$ that that for every
$\phi_{t}\in{\mathcal{R}}_{3}$, the points converging to the Lyapunov stable
chain recurrent calsses of $\phi_{t}$ is everywhere dense. Take
${\mathcal{R}}={\mathcal{R}}_{1}\cap{\mathcal{R}}_{2}\cap{\mathcal{R}}_{3}$.
Combining the discussion above, for every flow $\phi_{t}\in{\mathcal{R}}$, it
has only finitely many attractors, the union of the basins of these attractors
are open and dense in the ambient manifold. We conclude the proof. ∎

###### Proof of Theorem D.

By Proposition 2.21, there are attracting neighborhood $U$ of $\Lambda$, a
$\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of $\phi_{t}$ and
$\delta>0$, such that the flows in ${\mathcal{U}}$ are all $\delta$-entropy
expansive. By Lemma 2.17, $h_{top}(.|_{U})$ varies upper semi-continuous
respect the flows contained in ${\mathcal{U}}$.

Choose $U$ sufficiently small, such that for any flow
$\phi^{{}^{\prime}}_{t}\in{\mathcal{U}}$, its maximal invariant set contained
in $U$ is sectional-hyperbolic. Then by Theorem 2.8,
$h_{top}(\phi_{t}^{{}^{\prime}}|_{U})>0$. Applying Lemma 2.16, there is a
maximal measure $\mu^{{}^{\prime}}$ for the map $\phi^{{}^{\prime}}|_{U}$,
which is clearly non-trivial. Then $\operatorname{supp}(\mu^{{}^{\prime}})$
admits a sectional-hyperbolic splitting $E^{s}\oplus F^{cu}$, which implies
that $\mu^{{}^{\prime}}$ is a hyperbolic measure with index $dim(E^{s})$.

Apply Theorem 2.22, for any $\varepsilon>0$, there is a horse-shoe
$\Lambda_{\varepsilon}$ of $Y$ contained in $U$ with topological entropy
larger than $h_{top}(\phi_{t}^{{}^{\prime}}|_{U})-\varepsilon$. Because the
topological entropy of a horse-shoe varies continuously, we conclude the proof
of lower semi-continuation of the topological entropy and complete the proof
of Theorem D. ∎

##  Appendix A Proof of Theorem 5.1

For $x\in M\setminus{\rm Sing}(X)$ and $v\in T_{x}M$, the orthogonal
projection of $\Phi_{t}(v)$ on ${\mathcal{N}}_{\phi_{t}(x)}$ along the flow
direction, denoted by $\psi_{t}:{\mathcal{N}}_{x}\to{\mathcal{N}}_{x_{t}}$,
was defined in Subsection 2.3:

$\psi_{t}(v)=\pi_{\phi_{t}(x)}(\Phi_{t}(v))=\Phi_{t}(v)-\frac{<\Phi_{t}(v),X(\phi_{t}(x))>}{\|X(\phi_{t}(x))\|^{2}}X(\phi_{t}(x)).$

In this section, we need another flow
$\psi^{*}_{t}:{\mathcal{N}}\rightarrow{\mathcal{N}}$, which is called scaled
linear Poincaré flow:

$\psi^{*}_{t}(v)=\frac{\|X(x)\|}{\|X(\phi_{t}(x))\|}\psi_{t}(v)=\frac{\psi_{t}(v)}{\|\Phi_{t}|_{<X(x)>}\|},$

######  Lemma A.1.

$\psi^{*}_{t}$ is a bounded cocycle on ${\mathcal{N}}_{\Lambda}$ in the
following sense: for any $\tau>0$, there is $C_{\tau}>0$ such that for any
$t\in[-\tau,\tau]$,

$\|\psi^{*}_{t}\|\leq C_{\tau}.$

Moreover, let $\mu$ be a non-trivial ergodic measure supported on $\Lambda$,
then the two cocycles $\psi_{t}$ and $\psi^{*}_{t}$ have the same Lyapunov
exponents and Oseledets splitting.

###### Proof.

The uniformly boundness comes from the boundness of $\Phi_{t}$ for
$t\in[-\tau,\tau]$.

For $\mu$ almost every $x$, we have $\lim\frac{\log\|\Phi_{t}(X(x))\|}{t}=0$.
This implies that both $\psi_{t}$ and $\psi^{*}_{t}$ have the same Lyapunov
exponents and Oseledets splitting.

∎

###  A.1. Pesin block

######  Lemma A.2.

$X|_{\Lambda\setminus{\rm Sing}(X_{t})}\subset F$.

###### Proof.

Because $\phi_{t}|_{\operatorname{supp}(\mu)}$ is transitive (Birkhoff ergodic
theorem), by Lemma 2.13, we have either $X|_{\Lambda\setminus{\rm
Sing}(X_{t})}\subset E$ or $X|_{\Lambda\setminus{\rm Sing}(X_{t})}\subset F$.

Since $\mu$ is hyperbolic with index $dim(E)$, by Theorem 2.19, $f$ has
$dim(E)$ number of negative exponents and a vanishing center exponent which
corresponds to the flow direciton. The Lyapunov exponents restricted on $E(x)$
is clearly smaller than the exponents on $F(x)$. Hence, $E(x)$ coincides to
the subspace spanned by the sunbundles in the Oseledets splitting
corresponding to the negative exponents; and $F(x)$ is the subspace spanned by
the flow direction and the sunbundles in the Oseledets splitting corresponding
to the positive exponents. This implies that $X(x)\in F(x)$, and by the
transitivity, we conclude that $X|_{\Lambda\setminus{\rm Sing}(X_{t})}\subset
F$.

∎

######  Lemma A.3.

For both of $\psi_{t}$ and $\psi^{*}_{t}$, $\pi(E)\oplus\pi(F)$ is a dominated
splitting on ${\mathcal{N}}_{\Lambda\setminus{\rm Sing}(X_{t})}$, which is
also the Oseledets splitting of $\mu$ corresponding to the negative exponents
and positive exponents.

###### Proof.

Let us first prove that $\pi(E)\oplus\pi(F)$ is a dominated splitting for
$\psi_{t}$. The proof of $\psi^{*}_{t}$ follows immediately from the relation
between $\psi_{t}$ and $\psi^{*}_{t}$.

Consider $\psi_{t}$, $t<0$. Because $E\oplus F$ is a dominated splitting for
the flow, there are $C,c_{0}>0$ such that for any $t<0$, $u$ and $v$ unit
vectors contained in $E_{x}$ and $F_{x}$ for $x\in\Lambda\setminus{\rm
Sing}(X_{t})$, we have

$\frac{\|\Phi_{t}(u)\|}{\|\Phi_{t}(v)\|}>C\exp^{-c_{0}t}.$

Because $E\oplus F$ is a dominated splitting for flow, the angles between
these two subbundles are uniformly bounded from below. By Lemma A.2,
$X|_{\operatorname{supp}(\mu)\setminus{\rm Sing}(X_{t})}\subset F$, for any
$u\neq 0\in E$, there is $C^{*}>0$ such that

$\frac{1}{C^{*}}<\frac{\|\Phi_{t}(u)\|}{\|\psi_{t}(u)\|}<\frac{1}{C^{*}}.$

From the definition of $\psi_{t}$, which is a projection, we have
$\|\psi_{t}(v)\|\leq\|\Phi_{t}(v)\|$. Hence,

$\frac{\|\psi_{t}(u)\|}{\|\psi_{t}(v)\|}>C\exp^{-c_{0}t}/C^{*},$

which means that, $\pi(E)\oplus\pi(F)$ is a dominated splitting for
$\psi_{t}$.

Because $\mu$ has index $dim(E)$ and $\pi(E)\oplus\pi(F)$ is a dominated
splitting for $\psi_{t}$, by Theorem 2.19, we conclude that
$\pi(E)\oplus\pi(F)$ is the Oseledets splitting corresponding to the negative
and positive Lyapunov exponents of $\mu$ for both $\psi_{t}$ and
$\psi^{*}_{t}$.

∎

######  Definition A.4.

The orbit arc $\phi_{[0,T]}(x)$ is called $(\lambda,T_{0})^{*}$ quasi
hyperbolic with respect to a direct sum splitting
${\mathcal{N}}_{x}=E(x)\oplus F(x)$ and the scaled linear Poincaré flow
$\psi^{*}_{t}$ if there exists $0<\lambda<1$ and a partition

$0=t_{0}<t_{1}<\dots<t_{l}=T,\;\;\;\text{where}\;\;t_{i+1}-t_{i}\in[T_{0},2T_{0}]$

such that for $k=0,1,\dots,l$, we have

$\prod_{i=0}^{k-1}\|\psi^{*}_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(E(x))}\|\leq\lambda^{k};\;\;\prod_{i=k}^{l-1}m(\psi^{*}_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(F(x))})\geq\lambda^{k-l},$

and

$\frac{\|\psi^{*}_{t_{k+1}-t_{k}}|_{\psi_{t_{k}}(E(x))}\|}{m(\psi^{*}_{t_{k+1}-t_{k}}|_{\psi_{t_{k}}(F(x))})}\leq\lambda^{2}.$

Now we state a $\operatorname{C}^{1}$ version of Pesin block.

######  Lemma A.5.

There are $L^{{}^{\prime}},\eta,T_{0}>0$ and a positive $\tilde{\mu}_{1}$
measure subset $\Lambda_{0}\subset\Lambda\setminus{\rm Sing}(X)$, such that
for any $x,f^{n}(x)\in\Lambda_{0}$ with $n>L^{{}^{\prime}}$, $\phi_{[0,n]}(x)$
is $(\eta,T_{0})^{*}$ quasi hyperbolic with respect to a direct sum splitting
${\mathcal{N}}_{x}=E(x)\oplus F(x)$ and the scaled linear Poincaré flow
$\psi^{*}_{t}$.

###### Proof.

By Lemma A.3 and Theorem 2.19, on $\tilde{\mu}$ almost every $x$,
$\pi(E(x))\oplus\pi(F(x))$ is the Oseledets splitting of $f$ for
$\psi^{*}_{t}$ corresponding to the negative and positive exponents. Then by
subadditive ergodic theorem, there is $a<0$ such that

$\lim_{t\to\infty}\frac{1}{t}\int\log\|\psi_{t}^{*}|_{E}\|d\tilde{\mu}<a\;\;\text{and}\;\;\lim_{t\to-\infty}\frac{1}{|t|}\int\log\|\psi_{t}^{*}|_{F}\|d\tilde{\mu}<a.$

Taking $N_{0}$ sufficiently large, we have

$\frac{1}{N_{0}}\int\log\|\psi_{N_{0}}^{*}|_{E}\|d\tilde{\mu}<a\;\;\text{and}\;\;\frac{1}{N_{0}}\int\log\|\psi_{-N_{0}}^{*}|_{F}\|d\tilde{\mu}<a.$

Consider the ergodic decomposition of $\tilde{\mu}$ for $f^{N_{0}}$:

$\tilde{\mu}=\frac{1}{k_{0}}\big{(}\tilde{\mu}_{1}+\dots+\tilde{\mu}_{k_{0}}\big{)}.$

By changing the order, we may assume that:

$\frac{1}{N_{0}}\int\log\|\psi_{N_{0}}^{*}|_{E}\|d\tilde{\mu}_{1}<a\;\;\text{and}\;\;\frac{1}{N_{0}}\int\log\|\psi_{-N_{0}}^{*}|_{F}\|d\tilde{\mu}_{1}<a.$

Applying Birkhoff ergodic theorem on $f^{N_{0}}$, for $\tilde{\mu}_{1}$ almost
every $x$,

$\lim\frac{1}{mN_{0}}\sum_{i=0}^{m-1}\log\|\psi_{N_{0}}^{*}|_{E(f^{iN_{0}}(x))}\|<a$

and

$\lim\frac{1}{mN_{0}}\sum_{i=0}^{m-1}\log\|\psi_{-N_{0}}^{*}|_{F(f^{-iN_{0}}(x))}\|<a.$

There is $n_{x}>0$ such that for any $m>n_{x}$,

$\frac{1}{mN_{0}}\sum_{i=0}^{m-1}\log\|\psi_{N_{0}}^{*}|_{E(f^{iN_{0}}(x))}\|<a\;\;\text{and}\;\;\frac{1}{mN_{0}}\sum_{i=0}^{m-1}\log\|\psi_{-N_{0}}^{*}|_{F(f^{-iN_{0}}(x))}\|<a.$

Choose $N_{1}$ such that the set $\Lambda^{{}^{\prime}}=\\{x;n(x)<N_{1}\\}$
has positive $\tilde{\mu}_{1}$ measure. Let
$\Lambda_{0}\subset\Lambda^{{}^{\prime}}\setminus{\rm Sing}(X)$ be compact and
has positive $\tilde{\mu}_{1}$ measure. It follows immediately that
$\tilde{\mu}(\Lambda_{0})>0$. By Lemma A.1, we may define

$K=\max_{|t|\leq N_{0},y\in\Lambda\setminus{\rm
Sing}(X)}\\{\sup\\{\psi_{t}^{*}|_{E(y)}\\};\sup\\{\psi_{t}^{*}|_{F(y)}\\}\\}.$

Choose $N_{2}$ sufficiently large and $b<0$ such that

$\frac{N_{2}+N_{0}}{N_{0}}a+3K<b<0.$

We claim that for any sequence $n_{1}<n_{2}\cdots<n_{l}$ satisfying $N_{2}\leq
n_{i+1}-n_{i}\leq N_{2}+N_{0}$ for $0\leq i\leq l-1$:

$\frac{1}{l}\sum_{i=0}^{l-1}\log\|\psi^{*}_{n_{i+1}-n_{i}}|_{E(f^{n_{i}}(x))}\|<b.$

A similar formula also holds for $F$ bundle. Let $L^{{}^{\prime}}>0$ be
sufficiently large, such that for any $n>L^{{}^{\prime}}$, there always exists
a sequence $n_{1}<n_{2}\cdots<n_{l}$ satisfying $N_{2}\leq n_{i+1}-n_{i}\leq
N_{2}+N_{0}$ for each $0\leq i\leq l-1$. Then by the above claim, we conclude
the proof of this lemma.

It remains to prove the claim. For each $0\leq i<l$, denote

$k_{i}=[\frac{n_{i+1}}{N_{0}}]-[\frac{n_{i}}{N_{0}}]-1,n^{{}^{\prime}}_{i}=([\frac{n_{i}}{N_{0}}]+1)N_{0}\;\;\text{and}\;\;n^{*}_{i+1}=[\frac{n_{i+1}}{N_{0}}]N_{0}.$

Then $n^{*}_{i}\leq n_{i}\leq n^{{}^{\prime}}_{i}$,
$n^{*}_{i+1}-n^{{}^{\prime}}_{i}=k_{i}N_{0}$ and

$\psi^{*}_{n_{i+1}-n_{i}}|_{E(f^{n_{i}}(x))}=\psi^{*}_{n_{i+1}-n^{*}_{i+1}}|_{E(f^{n^{*}_{i+1}}(x))}\circ\psi^{*}_{k_{i}N_{0}}|_{E(f^{n_{i}^{{}^{\prime}}}(x))}\circ\psi^{*}_{n^{{}^{\prime}}_{i}-n_{i}}|_{E(f^{n_{i}}(x))}$

Observe that $n_{i+1}-n^{*}_{i+1}\leq N_{0}$ and
$n^{{}^{\prime}}_{i}-n_{i}\leq N_{0}$, which imply in particular that

$\begin{split}\log\|\psi^{*}_{n_{i+1}-n_{i}}|_{E(f^{n_{i}}(x))}\|&\leq
2K+\log\|\psi^{*}_{k_{i}N_{0}}|_{E(f^{n_{i}^{{}^{\prime}}}(x))}\|\\\ &\leq
3K+\log\|\psi^{*}_{n^{{}^{\prime}}_{i+1}-n^{{}^{\prime}}_{i}}|_{E(f^{n_{i}^{{}^{\prime}}}(x))}\|.\\\
\end{split}$

Because $n^{{}^{\prime}}_{0}=0$, we have that

$\begin{split}\frac{1}{l}\sum_{i=0}^{l-1}\log\|\psi^{*}_{n_{i+1}-n_{i}}|_{E(f^{n_{i}}(x))}\|&\leq\frac{1}{l}\sum_{i=0}^{l}\log\|\psi^{*}_{n^{{}^{\prime}}_{i+1}-n^{{}^{\prime}}_{i}}|_{E(f^{n_{i}^{{}^{\prime}}}(x))}\|+3K\\\
&\leq\frac{1}{l}\sum_{j=0}^{\frac{n^{{}^{\prime}}_{l+1}}{N_{0}}}\log\|\psi^{*}_{N_{0}}|_{E(f^{jN_{0}}(x))}\|+3K\\\
&\leq\frac{n^{{}^{\prime}}_{l+1}}{lN_{0}}a+3K\leq\frac{N_{2}+N_{0}}{N_{0}}a+3K\leq
b\\\ \end{split}$

∎

###  A.2. Liao’s shadowing lemma for scaled linear Poincaré flow

In this subsection we will introduce the Liao’s shadowing lemma for scaled
linear Poincaré flow of [26]. Because we need estimate the difference of the
time between the pseudo orbit and real orbit: the item (d) below, which only
follows from the proof, we provide here the idea of the proof:

######  Theorem A.6.

Given a compact set $\Lambda_{0}\subset\Lambda\setminus{\rm Sing}(X)$, and
$\eta>0,T_{0}>0$, for any $\varepsilon>0$ there exist $\delta>0$ and $L>0$,
such that for any $(\eta,T_{0})^{*}$ quasi hyperbolic orbit arc
$\phi_{[0,T]}(x)$ with respect to dominated splitting
${\mathcal{N}}_{x}=E(x)\oplus F(x)$ and the scaled linear Poincaré flow
$\psi_{t}^{*}$ which satisfies $x,\phi_{T}(x)\in\Lambda_{0}$ and
$d(x,\phi_{T}(x))\leq\delta$, there exists a point $p\in M^{d}$ and a
$\operatorname{C}^{1}$ strictly increasing function
$\theta:[0,T]\rightarrow\mathbb{R}$ such that

  * (a)

$\theta(0)=0$ and $1-\varepsilon<\theta^{{}^{\prime}}(t)<1+\varepsilon$;

  * (b)

$p$ is a periodic point with $\phi_{\theta(T)}(p)=p$;

  * (c)

$d(\phi_{t}(x),\phi_{\theta(t)}(p))\leq\varepsilon\|X(\phi_{t}(x))\|,t\in[0,T]$;

  * (d)

$d(\phi_{t}(x),\phi_{t}(p))\leq Ld(x,\phi_{T}(x))$.

  * (e)

$p$ has uniform size of stable manifold and unstable manifold.

###### Proof.

For simplicity, we assume $M$ is an open set in $R^{d}$, which implies that
for every regular point $x\in M$, ${\mathcal{N}}_{x}$ is a $d-1$-dimensional
hyperplane. Because $d(x,\phi_{T}(x))$ is sufficiently small, we may suppose
that $\phi_{T}(x)\in{\mathcal{N}}_{x}$.

The first step is to translate the problem into the shadowing lemma for a
sequence of maps: By [16][Lemmas 2.2, 2.3], there is $\beta$ depending on
$T_{0}$ such that the holonomy map induced by flow is well defined between

${\mathcal{P}}_{x,\phi_{t}(x)}:{\mathcal{N}}_{x}(\beta\|X(x)\|)\rightarrow{\mathcal{N}}_{\phi_{t}(x)}\;\;\;\text{for
any}\;\;\;t\in[T_{0},2T_{0}].$

This is conjugate to the following sequence of maps:

${\mathcal{P}}^{*}_{x,\phi_{t}(x)},{\mathcal{N}}_{x}(\beta)\rightarrow{\mathcal{N}}_{\phi_{t}(x)}:{\mathcal{P}}^{*}_{x,\phi_{t}(x)}(a)=\frac{{\mathcal{P}}_{x,\phi_{t}(x)}(a\|X(x)\|)}{\|X(\phi_{t}(x))\|}.$

The tangent map of ${\mathcal{P}}^{*}_{x,\phi_{t}(x)}$ is uniformly
continuous.

Take the sequence of times $0=t_{0}<t_{1}<\dots<t_{l}=T$ in the definition of
quasi hyperbolic orbit. We consider the local diffeomorphisms induced by the
holonomy maps:

${\mathcal{T}}_{i}={\mathcal{P}}^{*}_{\phi_{t_{i-1}}(x),\phi_{t_{i}}(x)}\;\;\;\text{for}\;\;\;1\leq
i\leq l-1;$

and
${\mathcal{T}}_{l}:{\mathcal{N}}_{\phi_{t_{l-1}}(x)}(\beta)\rightarrow{\mathcal{N}}_{x}$.
Because $d(x,\phi_{T}(x))$ is sufficiently small, the holonomy map
${\mathcal{T}}_{l}$ is well defined. For each $1\leq i\leq l$, we have

$T{\mathcal{T}}_{i}(0)=\psi^{*}_{t_{i}-t_{i-1}}|_{\phi_{t_{i-1}}(x)}.$

Denote $E_{i}=\psi_{t_{i}}(E(x))$ and $F_{i}=\psi_{t_{i}}(F(x))$. Then, by the
definition of quasi hyperbolic, there is $0<\lambda<1$ such that such that for
$k=1,\dots,l$, we have

$\prod_{i=1}^{k}\|{\mathcal{T}}_{i}\|\leq\lambda^{k};\;\;\prod_{i=k}^{l}m({\mathcal{T}}_{i})\geq\lambda^{k-l-1},$

and

$\frac{\|{\mathcal{T}}_{k}|_{E_{k}}\|}{m(\|{\mathcal{T}}_{k}|_{F_{k}})}\leq\lambda^{2}.$

Now we apply the version of Liao’s shadowing lemma on discrete quasi
hyperbolic maps ([15][Theorem 1.1], see also [23, 24, 25]). On the
$d(x,\phi_{T}(x))$ pseudo orbit
$\\{0_{x},0_{\phi_{t_{1}}(x)},\dots,0_{\phi_{t_{l-1}}(x)}\\}$, there is $L>0$
and a periodic orbit $p\in{\mathcal{N}}_{x}$ with period $l$ for
$\\{{\mathcal{T}}_{0},\cdots,{\mathcal{T}}_{l-1}\\}$, whose orbit
$Ld(x,\phi_{T}(x))$-shadows the pseudo orbit.

Using the version of flow tubular theorem [16][Lemmas 2.2], one may prove (a),
(b) and (c) above. Now let us focus on the proof of (d) and (e). We follow the
proof of [15]. By Lemma 3.1 of [15], there is a sequence of positive numbers
$\\{c_{i}\\}_{i=1}^{l}$ called well adapted such that:

  * (i)

$g_{j}=\prod_{j=1}^{k}c_{j}\leq 1$, $k=1,\dots,l-1$ and
$g_{l}=\prod_{j=1}^{l}c_{j}=1$;

  * (ii)

denote $\tilde{{\mathcal{T}}}_{j}(a)=g_{j}^{-1}{\mathcal{T}}_{j}(g_{j-1}a)$,
then

$\|T\tilde{{\mathcal{T}}}_{j}(0)|_{E_{j}}\|\leq\lambda\;\;\text{and}\;\;m(T\tilde{{\mathcal{T}}}_{j}(0)|_{F_{j}})\geq\frac{1}{\lambda}.$

It was shown in [15] (p. 631) that the well adapted sequence is uniformly
bounded from above and from zero, and the sequence of hyperbolic maps
$\\{\tilde{{\mathcal{T}}}_{j}\\}_{j=1}^{l}$ are Lipschtez maps (with uniform
Lipschtez constant).

Denote by

$\Psi_{k}={\mathcal{T}}_{k}\cdots\circ{\mathcal{T}}_{1}\;\;\text{and}\;\;\tilde{\Psi}_{k}=\tilde{{\mathcal{T}}}_{k}\cdots\circ\tilde{{\mathcal{T}}}_{1}.$

It is easy to see that

(10)
$\begin{split}\Psi_{k}=g_{k}\tilde{\Psi}_{k}\;\;&\text{and}\;\;\Psi_{l}=\tilde{\Psi}_{l}.\\\
\end{split}$

Observe that $\\{0_{x},0_{\phi_{t_{1}}}(x),\cdots,0_{\phi_{t_{l-1}}(x)}\\}$ is
still a $d(x,\phi_{T}(x))$ pseudo orbit of sequence of hyperbolic
diffeomorphisms $\\{\tilde{{\mathcal{T}}}_{j}\\}_{j=0}^{l-1}$. By the standard
shadowing lemma for hyperbolic diffeomorphisms (see [15][Lemma 2.1]), there is
$L>0$ such that the pseudo orbit is $Ld(x,\phi_{T}(x))$-shadowed by periodic
orbit $\\{\tilde{p},\tilde{p}_{1},\dots,\tilde{p}_{l-1}\\}$, and this periodic
orbit has uniform size of stable and unstable manifold. By (10),
$\\{g_{k}\tilde{p}_{k}\\}_{k=1}^{l}$ is a pseudo orbit for the original
sequence of maps $\\{{\mathcal{T}}_{k}\\}_{k=0}^{l-1}$ and
$p_{0}=\tilde{p}_{0}$. Hence, $p_{0}$ has uniform size of stable manifold and
unstable manifold. We conclude (e).

It remains to prove (d). We claim that

(11) $\begin{split}\sum_{0\leq i\leq l-1}d(p_{i},0_{\phi_{t_{i}}(x)})&\leq
L^{*}d(x,\phi_{t_{l}}(x)).\\\ \end{split}$

By the definition of well adapted sequence, $g_{k}\leq 1$ for every $1\leq
k\leq l$, which implies that $d(p_{k},0_{\phi_{t_{k}}(x)})\leq
d(\tilde{p}_{k},0_{\phi_{t_{k}}(x)})$. Hence, to prove this claim, it suffices
to verify that there is $L^{*}>0$ such that

$\sum_{0\leq i\leq l-1}d(\tilde{p}_{i},0_{\phi_{t_{i}}(x)})\leq
L^{*}d(x,\phi_{t_{l}}(x)).$

This property can be obtained directly from the proof of shadowing lemma of
uniformly hyperbolic maps. Here we state a proof which depends on the
shadowing lemma:

###### Proof.

We will consider $F$-admissible manifolds, which are manifolds contained in
${\mathcal{N}}_{\phi_{t_{i}}(x)}$ ($0\leq i\leq l$) and satisfy:

  * •

tangent to the $F$ cone;

  * •

has uniform size.

Take an $F$ admissible manifold ${\mathcal{I}}_{0}^{F}\ni
0_{x}\subset{\mathcal{N}}_{x}$. Denote $a_{0}={\mathcal{I}}_{0}^{F}\pitchfork
W^{s}_{loc}(\tilde{p}_{0})$. By the hyperbolicity of
$\\{\tilde{{\mathcal{T}}}_{j}\\}_{j=1}^{l}$,
$\tilde{\Psi}_{1}({\mathcal{I}}_{0}^{F})$ contains an $F$ admissible manifold
${\mathcal{I}}_{1}^{F}\ni 0_{\phi_{t_{1}}(x)}$. Denote
$a_{1}={\mathcal{I}}^{F}_{1}\pitchfork W^{s}_{loc}(\tilde{p}_{1})$, then
$a_{1}=\tilde{{\mathcal{T}}}_{0}(a_{0})$. By induction, we define a sequence
of $F$ admissible manifolds ${\mathcal{I}}^{F}_{j}$, $1\leq j\leq l$ such that
$\tilde{\Psi}_{j}({\mathcal{I}}^{F}_{0})\supset{\mathcal{I}}^{F}_{j}$. Denote
$a_{j}={\mathcal{I}}^{F}_{j}\pitchfork W^{s}_{loc}(\tilde{p}_{j})$, then
$a_{j}=\tilde{\Psi}_{j}(a_{0})$. Moreover, for $0\leq j\leq l-1$,
$0_{\phi_{t_{j}}(x)}\in{\mathcal{I}}^{F}_{j}$.

Let $\gamma^{E}_{0}\subset W^{s}_{loc}(\tilde{p}_{0})$ which connects
$\tilde{p}_{0}$ and $a_{0}$ with minimal length, and
$\gamma^{F}_{l}\subset{\mathcal{I}}_{l}^{F}$ which connects $a_{l}$ and
$\tilde{\Psi}_{l}(0_{x})$ with minimal length. By the uniformly transversality
between the $F$ admissible manifolds and local stable manifolds, there is
$K>0$ such that

$\operatorname{length}(\gamma^{E}_{0})\leq Kd(x,\tilde{p}_{0})\leq
LKd(x,\phi_{T}(x))$

and

$\operatorname{length}(\gamma^{F}_{l})\leq Kd(\phi_{T}(x),\tilde{p}_{0})\leq
LKd(x,\phi_{T}(x)).$

For $1\leq i\leq l$, denote
${\mathcal{G}}_{i}:{\mathcal{N}}_{x}\to{\mathcal{N}}_{\phi_{t_{i}}}(x)$:

${\mathcal{G}}_{i}=\tilde{{\mathcal{T}}}_{i+1}^{-1}\circ\dots\circ\tilde{{\mathcal{T}}_{l}}^{-1}.$

Then
$\gamma_{i}=\tilde{\Psi}_{i}(\gamma^{E}_{0})\cup{\mathcal{G}}_{i}(\gamma^{F}_{l})$
is a piecewise smooth curve connecting $0_{\phi_{t_{i}}(x)}$ and
$\tilde{\Psi}_{i}(\tilde{p})$. Moreover,
$\tilde{\Psi}_{i}(\gamma^{E}_{0})\subset W^{s}_{loc}(\tilde{p}_{i})$ and
${\mathcal{G}}_{i}(\gamma^{F}_{l})\subset{\mathcal{I}}^{F}_{i}$. By the
hyperbolicity, $\tilde{\Psi}_{i}|_{\gamma^{E}_{0}}$ and
${\mathcal{G}}_{i}|_{\gamma^{F}_{l}}$ both are exponentially contracting.
There is $\lambda<1$ such that

$\operatorname{length}(\gamma_{i})<\lambda^{i}KLd(x,\phi_{T}(x))+\lambda^{l-i}KLd(x,\phi_{T}(x)).$

Hence, there is $L^{*}$ such that:

$\begin{split}\sum_{0\leq i\leq
l-1}d(\tilde{p}_{i},0_{\phi_{t_{i}}(x)})&\leq\sum\operatorname{length}(\gamma_{i})\\\
&\leq\sum_{0\leq i\leq l-1}(\lambda^{i}+\lambda^{l-i})KLd(x,\phi_{T}(x))\\\
&\leq L^{*}d(x,\phi_{T}(x)).\\\ \end{split}$

∎

Let us continue the proof. For any $x\in M\setminus{\rm Sing}(X)$ and
$t\in[T_{0},2T_{0}]$, one may define the cross time function
$t_{x,\phi_{t}(x)}:{\mathcal{N}}_{x}(\beta)\rightarrow\mathbb{R}$: for any
$a\in{\mathcal{N}}_{x}(\beta)$,
$\phi_{t_{x,\phi_{t}(x)}(a)}(a\|X(x)\|)\in{\mathcal{N}}_{\phi_{t}(x)}$. The
function $t_{x,\phi_{t}(x)}$ is smooth by the implicite function theorem, and
whose derivative is uniformly continuous (for example, bounded by $L_{2}$),
which can be deduced from the proof of [16][Lemmas 2.3]. Then

$\sum_{i=0}^{l-1}|(t_{i+1}-t_{i})-t_{\phi_{t_{i}}(x),\phi_{t_{i+1}}(x)}(p_{i})|<L^{*}L_{2}d(x,\phi_{T}(x)).$

Moreover, there is $L_{3}$ such that for any
$y,z\in{\mathcal{N}}_{x}(\beta|X(\phi_{i}(x))|)$, and $t_{i}\leq t<t_{i+1}$,
we have

$d(\phi_{s}(y),\phi_{s}(z))<L_{3}d(y,z).$

Now we are prepare to prove (d):

Suppose $t_{i}\leq t<t_{i+1}$, denote by $T_{i}=\sum_{j\leq
i-1}t_{\phi_{t_{j}}(x),\phi_{t_{j}+1}(x)}(p_{j})$.

$\displaystyle d(\phi_{t}(x),\phi_{t}(p))$ $\displaystyle=$ $\displaystyle
d(\phi_{t-t_{i}}(\phi_{t_{i}}(x)),\phi_{t-T_{i}}(p_{i}))$ $\displaystyle\leq$
$\displaystyle
d(\phi_{t-t_{i}}(\phi_{t_{i}}(x)),\phi_{t-t_{i}}(p_{i}))+d(\phi_{t-t_{i}}(p_{i}),\phi_{t-T_{i}}(p_{i}))$
$\displaystyle<$ $\displaystyle L_{3}d(\phi_{t_{i}}(x),p_{i})+|t_{i}-T_{i}|D$
$\displaystyle<$ $\displaystyle
L_{3}L^{*}d(x,\phi_{T}(x))+L_{2}L^{*}Dd(x,\phi_{T}(x)).$

Let $L=L_{3}L^{*}+L_{2}L^{*}D$, the proof is complete.

∎

###  A.3. Proof of Theorem 5.1

It is easy to see that Theorem 5.1 is a direct corollary of Lemma A.5 and
Theorem A.6.

##  Appendix B Proof of Theorem 2.26

The following property comes directly from the definition of Lyapunov stable
chain recurrent class, and will be used frequently during the proof.

######  Lemma B.1.

Let $\Lambda$ be a Lyapunov stable chain recurrent class of $\phi_{t}$. Then
for any $x\in\Lambda$, its unstable set is contained in $\Lambda$.

We also need two $\operatorname{C}^{1}$ generic properties for flows.

######  Proposition B.2.

There is a $\operatorname{C}^{1}$ residual subset ${\mathcal{R}}$ of flows,
such that for any flow $\phi_{t}\in{\mathcal{R}}$, and for any chain recurrent
class $C$ of $\phi_{t}$ which contains a periodic point $p$, $C$ coincides to
the homoclinic class of ${\rm Orb}(p)$. In particular, $C$ is transitive.

######  Proposition B.3.

There is a $\operatorname{C}^{1}$ residual subset ${\mathcal{R}}$ of flows,
such that for any flow $\phi_{t}\in{\mathcal{R}}$ and any non-trivial chain
recurrent class $C$ of $\phi_{t}$, suppose $C$ contains a hyperbolic
singularity $\sigma$ and a hyperbolic periodic point $p$, further suppose
there is a dominated splitting $E^{s}(\sigma)=E^{ss}(\sigma)\oplus
E_{1}^{c}(\sigma)$ on $\sigma$ where $E^{c}_{1}$ is 1-dimensional weak
contracting subbundle, then the strong stable manifold $W^{ss}(\sigma)$ of
$\sigma$ divides the stable manifold into two branches $W^{s,+}(\sigma)$ and
$W^{s,-}(\sigma)$, and if
${\mathcal{C}}\cap(W^{s,i}(\sigma)\setminus\sigma)\neq\emptyset$ for $i=+,-$,
then $W^{s,i}(\sigma)\cap W^{u}(p)\neq\emptyset$.

Proposition B.2 is the flow version of similar results of diffeomorphisms
stated in [6][Corollary 1.4, Remark 1.10]. Proposition B.3 can be proved by
applying the connecting lemma of [6] on a branch of stable manifold of the
singularity and the unstable manifold of the other periodic orbit.

###### Proof of Theorem 2.26:.

Let ${\mathcal{R}}_{2}$ be the residual subset of flows which are Kupka-Smale
and satisfies Propositions B.2 and B.3.

By Corollary B and Proposition B.2, $\Lambda$ coincides to the homoclinic
class of a hyperbolic orbit ${\rm Orb}(p)$ and is transitive. In order to
prove $\Lambda$ is an attractor, it remains to show that $\Lambda$ is an
attracting set. From the definition of Lyapunov stable chain recurrent class,
it suffices to prove that $\Lambda$ is isolated, i.e., it cannot be
approximated by other chain recurrent classes.

Let $\\{U_{n}\\}$ be the sequence of attracting neighborhoods of $\Lambda$ in
the definition of Lorenz-like class. Suppose by contradiction that $\Lambda$
is not isolated, i.e., there are chain recurrent classes ${\mathcal{C}}_{n}$
and $y^{n}\in{\mathcal{C}}_{n}$ such that $y^{n}\rightarrow\Lambda$. We may
assume that each ${\mathcal{C}}_{n}$ is contained in an attracting
neighborhood $U_{n}$. Thus, $\limsup_{n}{\mathcal{C}}_{n}\subset\Lambda$.

Because $\phi_{t}\in{\mathcal{R}}_{2}$ is Kupka-Smale, the singularities are
isolated. Hence, $\phi_{t}$ contains at most finitely many singularities. We
suppose $\cup_{n}{\mathcal{C}}_{n}$ contains no singularities. Because
${\mathcal{C}}_{n}$ also induces the singular hyperbolic splitting from
$\Lambda$, it contains no singularity implies is hyperbolic. Therefore $C_{n}$
contains a hyperbolic periodic orbit ${\rm Orb}(x^{n})$. Denote
$\Lambda_{0}\subset\Lambda$ the Hausdorff limit of ${\rm Orb}(x^{n})$, which
is a compact invariant subset.

We claim that $\Lambda_{0}$ contains a singularity. Suppose $\Lambda_{0}$
contains no singularities. Because it admits a singular hyperbolic splitting,
hence, $\Lambda_{0}$ is a hyperbolic set. For $n$ large enough, the two
hyperbolic sets ${\mathcal{C}}_{n}$ and $\Lambda_{0}$ are homoclinic related
to each other. This implies that $\Lambda_{0}$ and ${\mathcal{C}}_{n}$ are
contained in the same chain recurrent class, a contradiction with the
assumption that $\Lambda$ and ${\mathcal{C}}_{n}$ are different chain
recurrent classes.

Let $\sigma$ be a singularity contained in $\Lambda_{0}$. Then as in the proof
of Lemma 4.1, $\sigma$ admits a partially hyperbolic splitting $E^{ss}\oplus
E^{cs}_{1}\oplus E^{u}$ where $E^{cs}_{1}$ is a 1-dimensional weak stable
bundle and ${\mathcal{F}}^{ss}\cap\Lambda=\emptyset$. Then
${\mathcal{F}}^{ss}$ divides $W^{s}(\sigma)$ into two branches
$W^{s,+}(\sigma)$ and $W^{s,-}(\sigma)$.

By modifying $x^{n}$ to a different point in the same periodic orbit, we
assume that $x^{n}\rightarrow\sigma$. Fix a small $\varepsilon$, for each
$x^{n}$, let $t_{n}<0$ be the time satisfying

$(x^{n})_{t_{n}}\in\partial(B_{\varepsilon}(\sigma))\;\;\text{and}\;\;(x^{n})_{[t_{n},0]}\subset
B_{\varepsilon}(\sigma).$

Denote $z^{n}=(x^{n})_{t_{n}}$. Replacing by a subsequence, suppose
$lim_{n}z^{n}=z^{0}$, then $z^{0}\in W^{s}(\sigma)$. We may further suppose
$y^{0}\in W^{s,+}(\sigma)$.

By Proposition B.3, $W^{u}({\rm Orb}(p))\pitchfork
W^{s,+}(\sigma)\neq\emptyset$. Denote $a$ an intersecting point. There is $t$
such that $a\in W^{uu}(p_{t})$. Consider a disk $D\ni a\subset W^{uu}(p_{t})$.
Then by $\lambda$ lemma, $\phi_{t}(D)\rightarrow W^{u}(\sigma)$ from the
‘right’ and $\\{\phi_{t}(D);t>0\\}$ is a submanifold tangent to the $F^{cu}$
bundle with dimension $dim(F^{cu})$ . Therefore for $n$ large enough,
${\mathcal{F}}^{s}(x^{n})\pitchfork\\{\phi_{t}(D);t>0\\}\neq\emptyset$.
Because $\Lambda$ is a Lyapunov stable chain recurrent class, by Lemma B.1,
$\phi_{t}(D)\subset\Lambda$. Thus
${\mathcal{F}}^{s}(x^{n})\cap\Lambda\neq\emptyset$. In particular, this
implies that

$d(\phi_{t}(x^{n}),\Lambda)=d(\phi_{t}(x),\phi_{t}(\Lambda))\to 0.$

Which only occurs when ${\mathcal{C}}^{n}\cap\Lambda\neq\emptyset$, a
contradiction. This completes the proof. ∎

## References

  * [1] F. Abdenur, C. Bonatti, S. Crovisier.  Nonuniform hyperbolicity for C1-generic diffeomorphisms.  Israel J. Math. 183: 1–60, 2011.
  * [2] V. S. Afraimovich, V. V. Bykov, and L. P. Shil’nikov.  On the appearence and structure of the Lorenz attractor.  Dokl. Acad. Sci. USSR, 234:336–339, 1977.
  * [3] V. Araújo and M.J. Pacifico.  Three-dimensional flows.  Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] $26^{\circ}$ Col’oqio Brasileiro de Mathemática.[26th Brazilian Mathematics Colloquium] Instituto de Mathem’atica Pura e Aplicada (IMPA), Rio de Janeiro, 2007.
  * [4] V. Araujo, M. J. Pacifico, E. Pujals, and M. Viana.  Singular-hyperbolic attractors are chaotic.  Trans. Amer. Math. Soc., 361:2431–2485, 2009.
  * [5] A. Arroyo and E. Pujals.  Dynamical properties of singular-hyperbolic attractors.  Discrete Contin. Dyn. Syst. 19 (2007), no. 1, 67–87.
  * [6] C. Bonatti and S. Crovisier.  Recurrence et généricité.  Inventiones Math. 158 (2004), no 1, 33-104.
  * [7] C. Bonatti, L. J. Díaz, and E. Pujals.  A ${C}^{1}$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources.  Annals of Math., 158:355–418, 2003.
  * [8] C. Bonatti, A. Pumariño, and M. Viana.  Lorenz attractors with arbitrary expanding dimension.  In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 39–44. World Sci. Publishing, 2000.
  * [9] R. Bowen.  Entropy expansive maps.  Trans. Am. Math. Soc., 164:323–331, 1972.
  * [10] D. Burguet, G. Liao and J. Yang.  Asymptotic h-expansiveness rate of $C^{\infty}$ maps.  Preprint www.arxiv.org.
  * [11] K. Burns and A. Wilkinson.  On the ergodicity of partially hyperbolic systems.  Annals of Math., 171: 451–489, 2010.
  * [12] W. Cowieson and L. S. Young.  SRB measures as zero-noise limits.  Ergod. Th. Dynamics. Systems. 25(4):1115–1138, 2005.
  * [13] L. J. Díaz, T. FISHER, M. J. Pacifico, and J. L. Vieitez.  Entropy-expansiveness for partially hyperbolic diffeomorphisms.  Discrete Contin. Dyn. Syst. 32(12):4195-4207, 2012.
  * [14] T. Downarowicz and S. Newhouse.  Symbolic extensions and smooth dynamical systems.  Inventiones Math., 160:453–499, 2005.
  * [15] S. Gan  A generalized shadowing lemma.  Discrete Contin. Dyn. Syst., 8 (2002), no. 3, 627–632.
  * [16] S. Gan and D. Yang  Morse-Smale systems and horseshoes for three dimensional singular flows.  www.arxiv.org
  * [17] J. Guckenheimer.  A strange, strange attractor.  In The Hopf bifurcation theorem and its applications, pages 368–381. Springer Verlag, 1976.
  * [18] J. Guckenheimer and R. F. Williams.  Structural stability of Lorenz attractors.  Publ. Math. IHES, 50:59–72, 1979.
  * [19] M. Hirsch, C. Pugh, and M. Shub.  Invariant manifolds, volume 583 of Lect. Notes in Math. Springer Verlag, 1977.
  * [20] A. Katok.  Lyapunov exponents, entropy and periodic points of diffeomorphisms.  Publ. Math. IHES., 51: 137–173, 1980.
  * [21] M. Li, S. Gan and L. Wen.  Robustly transitive singular sets via approach of extended linear Poincaré flow,  Discrete Contin. Dyn. Syst., 13:239–269, 2005.
  * [22] G. Liao, M. Viana, and J. Yang.  Entropy conjecture for diffeomorphisms away from tangencies.  J. European Math. Soc.
  * [23] S.T. Liao.  An existence theorem for periodic orbits.  Acta Sci. Natur. Univ. Pekinensis, 1: 1–20, 1979.
  * [24] S. T. Liao.  Certain uniformity properties of differential systems and a generalization of an existence theorem for periodic orbits.  Acta Sci. Natur. Univ. Pekinensis, 2: 1–19, 1985.
  * [25] S. T. Liao.  The qualitative theory of differential dynamical systems.  Science Press, 1996\.
  * [26] S. T. Liao.  On $(\eta,d)$-contractible orbits of vector fields.  Systems Science and Mathematica Sciences., 2: 193–227, 1989.
  * [27] E. N. Lorenz.  Deterministic nonperiodic flow.  J. Atmosph. Sci., 20:130–141, 1963.
  * [28] M.I. Malkin.  On the continuity of the entropy of discontinuous maps of the interval.  In Leontovich-Andronova, E.A. (ed.): Metody Kachestvennoi Teorii Diferents. Uravnenij: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ., 1982, 35-47.  [English transl.: Sel. Math. Sov. 8, No. 2, 131-139 (1989)] Abl. 572.54035
  * [29] C. Morales and M. J. Pacifico.  Lyapunov stability of $\omega$-limit sets.  Discrete Contin. Dyn. Syst., 8:671–674, 2002.
  * [30] C. Morales, M. J. Pacifico, and E. Pujals.  On ${\mathcal{C}}^{1}$ robust singular transitive sets for three-dimensional flows.  C. R. Acad. Sci. Paris Sér. I Math., 326:81–86, 1998.
  * [31] C. Morales, M. J. Pacifico, and E. R. Pujals.  Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers.  Ann. of Math, 160:375–432, 2004.
  * [32] S. Newhouse.  Continuity properties of entropy.  Annals of Math., 129:215–235, 1990.  Errata in Annals of Math. 131:409–410, 1990.
  * [33] R. Saghin and Z. Xia.  The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center.  Topology Appl., 157:29–34, 2010.
  * [34] B. Saltzman.  Finite amplitude free convection as an initial value problem.  J. Atmos. Sci., 19: 329–341, 1962.
  * [35] Y. Shi S. Gan and L. Wen.  On the singular hyperbolicity of star flows.  To appear in J. Mod. Dyn.
  * [36] S.V. Shlyachkov.  On the entropy of piecewise-monotone expanding maps of the interval.  In Leontovich-Andronova, E.A. (ed.) Metody Kachestvennoi Teorii Diferents. Uravnenij: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ., 1988, 91-97.
  * [37] M. Shub.  Dynamical systems, filtrations and entropy.  Bull. Amer. Math. Soc., 80:27–41, 1974.
  * [38] M. Shub and D. Sullivan.  Homology theory and dynamical systems.  Lecture Notes in Mathematics, 468:36–39, 1975.
  * [39] W. Tucker.  The Lorenz attractor exists.  C. R. Acad. Sci. Paris Sér. I Math., 328: 1197–1202, 1999
  * [40] W. Tucker.  A rigorous ODE solver and Smale’s 14th problem,  Found. Comput. Math., 2: 53–117, 2002
  * [41] M. Viana,  What’s new on Lorenz strange attractors?,  Math. Intelligencer ,The Mathematical Intelligencer, 22: 6–19, 2000
  * [42] M. Viana and J. Yang.  $cu$-equilibrium states.  Preprint, 2014.
  * [43] R.F. Williams.  The structure of the Lorenz attractor.  Publ. Math. IHES, 50:73–99, 1979.
  * [44] J, Yang.  Ergodic measures far away from tangencies.  to appear in Discrete and Continuous Dynamical Systems. Series A.
  * [45] J. Yang.  online video at: http://video.impa.br/index.php?page=international-conference-on-dynamical-system
  * [46] Y. Yomdin.  Volume growth and entropy.  Israel J. Math., 57:285–300, 1987.
  * [47] S. Zhu, S. Gan and L. Wen.  Indices of sigularities of robustly transitive sets.  Discrete Contin. Dyn. Syst. 21:945-957, 2008.