# An entropy dichotomy for singular star flows

Maria José Pacifico, Fan Yang and Jiagang Yang  Instituto de Matemática,
Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de
Janeiro, RJ, Brazil<EMAIL_ADDRESS>Department of Mathematics, Michigan
State University, East Lansing, MI, USA<EMAIL_ADDRESS>Department of
Mathematics, Southern University of Science and Technology of China,
Guangdong, China; and Departamento de Geometria, Instituto de Matemática e
Estatística, Universidade Federal Fluminense, Niterói, Brazil<EMAIL_ADDRESS>
###### Abstract.

We show that non-trivial chain recurrent classes for generic $C^{1}$ star
flows satisfy a dichotomy: either they have zero topological entropy, or they
must be isolated. Moreover, chain recurrent classes for generic star flows
with zero entropy must be sectional hyperbolic, and cannot be detected by any
non-trivial ergodic invariant probability. As a result, we show that $C^{1}$
generic star flows have only finitely many Lyapunov stable chain recurrent
classes.

M.J.P. and J.Y. are partially supported by CNPq, FAPERJ and PROEX-CAPES. J.Y.
is partially supported by NSFC 11871487 of China. F.Y. would like to thank the
hospitality of Southern University of Science and Technology of China (SUSTC),
where part of this work is done.

##  1\. Introduction

A milestone towards the final proof of the stability conjecture for
diffeomorphisms is the independent discovery of the star systems by Liao [19]
and Mañé [22]. Recall that a diffeomorphism with star condition is a $C^{1}$
diffeomorphism for which all the periodic orbits of all nearby diffeomorphisms
are hyperbolic (orbit-wise, not uniformly); in other words, there is no
periodic orbit bifurcation. It was first proven in [10, 19] that
$\Omega$-stability implies the star condition, and later proven by Hayashi
[16] with the help of the connecting lemma that star condition is equivalent
to Axiom A with the no-cycle condition. For smooth vector fields without
singularities, the same result was obtained by Gan and Wen [13] using a
different approach.

For singular flows, that is, flows exhibiting singularities, the situation is
much more complicated. Here, let us introduce the precise definition of the
star condition for singular flows, taking into account the presence of the
singularities.

######  Definition 1.

A $C^{1}$ vector field $X$ is a star vector field if there exists a
neighborhood $X\in{\mathcal{U}}\subset\mathscr{X}^{1}(M)$ such that for all
vector fields $Y\in{\mathcal{U}}$, all the critical elements of $Y$ i.e.,
singularities and periodic orbits, are hyperbolic. The collection of $C^{1}$
star vector fields is denoted by $\mathscr{X}^{1}_{*}(M)$.

It is well known that singular star flows can exhibit rich dynamics and
bifurcation phenomenon comparing to their non-singular counterpart. Certain
singular star flows, such as the famous Lorenz flows, do not have hyperbolic
nonwandering sets [14] and are not structural stable [15]. There are also
examples where the set of the periodic orbits is not dense in the nonwandering
set [9], and where the nonwandering set is Axiom A but no-cycle condition
fails [17].

In order to properly describe the hyperbolicity of an invariant set that
contains singularity, Morales, Pacifico and Pujals [24] proposed the notion of
singular hyperbolicity for three-dimensional flows. They require the flow to
have a one-dimensional uniformly contracting direction, and a two-dimensional
subbundle containing the flow direction of regular points, on which the flow
is volume expanding. This is later generalized to higher dimensions as
sectional hyperbolicity [18, 23]. See Definition 3 in the next section for
more details.

Next, let us introduce the notion of chain recurrent classes, which plays an
important role in the study of the stability conjecture. Roughly speaking,
chain recurrent classes are the largest non-trivial invariant set of a
topological dynamical system, outside which the system behaves like a gradient
system.

######  Definition 2.

For $\varepsilon>0,T>0$, a finite sequence $\\{x_{i}\\}_{i=0}^{n}$ is called
an $(\varepsilon,T)$-chain if there exists $\\{t_{i}\\}_{i=0}^{n-1}$ such that
$t_{i}>T$ and $d(\phi_{t_{i}}(x_{i}),x_{i+1})<\varepsilon$ for all
$i=0,\ldots,n-1$. We say that $y$ is chain attainable from $x$, if there
exists $T>0$ such that for all $\varepsilon>0$, there exists an
$(\varepsilon,T)$-chain $\\{x_{i}\\}_{i=0}^{n}$ with $x_{0}=x$ and $x_{n}=y$.
It is straightforward to check that chain attainability is an equivalent
relation on the closure of the set ${\\{x:\forall t>0,\mbox{ there is an
}(\varepsilon,T)\mbox{-chain with }x_{0}=x_{n}=x\mbox{ and
}\sum_{i}t_{i}>t\\}}$. Each equivalent class under this relation is then
called a chain recurrent class. A chain recurrent class $C$ is non-trivial if
it is not a singularity nor a periodic orbit.

As the first step towards the complete understanding of singular star flows,
the following conjecture was raised in [29].

######  Conjecture 1.1.

(Generic) singular star flows have only finitely many chain recurrent classes,
all of which are sectional hyperbolic for $X$ or $-X$.

A partial affirmative answer to this conjecture was obtained in [28], where it
is proven that for generic star flows, every non-trivial Lyapunov stable chain
recurrent class is sectional hyperbolic. In fact, they obtained a complete
characterization on the stable indices of singularities in the same chain
recurrent classes. This result is crucial to our current paper, and is
explained in more detail in Section 2.4.

A breakthrough was later obtained by da Luz and Bonatti [3, 8] which gives a
negative answer to the second half of this conjecture. They construct an
example which has two singularities with different indices robustly contained
in the same chain recurrent class. As a result, such classes cannot be
sectional hyperbolic. In [3] they propose the notion of multi-singular
hyperbolicity and prove that, generically, star condition is equivalent to
multi-singular hyperbolicity [3, Theorem 3]. To avoid technical difficulty, we
will not provide the definition of multi-singular hyperbolicity here, and
invite the interested readers to [7] for a simpler yet useful definition.

However, the first half of Conjecture 1.1 is still left unanswered: must a
(generic) singular star flow have only finitely many chain recurrent classes?
The goal of the current paper is to provide a partial answer to this question,
using the recent progress in the entropy theory [26]:

######  Theorem A.

There is a residual set ${\mathcal{R}}$ of $C^{1}$ star flows, such that for
every $X\in{\mathcal{R}}$ and every non-trivial chain recurrent class $C$ of
$X$, we have

  1. (1)

if $h_{top}(X|_{C})>0$, then $C$ contains some periodic point $p$ and is
isolated;

  2. (2)

if $h_{top}(X|_{C})=0$, then $C$ is sectional hyperbolic for $X$ or $-X$, and
contains no periodic orbits. In this case, every ergodic invariant measure
$\mu$ with $\operatorname{supp}(\mu)\subset C$ must satisfy
$\mu=\delta_{\sigma}$ for some $\sigma\in{\rm Sing}(X)\cap C$.

In other words, a chain recurrent class with zero topological entropy cannot
be detected by any non-trivial invariant ergodic probability measure, in the
sense that it can only support point masses of singularities. We call such
chain recurrent classes singular aperiodic classes.

Note that in the second case, $C$ cannot be Lyapunov stable due to [26,
Corollary D]. Also note that the example constructed by Bonatti and da Luz
belongs to the first case and is isolated.

An immediate corollary of Theorem A is:

######  Corollary B.

$C^{1}$ generic star flows have only finitely many Lyapunov stable chain
recurrent classes.

Now the first half of Conjecture 1.1, namely the finiteness of chain recurrent
classes for singular star flows, is reduced to the following conjecture:

######  Conjecture 1.2.

For (generic) singular star flows, singular aperiodic classes do not exist.
Consequently, (generic) singular star flows have only finitely many chain
recurrent classes, all of which are homoclinic classes of some periodic
orbits.

#### Organization of the paper

In Section 2 we provide the readers with preliminaries on singular flows:
dominated splitting, (extended and scaled) linear Poincaré flows and Liao’s
shadowing lemma. We also include in Section 2.4 a classification on the
singularities in the same chain recurrent classes, which is taken from [28].
Section 3 contains a detailed analysis on the dynamics near Lorenz-like
singularities and, more importantly, on the transverse intersection between
invariant manifolds of nearby periodic orbits with that of points in the
class. Finally, we provide the proof of Theorem A in Section 4 by showing that
every chain recurrent class with positive topological entropy must contain a
periodic orbit and, consequently, contains all periodic orbits that are
sufficiently close to the class.

##  2\. Preliminaries

Throughout this paper, $X$ will be a $C^{1}$ vector field on a $d$-dimensional
compact manifold $M$. Denote by ${\rm Sing}(X)$ (sometimes we also write ${\rm
Sing}(\phi_{t})$) the set of singularities of $X$, $\phi_{t}$ the flow
generated by $X$, and $f=\phi_{1}$ the time-one map of $\phi_{t}$. We will
write $\Phi_{t}$ for the tangent flow, i.e., $\Phi_{t}=D\phi_{t}:TM\to TM$.

This section includes the necessary background for the proof of Theorem A.
Most notably, we will introduce the scaled and extended linear Poincaré flow
by Liao [20, 21], and the shadowing lemma which was first introduced by Liao
[20, 21], and further developed by Gan [11]. We also collect some previously
established results on generic singular star flows in Section 2.4, most of
which can be found in [28] and [18].

###  2.1. Dominated splitting and invariant cones

A dominated splitting for a flow $\phi_{t}$ is defined similarly to the case
of diffeomorphisms. The invariant set $\Lambda$ admits a dominated splitting
$T_{\Lambda}M=E\oplus F$ if this splitting is invariant under $\Phi_{t}$, and
if 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

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

We invite the readers to [4, Appendix B] and [1] for more properties on the
dominated splitting. The next lemma states the relation between dominated
splitting for the flow and its time-one map.

######  Lemma 2.1.

[26, Lemma 2.6] Let $\Lambda$ be an invariant set. A splitting
$T_{\Lambda}M=E\oplus F$ 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
$X|_{\Lambda\setminus{\rm Sing}(X)}\subset F.$

######  Definition 3.

A compact invariant set $\Lambda$ of a flow $X$ is called sectional
hyperbolic, if it admits a dominated splitting $E^{s}\oplus F^{cu}$, such that
$E^{s}$ is uniformly contracting, and $F^{cu}$ is sectional-expanding: there
are constants $C,\lambda>0$ such that for every $x\in\Lambda$ and any subspace
$V_{x}\subset F^{cu}_{x}$ with $\dim V_{x}\geq 2$, we have

$|\det D\phi_{t}(x)|_{V_{x}}|\geq Ce^{\lambda t}\mbox{ for all }t>0.$

We call $\lambda$ the sectional volume expanding rate on $F^{cu}$.

######  Remark 2.2.

If the dominated splitting $E\oplus F$ is sectional hyperbolic, then
$\Phi_{t}$ on $E$ is uniformly contracting by definition. Since the flow speed
$\|X(x)\|$ is bounded and thus cannot be backward exponentially expanding, we
must have $X|_{\Lambda\setminus{\rm Sing}(X)}\subset F$. For more detail, see
[26, Lemma 3.10].

Let $E\oplus F$ be a dominated splitting for the flow $\phi_{t}$. For $a>0$
and $x\in M$, a $(a,F)$-cone on the tangent space $T_{x}M$ is defined as

$C_{a}(F_{x})=\\{v:v=v_{E}+v_{F}\mbox{ where }v_{E}\in E,v_{F}\in F\mbox{ and
}\|v_{E}\|<a\|v_{F}\|\\}\cup\\{0\\}.$

When $a$ is sufficiently small, the cone field $C_{a}(F_{x})$, $x\in M$, is
forward invariant by $\Phi_{1}$, i.e., there is $\lambda<1$ such that for any
$x\in M$, $\Phi_{1}(C_{a}(F_{x}))\subset C_{\lambda a}(F_{f(x)})$. Similarly,
we can define the $(a,E)$-cone $C_{a}(E_{x})$, which is backward invariant by
$\Phi_{1}$. When no confusing is caused, we call the two families of cones by
$F$ cones and $E$ cones.

For a $C^{1}$ disk with dimension at most $\dim F$, we say that $D$ is tangent
to the $(a,F)$-cone if for any $x\in D$, $T_{x}D\subset C_{a}(F_{x})$. The
same can be said for the $(a,E)$-cone if $T_{x}D\subset C_{a}(E_{x})$ for
every $x\in D$.

###  2.2. Scaled linear Poincaré flows and a shadowing lemma by Liao

In this section, $\mu$ will be a non-trivial ergodic measure with a dominated
splitting $E\oplus F$ for the time-one map $f=\phi_{1}$ on
$\operatorname{supp}\mu$.

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

$N_{\Lambda}=\bigcup_{x\in\Lambda\setminus{\rm Sing}(X)}N_{x},$

where $N_{x}$ is the orthogonal complement of the flow direction $X(x)$, i.e.,

$N_{x}=\\{v\in T_{x}M:v\perp X(x)\\}.$

Denote the orthogonal projection of $T_{x}M$ to $N_{x}$ by $\pi_{x}$ and the
projection of $T_{\Lambda}M$ to $N_{\Lambda}$ by $\pi$. Given $v\in N_{x}$ for
a regular point $x\in M\setminus{\rm Sing}(X)$ and recalling that $\Phi_{t}$
is the tangent flow, we can define $\psi_{t}(v)$ as the orthogonal projection
of $\Phi_{t}(v)$ onto $N_{\phi_{t}(x)}$, 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 the Oseledets theorem:

######  Proposition 2.3.

For $\mu$ almost every $x$, there exists $k=k(x)\in{\mathbb{N}}$ and real
numbers

$\hat{\lambda}_{1}(x)>\cdots>\hat{\lambda}_{k}(x)$

and a $\psi_{t}$ invariant measurable splitting on the normal bundle:

$N_{x}=\hat{E}^{1}_{x}\oplus\cdots\oplus\hat{E}^{k}_{x},$

such that

$\lim_{t\to\pm\infty}\frac{1}{t}\log\|\psi_{t}(v_{i})\|=\hat{\lambda}_{i}(x)\mbox{
for every non-zero vector }v_{i}\in\hat{E}^{i}_{x}.$

Now we state the relation between Lyapunov exponents and the Oseledets
splitting for $\psi_{t}$ and for $f=\phi_{1}$:

######  Theorem 2.4.

For $\mu$ almost every $x$, denote by $\lambda_{1}(x)>\cdots>\lambda_{k}(x)$
the Lyapunov exponents and

$T_{x}M=E^{1}_{x}\oplus\cdots\oplus E^{k}_{x}$

the Oseledets splitting of $\mu$ for $f$. Then

$N_{x}=\pi_{x}(E^{1}_{x})\oplus\cdots\oplus\pi_{x}(E^{k}_{x})$

is the Oseledets splitting of $\mu$ for the linear Poincaré flow $\psi_{t}$.
Moreover, the Lyapunov exponents of $\mu$ (counting multiplicity) for
$\psi_{t}$ is the subset of the exponents for $f$ obtained by removing one of
the zero exponent which comes from the flow direction.

######  Definition 4.

A non-trivial measure $\mu$ is called a hyperbolic measure for the flow
$\phi_{t}$ if it is an ergodic measure of $\phi_{t}$ and all the Lyapunov
exponents for the linear Poincaré flow $\psi_{t}$ are non-vanishing. In other
words, if we view $\mu$ as an invariant measure for the time-one map $f$, then
$\mu$ has exactly one exponent which is zero, given by the flow direction. We
call the number of the negative exponents of $\mu$, counting multiplicity, its
_index_.

The scaled linear Poincaré flow, which we denote by $\psi^{*}_{t}$, is the
normalization of $\psi_{t}$ using the flow speed:

(1)
$\psi^{*}_{t}(v)=\frac{\|X(x)\|}{\|X(\phi_{t}(x))\|}\psi_{t}(v)=\frac{\psi_{t}(v)}{\|\Phi_{t}|_{<X(x)>}\|}.$

We collect the following elementary properties of $\psi_{t}^{*}$.

The next lemma is first observed by Liao [20], see also [12] and [26, Section
4] for the proof.

######  Lemma 2.5.

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

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

Furthermore, for every non-trivial ergodic measure $\mu$, the cocycles
$\psi_{t}$ and $\psi^{*}_{t}$ have the same Lyapunov exponents and Oseledets
splitting.

Next we describe the (quasi-)hyperbolicity for the scaled linear Poincaré flow
$\psi^{*}_{t}$.

######  Definition 5.

For $T_{0}>0$, $\lambda\in(0,1)$, an orbit segment $\\{\phi_{t}(x)\\}_{[0,T]}$
is called $(\lambda,T_{0})$-forward contracting for the bundle $E\subset
N_{x}$, if there exists a partition

$0=t_{0}<t_{1}<\cdots<t_{n}=T,\mbox{\hskip 5.69046pt where
}t_{i+1}-t_{i}\in[T_{0},2T_{0}],$

such that for all $k=1,\ldots,n-1$,

(2)
$\prod_{i=0}^{k-1}\|\psi^{*}_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(E)}\|\leq\lambda^{k}.$

Similarly, an orbit segment $\\{\phi_{t}(x)\\}_{[0,T]}$ is called
$(\lambda,T_{0})$-backward contracting for the bundle $E\subset N_{x}$, if the
orbit segment $\\{\phi_{-t}(\phi_{T}(x))\\}_{[0,T]}$ is
$(\lambda,T_{0})$-forward contracting for the flow $-X$.

######  Definition 6.

For $T_{0}>0,\lambda\in(0,1)$, the orbit segment $\\{\phi_{t}(x)\\}_{[0,T]}$
is called $(\lambda,T_{0})^{*}$ quasi-hyperbolic with respect to a splitting
$N_{x}=E^{N}_{x}\oplus F^{N}_{x}$ and the scaled linear Poincaré flow
$\psi^{*}_{t}$, if it is $(\lambda,T_{0})$-forward contracting for
$E_{x}^{N}$, $(\lambda,T_{0})$-backward contracting for $F_{x}^{N}$, and
satisfies:

(3)
$\frac{\|\psi^{*}_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(E^{N}_{x})}\|}{m(\psi^{*}_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(F^{N}_{x})})}\leq\lambda^{2}.$

######  Definition 7.

Assume that the scaled linear Poincaré flow has a dominated splitting $E\oplus
F$. A point $x$ is called a $(\lambda,T_{0})$-forward hyperbolic time for the
bundle $E\subset N_{x}$, if the infinite orbit $\phi_{[0,+\infty)}$ is
$(\lambda,T_{0})$-forward contracting. In this case the partition is taken as

$0=t_{0}<t_{1}<\cdots<t_{n}<\ldots,\mbox{\hskip 5.69046pt where
}t_{i+1}-t_{i}\in[T_{0},2T_{0}],$

and (2) is stated for all $k\in{\mathbb{N}}$. Similarly, $x$ is called a
$(\lambda,T_{0})$-backward hyperbolic time for the bundle $F\subset N_{x}$, if
it is a forward hyperbolic time for the bundle $F$ and for the flow $-X$. $x$
is called a two-sided hyperbolic time, if it is both a forward and backward
hyperbolic time.

The following lemma states that two consecutive orbit segments that are both
$(\lambda,T_{0})$-forward contracting can be “glued together” to form a
$(\lambda,T_{0})$-forward contracting orbit segment. The same can be said for
backward contracting orbit segments by considering the flow $-X$.

######  Lemma 2.6.

The following statements hold:

  1. (1)

Assume that $\\{\phi_{t}(x)\\}_{[0,T_{1}]}$ is $(\lambda,T_{0})$-forward
contracting for the bundle $E$, and
$\\{\phi_{t}(\phi_{T_{1}}(x))\\}_{[0,T_{2}]}$ is $(\lambda,T_{0})$-forward
contracting for the bundle $\psi_{T_{1}}(E)$. Then
$\\{\phi_{t}(x)\\}_{[0,T_{1}+T_{2}]}$ is $(\lambda,T_{0})$-forward contracting
for the bundle $E$.

  2. (2)

Assume that the scaled linear Poincaré flow has a dominated splitting $E\oplus
F$. Let $\\{\phi_{t}(x)\\}_{[0,T_{1}]}$ be $(\lambda,T_{0})$-forward
contracting for the bundle $E$, and assume that $\phi_{T_{1}}(x)$ is a
$(\lambda,T_{0})$-forward hyperbolic time for the bundle $E$. Then $x$ is a
$(\lambda,T_{0})$-forward hyperbolic time for the bundle $E$.

The proof is standard and thus omitted.

By the classic work of Liao [20], there exists
$\delta=\delta(\lambda,T_{0})>0$ such that if $x$ is a
$(\lambda,T_{0})$-backward hyperbolic time, then $x$ has unstable manifold
with size $\delta\|X(x)\|$. Similarly, if $x$ is a $(\lambda,T_{0})$-forward
hyperbolic time then it has stable manifold with size $\delta\|X(x)\|$. In
both cases, we say that $x$ has unstable/stable manifold up to the flow speed.

The next lemma can be seen as a $C^{1}$ version of the Pesin theory for flows.
The proof can be found in [26, Section 4]

######  Lemma 2.7.

Let $\mu$ be a hyperbolic measure for the flow $\phi_{t}$. For almost every
ergodic component $\tilde{\mu}$ of $\mu$ with respect to $f=\phi_{1}$, there
are $L^{\prime},\eta,T_{0}>0$ and a compact set
$\Lambda_{0}\subset\operatorname{supp}\mu\setminus{\rm Sing}(X)$ with positive
$\tilde{\mu}$ measure, such that for every $x$ satisfying
$f^{n}(x)\in\Lambda_{0}$ for $n>L^{\prime}$, the orbit segment
$\\{\phi_{t}(x)\\}_{[0,n]}$ is $(\eta,T_{0})^{*}$ quasi-hyperbolic with
respect to the splitting $N_{x}=\pi_{x}(E_{x})\oplus\pi_{x}(F_{x})$ and the
scaled linear Poincaré flow $\psi^{*}_{t}$.

Next. we introduce a shadowing lemma by Liao [20] for the scaled linear
Poincaré flow. See [11] and [26] for the current version.

######  Lemma 2.8.

Given a compact set $\Lambda_{0}$ with $\Lambda_{0}\cap{\rm
Sing}(X)=\emptyset$ and $\eta\in(0,1),T_{0}>0$, for any $\varepsilon>0$ there
exists $\delta>0$, $L>0$ and $\delta_{0}>0$, such that for any
$(\eta,T_{0})^{*}$ quasi-hyperbolic orbit segment $\\{\phi_{t}(x)\\}_{[0,T]}$
with respect to a dominated splitting $N_{x}=E_{x}\oplus F_{x}$ and the scaled
linear Poincaré flow $\psi^{*}_{t}$, if $x,\phi_{T}(x)\in\Lambda_{0}$ with
$d(x,\phi_{T}(x))<\delta$, then there exists a point $p$ and a $C^{1}$
strictly increasing function $\theta:[0,T]\to{\mathbb{R}}$, such that

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

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

  5. (e)

$p$ has stable and unstable manifold with size at least $\delta_{0}$.

  6. (f)

if $\Lambda_{0}\subset\Lambda$ for a chain recurrent class $\Lambda$, then
$p\in\Lambda$.

###  2.3. Extended linear Poincaré flows

Note that the (scaled) linear Poincaré flow is only defined at the regular
points $M\setminus{\rm Sing}(X)$. To solve this issue, we introduce the
extended linear Poincaré flow, which is a useful tool developed by Liao [20,
21] and Li et al [18] to study hyperbolic singularities.

Denote by

$G^{1}=\\{L:L\mbox{ is a 1-dimensional subspace of }T_{x}M,x\in M\\}$

the Grassmannian manifold of $M$. Given a $C^{1}$ flow $\phi_{t}$, the tangent
flow $\Phi_{t}$ acts naturally on $G^{1}$ by mapping each $L$ to
$\Phi_{t}(L)$.

Write $\beta:G^{1}\to M$ and $\xi:TM\to M$ the bundle projection. The pullback
bundle of $TM$:

$\beta^{*}(TM)=\\{(L,v)\in G^{1}\times TM:\beta(L)=\xi(v)\\}$

is a vector bundle over $G^{1}$ with dimension $\dim M$. The tangent flow
$\Phi_{t}$ lifts naturally to $\beta^{*}(TM)$:

$\Phi_{t}(L,v)=(\Phi_{t}(L),\Phi_{t}(v)).$

Recall that the linear Poincaré flow $\psi_{t}$ projects the image of the
tangent flow to the normal bundle of the flow direction. The key observation
is that this projection can be defined not only w.r.t the bundle perpendicular
to the flow, but to the orthogonal complement of any section $\\{L_{x}:x\in
M\\}\subset G^{1}$.

To be more precise, given ${\mathcal{L}}=\\{L_{x}:x\in M\\}$ we write

${\mathcal{N}}_{\mathcal{L}}=\\{(L_{x},v)\in\beta^{*}(TM):v\perp L_{x}\\}.$

Then ${\mathcal{N}}$, consisting of vectors perpendicular to $L$, is a sub-
bundle of $\beta^{*}(TM)$ over $G^{1}$ with dimension $\dim M-1$. The extended
linear Poincaré flow is then defined as

$\psi_{t}:{\mathcal{N}}_{\mathcal{L}}\to{\mathcal{N}}_{\mathcal{L}},\\\
\psi_{t}(L_{x},v)=\pi(\Phi_{t}(L_{x},v)),$

where $\pi$ is the orthogonal projection from fibres of $\beta^{*}(TM)$ to the
corresponding fibres of ${\mathcal{N}}$ along ${\mathcal{L}}$.

If we consider the the map

$\zeta:{\rm Reg}(X)\to G^{1}$

that maps every regular point $x$ to the unique $L_{x}\in G^{1}$ with
$\beta(L_{x})=x$ such that $L_{x}$ is generated by the flow direction at $x$,
then the extended linear Poincaré on $\zeta({\rm Reg}(X))$ can be naturally
identified with the linear Poincaré flow defined earlier. On the other hand,
given any invariant set $\Lambda$ of the flow $\phi_{t}$, consider the set:

$\tilde{\Lambda}=\overline{\zeta(\Lambda\cap{\rm Reg}(X))}.$

If $\Lambda$ contains no singularity, then $\tilde{\Lambda}$ can be seen as a
natural copy of $\Lambda$ in $G^{1}$ equipped with the direction of the flow
on $\Lambda$. If $\sigma\in\Lambda$ is a singularity, then $\tilde{\Lambda}$
contains all the direction in $\beta^{-1}(\sigma)$ that can be approximated by
the flow direction at regular points in $\Lambda$. The extended Poincaré flow
restricted to $\tilde{\Lambda}$ can be seen as the continuous extension of the
linear Poincaré flow on $\Lambda$. The same treatment can be applied to the
scaled linear Poincaré flow $\psi_{t}^{*}$.

###  2.4. Classification of chain recurrent classes and singularities for
generic star flows

In this subsection we recap the main result in [28] on $C^{1}$ generic star
flows. We begin with the following classification on the singularities.

######  Definition 8.

Let $\sigma$ be a hyperbolic singularity contained in a non-trivial chain
recurrent class $C(\sigma)$. Assume that the Lyapunov exponents of $\sigma$
are:

$\lambda_{1}\leq\cdots\leq\lambda_{s}<0<\lambda_{s+1}\leq\cdots\leq\lambda_{\dim
M}.$

Write ${\rm Ind}(\sigma)=s$ for the stable index of $\sigma$. We say that

  1. (1)

$\sigma$ is Lorenz-like, if $\lambda_{s}+\lambda_{s+1}>0$,
$\lambda_{s-1}<\lambda_{s}$ (this implies that $e^{\lambda_{s}}<1$ is a real
single eigenvalue of $\Phi_{t}|_{T_{\sigma}M}$), and
$W^{ss}(\sigma)\cap\\{\sigma\\}=\\{\sigma\\}$, where $W^{ss}(\sigma)$ is the
stable manifold of $\sigma$ corresponding to
$\lambda_{1},\ldots,\lambda_{s-1}$; regular orbits in $C(\sigma)$ can only
approach $\sigma$ along $E^{cu}(\sigma)$ cone, where $E^{cu}$ is the
$\Phi_{t}$-invariant subspace correspond to $\lambda_{s},\ldots,\lambda_{\dim
M}$;

  2. (2)

$\sigma$ is reverse Lorenz-like,111In [28] and [7] both cases are called
Lorenz-like. Here we distinguish between the two since our main argument are
different in each case. See the proof of Lemma 4.6 for more details. if it is
Lorenz-like for $-X$; in this case, regular orbits in $C(\sigma)$ can only
approach $\sigma$ along $E^{cs}(\sigma)$ cone. See Figure 1.

Figure 1. Lorenz-like and reverse Lorenz-like singularities

Then it is shown in [18] and [28] that (all the theorems are labeled according
to [28]):

  * •

for a star vector field $X$, if a chain recurrent class $C$ is non-trivial,
then every singularity in $C$ is either Lorenz-like or reverse Lorenz-like
(Theorem 3.6); the original proof can be found in [18];

  * •

there exists a residual set ${\mathcal{R}}\subset\mathscr{X}^{1}_{*}(M)$ such
that for every $X\in{\mathcal{R}}$, if a periodic orbit $p$ is sufficiently
close to a singularity $\sigma$, then:

    * –

when $\sigma$ is Lorenz-like, the index of the $p$ must be ${\rm
Ind}(\sigma)-1$;

    * –

when $\sigma$ is reverse Lorenz-like, the index of the $p$ is ${\rm
Ind}(\sigma)$ (Lemma 4.4);

furthermore, the dominated splitting on $\sigma$ induced by such periodic
orbits coincides with the hyperbolic splitting on $\sigma$ (proof of Theorem
3.7);

  * •

For every chain recurrent class $C$ there exists an integer ${\rm Ind}_{C}>0$,
such that every periodic orbit contained in a sufficiently small neighborhood
of $C$ has the same stable index which equals ${\rm Ind}_{C}$ (Theorem 5.7);

  * •

combine the previous two results, we see that all the singularity in $C$ has
index either ${\rm Ind}_{C}+1$ (in which case it must be Lorenz-like) or ${\rm
Ind}_{C}$ (reverse Lorenz-like);

  * •

if all the singularities in $C$ are Lorenz-like, then $C$ is sectional
hyperbolic (Theorem 3.7); if all the singularities in $C$ are reverse Lorenz-
like, then $C$ is sectional hyperbolic for $-X$ (Theorem 3.7);

  * •

if $C$ contains singularity with different indices (note that they can only
differ by one), then there is no sectional hyperbolic splitting on $C$; one of
such examples was constructed by Bonatti and da Luz [3, 8].

##  3\. Flow orbits near singularities

This section contains some general results on hyperbolic singularities of
$C^{1}$ vector fields (Section 3.1), and on Lorenz-like singularities for star
flows (Section 3.2). The key results are Lemma 3.2 and 3.4, which state that
the time near a singularity where the orbit is “make the turn” is bounded from
above. For Lorenz-like singularities of star flows, we prove in Lemma 3.12
that for a periodic orbit ${\rm Orb}(p)$ approaching a singularity $\sigma$
while exhibiting backward hyperbolic times, the unstable manifold of ${\rm
Orb}(p)$ must transversally intersect with $W^{s}(\sigma)$. Then Lemma 3.13
deals with the case where a sequence of forward hyperbolic times approaches a
Lorenz-like singularity. These two results will allow us to show in the next
section that such periodic orbit must be in the same chain recurrent class as
$\sigma$.

###  3.1. Flow orbits near hyperbolic singularities

In this section we will establish some geometric properties for flow orbits in
a small neighborhood of a hyperbolic singularity. Our result applies to all
$C^{1}$ vector fields $X$ which are not necessarily star.

For this purpose, let $\sigma$ be a hyperbolic singularity with the hyperbolic
splitting $E^{s}_{\sigma}\oplus E^{u}_{\sigma}$. Without loss of generality,
we can think of $\sigma$ as the origin in ${\mathbb{R}}^{n}$, and assume that
$E^{s}_{\sigma}$ and $E^{u}_{\sigma}$ are perpendicular (which is possible if
one changes the metric). In particular, we will assume that
$E^{s}_{\sigma}={\mathbb{R}}^{s}$ is the $s$-dimensional subspace of
${\mathbb{R}}^{n}$ with the last $\dim M-s$ coordinates being zero. Here
$s=\dim E^{s}_{\sigma}$ is the stable index of $\sigma$. Similarly,
$E^{u}_{\sigma}$ is the subspace of ${\mathbb{R}}^{n}$ where the first $s$
coordinates are zero. As before we will write $f=\phi_{1}$ for the time-one
map of the flow.

Since the vector filed $X$ is $C^{1}$, we can take a neighborhood
$U=B_{r}(\sigma)$ with $r$ small enough, such that:

  * •

the flow in $U$ can be written as

(4) $\phi_{t}(x)=e^{At}x+C^{1}\mbox{ small perturbation},$

where $A$ is a matrix no eigenvalue on the imaginary axis;

  * •

for $x\in U$, the tangent map $Df_{x}=\Phi_{1}|_{T_{x}M}$ are small
perturbations of the hyperbolic matrix $e^{A}$, with eigenvalues bounded away
from $1$.

For each $x\in U$, denote by $x^{s}$ its distance to $E^{u}_{\sigma}$ and
$x^{u}$ its distance to $E^{s}_{\sigma}$. Then for every $\alpha>0$ small, we
define the $\alpha$-cone on the manifold, denote by $D^{i}_{\alpha}(\sigma)$,
$i=s,u$, as follows:

$D^{s}_{\alpha}(\sigma)=\\{x\in U:x^{u}<\alpha x^{s}\\},\hskip
28.45274ptD^{u}_{\alpha}(\sigma)=\\{x\in U:x^{s}<\alpha x^{u}\\}.$

Note that the hyperbolic splitting $T_{\sigma}M=E^{s}_{\sigma}\oplus
E^{u}_{\sigma}$ can be extended to $U$ in a natural way: for each $x\in U$,
put $E^{s}(x)$ as the $s$-dimensional hyperplane that is parallel to
$E^{s}_{\sigma}$; the same can be done for $E^{u}(x)$. This allows us to
consider the $\alpha$-cones $C_{\alpha}(E^{i})$, $i=s,u$, on the tangent
bundle as defined in Section 2.1. The next lemma easily follows from the
smoothness of the vector field $X$ and the hyperbolicity of $\sigma$:

######  Lemma 3.1.

There exists $L\geq 1$, such that for all $\alpha>0$ small enough,

  1. (1)

for every $x\in{\rm Cl}(D^{s}_{\alpha}(\sigma))$, we have $X(x)\in
C_{L\alpha}(E^{s})$;

  2. (2)

for every $x\in U$, if $X(x)\in C_{\alpha}(E^{s})$, we have $x\in
D^{s}_{L\alpha}(\sigma)$.

Moreover, the same holds for $D^{u}_{\alpha}(\sigma)$ and $C_{\alpha}(E^{u})$.

Let us fix some $\alpha>0$ small enough that will be specified at the end of
this section. Note that if $x\in U\setminus(D^{s}_{\alpha}(\sigma)\cup
D^{u}_{\alpha}(\sigma))$, we lose control on the direction of $X(x)$. One can
think of the region $U\setminus(D^{s}_{\alpha}(\sigma)\cup
D^{u}_{\alpha}(\sigma))$ as the place where the flow is ‘making the turn’ from
the $E^{s}$ cone to the $E^{u}$ cone. The next lemma states that the time that
an orbit segment spend in this region is uniformly bounded. To this end, we
write, for each $x\in U$,

$t^{+}(x)=\sup\\{t>0:\phi_{[0,t]}(x)\subset U\\},\hskip
14.22636ptt^{-}(x)=\sup\\{t>0:\phi_{[-t,0]}(x)\subset U\\}.$

Then the orbit segment $\phi_{(-t^{-},t^{+})}(x)$ contains $x$ and is
contained in $U$. With slight abuse of notation, we will frequently drop the
depends of $t^{\pm}(x)$ on $x$.

######  Lemma 3.2.

Let $\sigma$ be a hyperbolic singularity for a $C^{1}$ vector field $X$. Then
for every $\alpha>0$ small enough, there exists $T_{\alpha}>0$ such that for
every $r>0$ small enough and every $x\in U=B_{r}(x)$, the set

$T(x):=\\{t\in(-t^{-},t^{+}):\phi_{t}(x)\notin D^{s}_{\alpha}(\sigma)\cup
D^{u}_{\alpha}(\sigma)\\}$

has Lebesgue measure bounded by $T_{\alpha}$.

###### Proof.

Recall that the Lebesgue measure on the interval $(-t^{-},t^{+})$ corresponds
to the length of open intervals. Below we will prove that the set in question
is contained in a subinterval of $(-t^{-},t^{+})$ whose length is bounded from
above.

We take a small neighborhood $x\in V\subset U$, such that for every $y\in V$
it holds

$\phi_{(-t^{-}(y),t^{+}(y))}(y)\cap
D_{\alpha}^{*}(\sigma)\neq\emptyset,*=s,u.$

Note that if a orbit segment $\phi_{(-t^{-},t^{+})}(x)$ does not intersect
with $V$, then $t^{-}+t^{+}$ must be bounded. Therefore we only need to prove
the lemma for orbit segments that intersect with $V$.

Shrinking $V$ if necessary, we may assume that

$\phi_{(-t^{-},t^{+})}(x)\cap V\neq\emptyset\implies\phi_{-t^{-}}(x)\in
D^{s}_{\alpha}(\sigma),\phi_{t^{+}}(x)\in D^{u}_{\alpha}(\sigma).$

We will also assume, by changing to a different point on the same orbit
segment if necessary, that $x\in V\setminus(D^{s}_{\alpha}(\sigma)\cup
D^{u}_{\alpha}(\sigma))$. For such an orbit segment
$\phi_{(-t^{-},t^{+})}(x)$, define

$t^{s}=t^{s}(x)=\sup\\{t>0:\phi_{(-t^{-},-t)}(x)\subset
D^{s}_{\alpha}(\sigma)\\},$

and

$t^{u}=t^{u}(x)=\sup\\{t>0:\phi_{(t,t^{+})}(x)\subset
D^{u}_{\alpha}(\sigma)\\}.$

Clearly we have $T(x)\subset(-t^{s},t^{u})$. Below we will show that
$t^{s}+t^{u}$ is bounded from above.

Writing $x^{s}=\phi_{-t^{s}}(x)$, $x^{u}=\phi_{t^{u}}(x)$, Lemma 3.1(2) shows
that

$X(x^{s})\notin C_{\alpha/L}(E^{s}),\,\,X(x^{u})\notin C_{\alpha/L}(E^{u}).$

In particular222Following our notation earlier,
$X(x^{*})=X(x^{*})^{s}+X(x^{*})^{u}\in E^{s}\oplus E^{u}$.,

(5)
$\frac{||X(x^{s})^{u}||}{||X(x^{s})^{s}||}>\alpha/L,\,\,\,\,\frac{||X(x^{u})^{u}||}{||X(x^{u})^{s}||}<L/\alpha.$

On the other hand, by the hyperbolicity of $\sigma$, there is $\lambda_{0}>1$
independent of $\alpha$ such that for each $x\in U\cap\phi_{-1}(U)$, and for
every $v\in T_{x}M$ it holds that

$\frac{||\Phi_{1}(v)^{u}||}{||\Phi_{1}(v)^{s}||}>\lambda_{0}\frac{||v^{u}||}{||v^{s}||}.$

Combine this with (5) and the observation that
$\Phi_{t^{s}+t^{u}}(X(x^{s}))=X(x^{u})$, we obtain

$\lambda_{0}^{t^{s}+t^{u}}<\frac{L^{2}}{\alpha^{2}},$

which implies that

$t^{s}+t^{u}<\frac{2\log L-2\log\alpha}{\log\lambda_{0}}:=T_{\alpha}.$

Also note that $T_{\alpha}$ can be made uniform for $r>0$ small enough, since
$L$ and $\lambda_{0}$ can be chosen independent of $r$. This concludes the
proof of the lemma. ∎

Now let us look at this lemma from the perspective of invariant measures.
Setting for $i=s,u$,

${\mathcal{L}}^{i}(\sigma)=\\{L\in G^{1}:\beta(L)=\sigma,L\mbox{ is parallel
to }E^{i}\\},$

then ${\mathcal{L}}^{i}$ are invariant under $\Phi_{t}|_{\beta^{-1}(\sigma)}$
(note that this is the tangent flow on $G^{1}$). Furthermore, the
hyperbolicity of $\sigma$ implies that ${\mathcal{L}}^{s}$ is a repelling set
while ${\mathcal{L}}^{u}$ is an attracting set.

Next we take a sequence of points $\\{x_{i}\\}\subset U$ with $x_{i}\to\sigma$
as $i\to\infty$. To simply notation, we will write
$t^{\pm}_{i}=t^{\pm}(x_{i})$. Note that $t^{\pm}_{i}\uparrow+\infty$. For
every $\varepsilon>0$ small enough, the time that the orbit segments
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ spend in the region $U\setminus
B_{\varepsilon}(\sigma)$ is uniformly bounded in $i$. As a result, the
empirical measures supported on these orbit segments behaves trivially:

(6)
$\nu_{i}=\frac{1}{t^{-}_{i}+t^{+}_{i}}\int_{-t^{-}_{i}}^{t^{+}_{i}}\delta_{\phi_{s}(x_{i})}\,ds\xrightarrow{i\to\infty}\delta_{\sigma},$

where $\delta_{\sigma}$ is the atomic measure on $\sigma$.

On the other hand, the map $\zeta:{\rm Reg}(X)\to G^{1}$ defined in Section
2.3 lifts any measure $\mu$ on $M$ with $\mu({\rm Sing})=0$ to a measure
$\zeta_{*}(\mu)$ on $G^{1}$. Now consider the lifted empirical measures:

(7) $\tilde{\nu}_{i}=\zeta_{*}(\nu_{i}).$

If we take any weak-* limit $\tilde{\mu}$ of $\\{\tilde{\nu}_{i}\\}$ (the
limit exists since $G^{1}$ is compact), $\tilde{\mu}$ must be invariant under
$\Phi_{t}$ and is supported on
${\mathcal{L}}^{s}\cup{\mathcal{L}}^{u}\subset\beta^{-1}(\sigma)$ since
${\mathcal{L}}^{s}\cup{\mathcal{L}}^{u}$ contains the non-wandering set of
$\Phi_{t}|_{\beta^{-1}(\sigma)}$. Write
${\mathcal{U}}_{\alpha}^{*}=\overline{\zeta(D^{*}_{\alpha}(\sigma))}$ for
$*=s,u$. Observe that by Lemma 3.1, ${\mathcal{U}}_{\alpha}^{*}$ each contains
a neighborhood of ${\mathcal{L}}^{*}(\sigma)$ in $G^{1}$, $*=s,u$.
Furthermore, we have
${\mathcal{U}}_{\alpha}^{s}\cap{\mathcal{U}}_{\alpha}^{u}=\emptyset$. Combine
this with Lemma 3.2, we obtain the following lemma:

######  Lemma 3.3.

For all $\alpha>0$ small, we have
$\tilde{\nu}_{i}({\mathcal{U}}_{\alpha}^{s}\cup{\mathcal{U}}_{\alpha}^{u})\to
1$ as $i\to N$.

The next lemma states that the time that orbit segments
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ spend in $D^{s}_{\alpha}(\sigma)$ and
$D^{u}_{\alpha}(\sigma)$ are comparable:

######  Lemma 3.4.

There is $a\in(0,\frac{1}{2})$ independent of $\alpha$, such that for every
sequence $\\{x_{i}\\}\subset U$ with $x_{i}\to\sigma$ and every weak*-limit
$\tilde{\mu}$ of the empirical measure $\tilde{\nu}_{i}$ defined using (6) and
(7), we have

$\tilde{\mu}({\mathcal{U}}_{\alpha}^{s})>a\mbox{ and
}\tilde{\mu}({\mathcal{U}}_{\alpha}^{u})>a.$

###### Proof.

We will show that for every orbit segment
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$, the time it spends in
$D^{*}_{\alpha}$, $*=s,u$ are comparable, with a ratio that is uniform in $i$
and $\alpha$.

First, note that since the vector field is $C^{1}$, the flow speed is a
Lipschitz function of $d(x,\sigma)$: there is $0<C_{1}<C_{2}$ such that

$\frac{\|X(x)\|}{d(x,\sigma)}\in(C_{1},C_{2}).$

To simplify notation, we write

$x_{e,i}=\phi_{-t^{-}_{i}}(x_{i})\mbox{, and }x_{l,i}=\phi_{t^{+}_{i}}(x_{i})$

for the end points of $\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ that enter and
leave the neighborhood $U$. By our construction, $x_{e,i},x_{l,i}\in\partial
U=\partial B_{r}(x)$. As a result, for every $i$ it holds

$\frac{\|X(x_{e,i})\|}{\|X(x_{l,i})\|}\in\left(\frac{C_{1}}{C_{2}},\frac{C_{2}}{C_{1}}\right).$

Denote by $t^{0}_{i}\in(-t^{-}_{i},t^{+}_{i})$ the time such that the point
$x^{0}_{i}=\phi_{t_{0}}(x_{i})$ satisfies $(x^{0}_{i})^{s}=(x^{0}_{i})^{u}$.
One could think of $x^{0}_{i}$ as the point on the orbit segment
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ where the flow speed is the lowest. We
parse each orbit segment $\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ into three
sub-segments (recall the definition of $t^{s}(x_{i})$ and $t^{u}(x_{i})$ in
Lemma 3.2. To simplify notation we will write $t_{i}^{*}=t^{*}(x_{i})$,
$*=s,u$):

  * •

write $x^{s}_{i}=\phi_{t^{s}_{i}}(x_{i})$ for the point on
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ that is on the boundary of
$D^{s}_{\alpha}(\sigma)$; then the orbit from $x_{e,i}$ to $x^{s}_{i}$ is
contained in $D^{s}_{\alpha}(\sigma)$;

  * •

write $x^{u}_{i}=\phi_{t^{u}_{i}}(x_{i})$ for the point on
$\phi_{(-t^{-}_{i},t^{+}_{i})}(x_{i})$ that is on the boundary of
$D^{u}_{\alpha}(\sigma)$; then the orbit from $x^{u}_{i}$ to $x_{l,i}$ is
contained in $D^{u}_{\alpha}(\sigma)$;

  * •

the orbit segment from $x^{s}_{i}$ to $x^{u}_{i}$ is outside
$D^{*}_{\alpha}(\sigma)$, $*=s,u$; by Lemma 3.2, $t^{u}_{i}-t^{s}_{i}\leq
T_{\alpha}$.

Note that $x^{0}_{i}$ is contained in the orbit segment from $x^{s}_{i}$ to
$x^{u}_{i}$. Since the flow time from $x^{0}_{i}$ to $x^{\pm}_{i}$ is bounded
by $T_{\alpha}$ and the flow is $C^{1}$, we obtain

$\frac{\|X(x^{u}_{i})\|}{\|X(x^{s}_{i})\|}=\frac{\|X(x^{u}_{i})\|}{\|X(x^{0}_{i})\|}\frac{\|X(x^{0}_{i})\|}{\|X(x^{s}_{i})\|}\in\left(||\Phi_{T_{\alpha}}||^{2},||\Phi_{T_{\alpha}}||^{-2}\right).$

For the orbit segment from $x_{e,i}$ to $x^{s}_{i}$, Lemma 3.1(1) shows that
$X(x)\in C_{L\alpha}(E^{s})$ for each $x$ in this orbit segment. Since the
flow speed is uniformly exponentially contracting in $C_{L\alpha}(E^{s})$
provided that $\alpha$ and $r$ are small enough, we see that the time length
of this orbit segment satisfies

$t^{-}_{i}-t^{s}_{i}=\mathcal{O}(\log\frac{\|X(x_{e,i})\|}{\|X(x_{i}^{-})\|}).$

Similarly,

$t^{+}_{i}-t^{u}_{i}=\mathcal{O}(\log\frac{\|X(x_{l,i})\|}{\|X(x^{u}_{i})\|}).$

Then the ratio is

$\frac{t^{-}_{i}-t^{s}_{i}}{t^{+}_{i}-t^{u}_{i}}=\mathcal{O}\left(\frac{\log\|X(x_{e,i})\|-\log\|X(x_{i}^{-})\|}{\log\|X(x_{l,i})\|-\log\|X(x^{u}_{i})\|}\right)=\mathcal{O}\left(\frac{\log\|X(x_{i}^{-})\|}{\log\|X(x^{u}_{i})\|}\right)=\mathcal{O}(1),$

where in the last equality we use the elementary fact that if $a_{i}\to
0,b_{i}\to 0$ such that $a_{i}/b_{i}$ is bounded from above and away from
zero, then $\log a_{i}/\log b_{i}\to 1.$

Finally, note that even though the ratio
$\frac{\|X(x^{u}_{i})\|}{\|X(x^{s}_{i})\|}$ depends on $\alpha$,
$\frac{t^{-}_{i}-t^{s}_{i}}{t^{+}_{i}-t^{u}_{i}}$ only depends on the
exponential contracting/expanding rate in $C_{L\alpha}(E^{*})$, which can be
made uniform for $\alpha$ small enough. This finishes the proof of the lemma.
∎

We conclude this subsection with the following lemma, which will be used later
to create transverse intersection between the unstable manifold of a periodic
orbit and the stable manifold of the singularity $\sigma$. As before, $s$ is
the stable index of $\sigma$.

######  Lemma 3.5.

For each $\beta>0$ small and $\delta>0$, there is $\alpha_{0}>0$ such that for
all $\alpha<\alpha_{0}$ and for every point $x\in D^{s}_{\alpha}(\sigma)$, let
$W(x)$ be a $(\dim M-s)$-dimensional submanifold that contains $x$ and is
tangent to $C_{\beta}(E^{u})$. If $\operatorname{diam}W(x)>\delta\|X(x)\|$,
then $W(x)\pitchfork W^{s}(\sigma)\neq\emptyset$.

###### Proof.

Since the flow speed at $x$ is a Lipschitz function of $d(x,\sigma)$, we see
that

$\operatorname{diam}W(x)>C_{1}\delta d(x,\sigma)>C_{1}C^{\prime}\delta x^{s}$

for some $C^{\prime}>0$. Here $C_{1}$ is the same constant as in the previous
lemma.

On the other hand, since $W(x)$ is tangent to the $\beta$-cone of $E^{u}$,
there is $C^{\prime\prime}>0$ such that if
$\operatorname{diam}W(x)>C^{\prime\prime}x^{u}=C^{\prime\prime}d(x,W^{s}(\sigma))$,
we must have $W(x)\pitchfork W^{s}(\sigma)\neq\emptyset$.

Since $x^{u}<\alpha x^{s}$ in the cone $D^{s}_{\alpha}$, the choice of
$\alpha<\alpha_{0}:=C_{1}C^{\prime}\delta/C^{\prime\prime}$ guarantees that
$\operatorname{diam}W(x)>CC^{\prime}\delta x^{s}>C^{\prime\prime}x^{u}$,
therefore $W(x)\pitchfork W^{s}(\sigma)\neq\emptyset$. This concludes the
proof of the lemma. ∎

###  3.2. Near Lorenz-like singularities

Now we turn our attention to Lorenz-like singularities for star flows. Assume
that $\sigma$ is a Lorenz-like singularity contained in a non-trivial chain
recurrent class $C=C(\sigma)$. The discussion below applies to reverse Lorenz-
like singularities if one considers the flow $-X$.

Let $E_{\sigma}^{cs}\oplus E_{\sigma}^{u}$ be the hyperbolic splitting on
$T_{\sigma}M$. We will write $E_{\sigma}^{ss}$ the subspace of $T_{\sigma}M$
corresponding to the exponents $\lambda_{1},\ldots,\lambda_{s-1}$ and
$E_{\sigma}^{c}$ the subspace of $T_{\sigma}M$ corresponding to the exponent
$\lambda_{s}$. Then $E_{\sigma}^{ss}\oplus E_{\sigma}^{c}=E_{\sigma}^{cs}$.

The discussion in the previous sub-section applies to $\sigma$ without any
modification (note that this time, we change the notation of $E^{s}$ to
$E^{cs}$ and ${\mathcal{L}}^{s}$ to ${\mathcal{L}}^{cs}$). Furthermore, we can
think of $\sigma$ as the origin in ${\mathbb{R}}^{n}$ with three bundles
$E_{\sigma}^{ss},E_{\sigma}^{c}$ and $E_{\sigma}^{u}$ perpendicular to one
another (which is possible if one changes the metric). These bundles can be
naturally extended to $U=B_{r}(x)$ as before. As a result, the cone field
$C_{\alpha}(E^{*})$ can be defined for $*=ss,c,u,cu$ and $cs$. The same can be
said about the cones on the manifold, $D_{\alpha}^{*}(\sigma)$.

It is proven in [25], [18] and [28] that
$W^{ss}(\sigma)\cap\operatorname{C}(\sigma)=\\{\sigma\\}$. Furthermore, it is
shown that if the orbit segment is taken inside $C(\sigma)$ (or if the orbit
segment belongs to a periodic orbit that is sufficiently close to
$C(\sigma)$), then it can only approach the singularity $\sigma$ along the
one-dimensional subspace $E^{c}$ in the following sense: write

${\mathcal{L}}^{c}=\\{L\in G^{1}:\beta(L)=\sigma,L\mbox{ is parallel to
}E^{c}\\},$

then ${\mathcal{L}}^{c}\subset{\mathcal{L}}^{cs}$ consists of a single point
in $G^{1}$. If we take $x_{i}\in C(\sigma)$ with $x_{i}\to\sigma$ and define
the empirical measure $\nu_{i}$ and its lift $\tilde{\nu}_{i}$ according to
(6) and (7), then any weak*-limit $\tilde{\mu}$ of $\\{\tilde{\nu}_{i}\\}$
must satisfy

$\operatorname{supp}\tilde{\mu}={\mathcal{L}}^{c}\cup{\mathcal{L}}^{u}.$

This observation leads to the following lemma, which is an improved version of
Lemma 3.3 and 3.4.

######  Lemma 3.6.

Let $\\{x_{i}\\}\subset C(\sigma)$, then the conclusion of Lemma 3.3 and Lemma
3.4 remain true with ${\mathcal{U}}^{s}(\alpha)$ replaced by a neighborhood
${\mathcal{U}}^{c}(\alpha)$ of ${\mathcal{L}}^{c}$ in $G^{1}$. The same can be
said if we take $\\{p_{i}\\}$ to be periodic points with $p_{i}\to\sigma$ but
not necessarily in $C(\sigma)$.

The proof remains unchanged and is thus omitted.

Now let us describe the hyperbolicity of the periodic orbits close to
$\sigma$. Recall that for a regular point $x$, $N_{x}\subset T_{x}M$ is the
orthogonal complement of the flow direction $X(x)$. Since $X$ is a star vector
field, every periodic orbit is hyperbolic. It is proven in [28, Theorem 3.7]
if $p_{n}$ is a sequence of periodic points near $C(\sigma)$ and approaches
$\sigma$, then the hyperbolic splitting $E^{s}\oplus E^{cu}$ (with
$E^{cu}=<X>\oplus E^{u}$) on ${\rm Orb}(p_{n})$ extends to a dominated
splitting on $\sigma$, which coincides with the splitting
$E^{ss}_{\sigma}\oplus E_{\sigma}^{cu}$ where
$E_{\sigma}^{cu}=E_{\sigma}^{c}\oplus E_{\sigma}^{u}$. Since we assume that
$E^{ss}$ is perpendicular to $E^{cu}$, and the flow direction is tangent to
the cone $C_{L\alpha}(E^{c})$ as the orbit approaches the singularity
$\sigma$, it follows that for $n$ large enough, the local stable manifold
$W^{s}(p^{\prime}_{n})$ (which has dimension $s-1$) for $p^{\prime}_{n}\in{\rm
Orb}(p_{n})\cap D^{cs}_{\alpha}(\sigma)$ is tangent to a $E^{ss}$ cone.

######  Remark 3.7.

It is tempting to argue that for $n$ large enough, $W^{u}(p^{\prime}_{n})$
must intersect transversally with $W^{cs}(\sigma)$. However, this is not
necessarily the case: as $p_{n}$ gets closer to the singularity $\sigma$, the
size of the invariant manifolds of $p$ will shrink with rates proportional to
the flow speed, as observed by Liao [20]. As a result, even if we have a
sequence of periodic points $p_{n}\to p\in W^{cs}(\sigma)$, there is still no
guarantee that the unstable manifold of $p_{n}$ will intersect with
$W^{cs}(\sigma)$. To solve this issue, we consider the hyperbolic times of
$p_{n}$ as defined in Definition 5. But before that, let us first estimate the
hyperbolicity of the orbit segment inside $D^{cs}_{\alpha}$ and
$D^{u}_{\alpha}$.

######  Lemma 3.8.

Let $\sigma$ be a Lorenz-like singularity, then for $r$ small enough, for
every orbit segment $\phi_{[0,T]}(x)\subset B_{r}(\sigma)\cap C(\sigma)$, the
scaled linear Poincaré flow $\psi^{*}_{t}|_{E^{ss}_{x}}$ is uniformly
contracting, where $E^{ss}_{x}\subset N_{x}$ is the stable subspace for the
scaled linear Poincaré flow. The same holds true for every periodic orbit
segment $\phi_{[0,T]}(p)\subset B_{r}(\sigma)$ but not necessarily in
$C(\sigma)$.

###### Proof.

We only need to estimate

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

for $v\in E^{ss}_{x}$.

We take $\varepsilon>0$ small enough such that
$\lambda_{s-1}+2\varepsilon<\lambda_{s}-\varepsilon<0$ (recall that
$\lambda^{s}<0$ is the largest negative exponent of $\sigma$). If we take
$r>0$ small enough, then inside $B_{r}(\sigma)$ we have

$\|\psi_{t}|_{E^{ss}_{x}}(v)\|\leq e^{(\lambda_{s-1}+\varepsilon)t}\|v\|.$

On the other hand, for $\frac{\|X(x)\|}{\|X(\phi_{t}(x))\|}$ we have (in the
worst case scenario, where the flow direction is tangent to the $E^{c}$ cone):

$\|X(\phi_{t}(x))\|\geq e^{\lambda_{s}-\varepsilon}\|X(x)\|.$

Indeed the flow speed is expanding while the direction is tangent to the
$E^{u}$ cone. While the orbit is neither in $D^{cs}_{\alpha}$ nor in the
$D^{u}_{\alpha}$, we lose all the estimate. However, the length of such orbit
segment is uniformly bounded due to Lemma 3.2, and can be safely ignored.

As a result, we get

$\|\psi^{*}_{t}(v)\|\leq
e^{(\lambda_{s-1}+\varepsilon-\lambda_{s}+\varepsilon)t}\|v\|\leq
e^{-\varepsilon t}\|v\|,$

and conclude the proof of the lemma. ∎

On the unstable subspace $E^{u}_{x}\subset N_{x}$, the situation is different:
when the orbit of $x$ moves away from $\sigma$, it can only do so along
$D^{u}_{\alpha}(\sigma)$. As a result, one loses the hyperbolicity along the
$E^{u}_{x}$ direction. On the other hand, when the orbit approaches $\sigma$,
the orbit segment will be ‘good’ (in the sense that it is quasi-hyperbolic
according to Definition 6) as long as the flow direction is tangent to the
$E^{c}$ cone. This is summarized in the next lemma:

######  Lemma 3.9.

Let $\sigma$ be a Lorenz-like singularity, then there exists
$\lambda\in(0,1),T_{0}>0$, $\alpha_{1}>0$ and $r>0$, such that if
$\alpha<\alpha_{1}$ and $x$ is a periodic orbit such that the orbit segment
$\phi_{[0,T]}(x)$ is contained in $B_{r}(\sigma)\cap{\rm
Cl}(D^{cs}_{\alpha}(\sigma))$, then the orbit segment is
$(\lambda,T_{0})$-backward contracting. If the orbit segment is in ${\rm
Cl}(D^{u}_{\alpha}(\sigma))$, then it does not have any sub-segment that is
backward contracting.

###### Proof.

Case 1. $\phi_{[0,T]}(x)\subset{\rm Cl}(D^{cs}_{\alpha}(\sigma))$.

To simplify notation we write $y=\phi_{T}(x)$ for the endpoint of the orbit
segment (note that it is the starting point for the same orbit segment under
$-X$). Take any $t\in[0,T]$, we will estimate

$\psi^{*}_{-t}(v)=\frac{\|X(y)\|}{\|X(\phi_{-t}(y))\|}\psi_{-t}(v),$

for $v\in E^{u}_{y}$.

Recall that $\lambda_{s+1}$ is the smallest positive exponent of $\sigma$.
Like in the previous lemma, we take $\varepsilon>0$ small such that
$\lambda_{s}+\varepsilon<0$, $\lambda_{s+1}-\varepsilon>0$, therefore
$-\lambda_{s+1}+\lambda_{s}+2\varepsilon<0$.

Then we can take $r>0$ and $\alpha_{1}>0$ small enough such that all the
lemmas in Section 3.1 hold; furthermore,

$\|\psi_{-t}(v)\|\leq e^{-(\lambda_{s+1}-\varepsilon)t}\|v\|,$

and

$\|X(\phi_{-t}(y))\|\geq e^{-(\lambda_{s}+\varepsilon)t}\|X(y)\|,$

since the orbit segment is in ${\rm Cl}(D^{cs}_{\alpha}(\sigma))$. This gives

$\|\psi^{*}_{-t}(v)\|\leq
e^{(-\lambda_{s+1}+\varepsilon+\lambda_{s}+\varepsilon)t}\|v\|=e^{(-\lambda_{s+1}+\lambda_{s}+2\varepsilon)t}\|v\|,$

which shows that the orbit segment is backward contracting.

Case 2. $\phi_{[0,T]}(x)\subset{\rm Cl}(D^{u}_{\alpha}(\sigma))$.

We take any orbit segment $\phi_{[0,T]}(x)$ in ${\rm
Cl}(D^{u}_{\alpha}(\sigma))$ and take any $y\in\phi_{[0,T]}(x)$. By Lemma 3.1
the flow direction is almost parallel to $E^{u}$ if we take $\alpha_{1}$ small
enough. As a result, we can take $v\in E^{u}_{y}$ such that $v$ is almost
parallel to $E^{c}(\sigma)$. For such $v$ we have

$\|\psi_{-t}(v)\|\geq e^{-(\lambda_{s}+\varepsilon)t}\|v\|,$

and the flow direction satisfies

$\|X(\phi_{-t}(y))\|\leq e^{(-\lambda_{s+1}+\varepsilon)t}\|X(y)\|.$

This shows that

$\|\psi^{*}_{-t}(v)\|\geq
e^{(-\lambda_{s}-\varepsilon+\lambda_{s+1}-\varepsilon)t}\|v\|=e^{(-\lambda_{s}+\lambda_{s+1}-2\varepsilon)t}\|v\|.$

For $\varepsilon$ small enough, $-\lambda_{s}+\lambda_{s+1}-2\varepsilon>0$.
As a result, $\psi^{*}_{-t}$ will never be contracting as long as the orbit
segment is contained in ${\rm Cl}(D^{u}_{\alpha}(\sigma))$. Therefore
$\phi_{[0,T]}(x)$ does not contain any sub-segment that is backward
contracting. ∎

######  Remark 3.10.

The previous two lemmas have similar formulations for reverse Lorenz-like
singularities.

Recall the definition of $t^{\pm}(x)$ from the previous section. Next, we
introduce the main lemmas in this section, which will enable us to solve the
issue mentioned in Remark 3.7 and create transverse intersection between the
invariant manifolds of $p$ with points in $C$.

######  Lemma 3.11.

Let $\sigma$ be a Lorenz-like singularity. Then for $r>0$ small enough, for
every $\lambda\in(0,1),T_{0}>0$ there exists $\alpha_{2}>0$ such that for all
$\alpha<\alpha_{2}$, if $y\in B_{r}(\sigma)\cap{\rm
Cl}(D_{\alpha}^{cs}(\sigma))$ is a $(\lambda,T_{0})$-backward hyperbolic time
on its orbit, then $W^{u}(y)\pitchfork W^{cs}(\sigma)\neq\emptyset$.

###### Proof.

Since $y$ is a $(\lambda,T_{0})$-backward hyperbolic time, according to the
classic work of Liao [20], there is $\delta>0$ such that $W^{u}(y)$ has size
$\delta\|X(y)\|$, tangent to $C_{\beta}(E^{u})$ for some $\beta>0$ (in fact,
tangent to $C_{\beta}(E^{uu}_{y})$ where $E^{uu}_{y}$ is the unstable subspace
in $N_{y}$; however, we may assume that $E^{uu}_{y}$ and $E^{u}$ are almost
parallel as long as the flow orbit remains in the cone ${\rm
Cl}(D^{cs}_{\alpha}(\sigma))$). Let $\alpha_{2}$ be given by Lemma 3.5 for
such $\beta$ and $\delta$, we see that $W^{u}(y)\pitchfork
W^{cs}(\sigma)\neq\emptyset$.

∎

######  Lemma 3.12 (Backward hyperbolic times near $\sigma$).

Let $\sigma$ be a Lorenz-like singularity, and $\\{p_{n}\\}$ be a sequence of
periodic points with $p_{n}\to\sigma$. For $\lambda\in(0,1),T_{0}>0$ and for
$\alpha\in(0,\min\\{\alpha_{1},\alpha_{2}\\})$, assume that the set

$H_{n}=\\{t\in(-t^{-}_{n},t^{+}_{n}):\phi_{t}(p_{n})\mbox{ is a
}(\lambda,T_{0})\mbox{-backward hyperbolic time.}\\}$

has positive density: there exists $a>0$ such that for every $n$,

$\nu_{n}(\\{\phi_{t}(p_{n}):t\in H_{n}\\})>a>0,$

where $t^{\pm}_{n}$ and $\nu_{n}$ are taken according to (6). Then there
exists $N>0$ such that

$W^{u}({\rm Orb}(p_{n}))\pitchfork W^{cs}(\sigma)\neq\emptyset,\mbox{ for all
}n>N.$

###### Proof.

Figure 2. Transverse intersection between $W^{u}({\rm Orb}(p_{n}))$ and
$W^{cs}(\sigma)$

Let $\alpha<\min\\{\alpha_{1},\alpha_{2}\\}$ where $\alpha_{1}$ is given by
Lemma 3.9. We claim that $\\{\phi_{t}(x):t\in H_{n}\\}\cap
D^{cs}_{\alpha}(\sigma)\neq\emptyset$.

To this end, we parse the orbit segment $\phi_{(-t^{-}_{i},t^{+}_{i})}(p_{n})$
into three consecutive parts like in the proof of Lemma 3.4:

$(-t^{-}_{n},t^{+}_{n})=(-t^{-}_{n},-t^{s}_{n})\cup(-t^{s}_{n},t^{u}_{n})\cup(t^{u}_{n},t^{+}_{n}),$

such that:

  * •

$-t^{s}_{n}$ is the first time in $(-t^{-}_{n},t^{+}_{n})$ such that
$\phi_{-t^{s}_{n}}(p_{n})\notin D^{cs}_{\alpha}(\sigma)$;

  * •

$t^{u}_{n}$ is the first time in $(-t^{-}_{n},t^{+}_{n})$ such that
$\phi_{t^{u}_{n}}(p_{n})\in D^{u}_{\alpha}(\sigma)$.

In other words, the orbit segment in $(t^{s}_{n},t^{u}_{n})$ is ‘making the
turn’. Then according to Lemma 3.2, $t^{u}_{n}+t^{s}_{n}$ is uniformly bounded
by $T_{\alpha}$. Therefore, we can take $n$ large enough such that

$\nu_{n}(\\{\phi_{t}(p_{n}):t\in(t^{s}_{n},t^{u}_{n})\\})<\frac{a}{2}.$

On the other hand, Lemma 3.9 states that for every
$t\in(t^{u}_{n},t^{+}_{n}),$ $\phi_{t}(p_{n})$ cannot be a backward hyperbolic
time, since any sub-segment contained in $\phi_{(t^{u}_{n},t)}(p_{n})$ cannot
be backward contracting. As a result, we have
$H_{n}\cap(t^{u}_{n},t^{+}_{n})=\emptyset$. It then follows that

$\nu_{n}(\\{\phi_{t}(p_{n}):t\in
H_{n}\cap(-t^{-}_{n},-t^{s}_{n})\\})>\frac{a}{2}.$

In other words, there is a backward hyperbolic time $y_{n}=\phi_{t}(p_{n})$
contained in $D^{cs}_{\alpha}(\sigma)$.

It follows from Lemma 3.11 that $W^{u}(y_{n})\pitchfork
W^{cs}(\sigma)\neq\emptyset$. See Figure 2. We conclude the proof of the
lemma. ∎

######  Lemma 3.13 (Forward hyperbolic times near $\sigma$).

Let $\sigma$ be a Lorenz-like singularity. For $\beta>0$ small enough,
$\lambda\in(0,1),T_{0}>0$, there exists $\alpha_{3}>0$ with the following
property:
For every $\alpha<\alpha_{3}$, let $D$ be a $(\dim E^{u}+1)$-dimensional disk
that contains $\sigma$ and is tangent to $C_{\beta}(E^{cu})$, and let $z$ be a
$(\lambda,T_{0})$-forward hyperbolic time that is contained in
$D_{\alpha}^{c}(\sigma)\cap B_{r}(\sigma)$ for some $r>0$ small enough. Then
we have

$W^{s}(z)\pitchfork D\neq\emptyset,\mbox{ for all $n$ large enough}.$

###### Proof.

Similar to Lemma 3.11, there is $\delta>0$ such that $W^{s}(z)$ has size
$\delta\|X(z)\|$, tangent to $C_{\beta^{\prime}}(E^{ss})$ for some
$\beta^{\prime}>0$ small enough.

Following the idea of Lemma 3.5, we have

$\operatorname{diam}W^{s}(z)>C_{1}\delta d(z,\sigma)>CC^{\prime}\delta x^{c}.$

On the other hand, since $D$ is tangent to $C_{\beta}(E^{cu})$ and satisfies

$\dim D+\dim W^{s}(z)=\dim E^{u}+1+\dim E^{ss}=\dim M,$

there exists $C^{\prime\prime\prime}>0$ such that whenever
$\operatorname{diam}W^{s}(z)>C^{\prime\prime}x^{ss}$ we must have
$W^{s}(z)\pitchfork D\neq\emptyset$. See Figure 3. Since $x^{ss}<\alpha x^{c}$
inside the center cone $D_{\alpha}^{c}(\sigma)$, the choice of
$\alpha_{3}=C_{1}C^{\prime}\delta/C^{\prime\prime\prime}$ satisfies the
requirement of the lemma.

Figure 3. Transverse intersection between $W^{s}(z)$ and $D$

∎

Henceforth, we assume that $r>0$ is small enough such that all the previous
lemmas hold. Let $\lambda\in(0,1),T_{0}>0$ be given by Lemma 3.9, and
$\alpha<\min\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$.

##  4\. Proof of Theorem A

This section contains the proof of Theorem A. The proof is three-fold:

  1. (1)

every chain recurrent class $C$ with positive entropy must contain a periodic
orbit; note that the converse is also true: for $C^{1}$ generic flows, a non-
trivial chain recurrent class containing a periodic point $p$ must coincide
with the homoclinic class of $p$ ([26, Proposition 4.8]; see also [2]),
therefore has positive topological entropy;

  2. (2)

there exists a neighborhood $U$ of $C$, such that every periodic orbit in $U$
must be contained in $C$;

  3. (3)

$C$ is isolated.

Recall that ${\mathcal{R}}$ is the residual set in $\mathscr{X}_{*}^{1}(M)$
described in Section 2.4. Denote by
${\mathcal{R}}_{0}\subset\mathscr{X}^{1}(M)$ the residual set of Kupka-Smale
vector fields. The next lemma takes care of Step (3) above.

######  Lemma 4.1.

For a $C^{1}$ vector field $X\in{\mathcal{R}}\cap{\mathcal{R}}_{0}$, let $C$
be a chain recurrent class with the following property: there exists a
neighborhood $U$ of $C$ such that every periodic orbit in $U$ is contained in
$C$.333Recall that for $C^{1}$ generic diffeomorphisms, every non-trivial
chain recurrent class is approximated by periodic orbits. See [6]. Then $C$ is
isolated.

###### Proof.

The proof is standard. Let $C_{n}$ be a sequence of distinct chain recurrent
classes approaching $C$ in the Hausdorff topology. Without loss of generality
we may assume that $C_{n}\subset U$, and $C_{n}\neq C$ for all $n$. Since $X$
has only finitely many singularities, we may assume that $U$ is small enough
such that all the singularities in $U$ are indeed contained in $C$. It then
follows that $C_{n}\cap{\rm Sing}(X)=\emptyset$ for all $n$ large enough.

Since $X$ is a star vector field, the main result of [13] shows that $C_{n}$
is uniformly hyperbolic. In particular, there exists periodic orbit ${\rm
Orb}(p_{n})\subset C_{n}\subset U$. By assumption we must have ${\rm
Orb}(p_{n})\subset C$, so $C_{n}=C$ which is a contradiction.

∎

Next, we turn our attention to Step (1) and (2).

###  4.1. $C$ contains a periodic orbit

Step (1) of the proof is carried out in the following two propositions, which
are of independent interest.

######  Proposition 4.2.

There exists a residual set $\tilde{\mathcal{R}}\subset\mathscr{X}_{*}^{1}(M)$
such that for every $X\in\tilde{{\mathcal{R}}}$, let $C$ be a chain recurrent
class of $X$ with singularities of different indices, then $C$ contains a
periodic point and has positive topological entropy.

###### Proof.

Let $X\in{\mathcal{R}}\cap{\mathcal{R}}_{0}$ as before. Following the
discussion in Section 2.4, let $\sigma^{+}\in C$ be a Lorenz-like singularity,
and $\sigma^{-}\in C$ be reverse Lorenz-like. We have

${\rm Ind}(\sigma^{+})-1={\rm Ind}(\sigma^{-})={\rm Ind}_{C},$

where ${\rm Ind}_{C}$ is also the stable index of the periodic orbits
sufficiently close to $C$. We will construct a periodic point $p$ such that
Lemma 3.11 can be applied at $\sigma^{+}$ for $X$, and at $\sigma^{-}$ for
$-X$. This shows that $p\in C$, which also implies that $C=H(p)$ where $H(p)$
is the homoclinic class of $p$, therefore has positive topological entropy, by
[2].

Given a periodic orbit $\gamma$, we denote by $\Pi(\gamma)$ its primary
period. By [28, Lemma 2.1] which is originally due to Liao, there is
$\tilde{\lambda}\in(0,1)$, $T>0$ such that for every periodic orbit $\gamma$
of $X$ with periodic $\Pi(\gamma)$ longer than $T$, we have

$\prod_{i=0}^{[\Pi(\gamma)/T]-1}\|\psi_{T}|_{N^{s}(\phi_{iT}(x))}\|\leq\tilde{\lambda}^{\Pi(\gamma)},$

and a similar estimate holds on $N^{u}$. Here $N^{s}\oplus N^{u}$ is the
hyperbolic splitting on $N_{x}$ for the linear Poincaré flow $\psi_{t}$. By
the Pliss Lemma [27], for every $\lambda\in(\tilde{\lambda},1)$ there are
$(\lambda,T_{0})$-backward hyperbolic times for $N^{u}$ along the orbit of
$\gamma$. Moreover, such points have positive density along the orbit of
$\gamma$.

Fix such $\lambda$ and $r>0$ small enough. For each $n>0$, consider the
following property:

(P(n)): there is a hyperbolic periodic orbit $p_{n}$, such that (see Figure
4):

  * •

the time that ${\rm Orb}(p_{n})$ spends inside $B_{r}(\sigma^{+})$ is more
than $(1/2-1/n)\Pi(p_{n})$;

  * •

the time that ${\rm Orb}(p_{n})$ spends inside $B_{r}(\sigma^{-})$ is more
than $(1/2-1/n)\Pi(p_{n})$;

  * •

the time that ${\rm Orb}(p_{n})$ spends outside $B_{r}(\sigma^{-})\cup
B_{r}(\sigma^{-})$ is less than $\frac{1}{n}\Pi(p_{n})$.

Clearly this is an open property in $\mathscr{X}^{1}(M)$. On the other hand,
note that $W^{cs}(\sigma^{+})$ and $W^{cu}(\sigma^{-})$ must have transverse
intersection due to the Kupka-Smale theorem. Since $\sigma^{\pm}$ are in the
same chain recurrent class, using the connecting lemma we can create an
intersection between $W^{u}(\sigma^{+})$ and $W^{s}(\sigma^{-})$, which in
turn creates a loop between $\sigma^{+}$ and $\sigma^{-}$. Then standard
perturbation technique (see [28] for instance) will allow one to create
periodic orbits that satisfy the conditions above. Therefore the following
property is generic:

(P’): there exists periodic orbits $\\{p_{n}\\}_{n}$ that approach both
$\sigma^{\pm}$, such that Property P(n) holds for every $n$.

Figure 4. The periodic points $\\{p_{n}\\}$ satisfying Property (P’)

As a result, passing to the generic subset ${\mathcal{R}}_{1}$ where property
(P’) holds, we may assume that $X$ itself has periodic orbits $\\{p_{n}\\}$
satisfying Property (P’).

Below we will show that there exists $(\lambda,T_{0})$-backward hyperbolic
times contained in ${\rm Orb}(p_{n})\cap B_{r}(\sigma^{+})\cap
D_{\alpha}^{cs}(\sigma^{+})$, for $n$ large enough. This allows us to apply
Lemma 3.11 to show that $W^{u}({\rm Orb}(p_{n}))$ has transverse intersection
with $W^{cs}(\sigma^{+})$. The same argument applied to the flow $-X$ shows
the transverse intersection between $W^{s}({\rm Orb}(p_{n}))$ and
$W^{cu}(\sigma^{-})$, thus $p_{n}$ is contained in $C$ for $n$ large enough.

For convenience, we assume that $p_{n}\to\sigma^{+}$ such that $p_{n}$ lies on
the boundary of $D^{cs}_{\alpha}(\sigma^{+})$, where
$\alpha\in(0,\min\\{\alpha_{1},\alpha_{2}\\})$ as specified at the end of the
previous section. Denote by

$p_{n}^{\sigma^{+}}=\phi_{t_{1}^{n}}(p_{n})$

where $t_{1}^{n}<0$ is the largest real number such that
$p_{n}^{\sigma^{+}}\in\partial B_{r}(\sigma^{+})$, and

$p_{n}^{\sigma^{-}}=\phi_{t_{2}^{n}}(p_{n})$

where $t^{n}_{2}=\sup\\{t<t_{1}^{n}:\phi_{t}(p_{n})\in B_{r}(\sigma^{-})\\}.$
See the Figure above. Since the orbit segment
$\phi_{[t_{2}^{n},t_{1}^{n}]}(p_{n})\subset M\setminus(B_{r}(\sigma^{+})\cup
B_{r}(\sigma^{-}))$, Property (P’) dictates that
$t^{n}_{1}-t^{n}_{2}<\frac{1}{n}\Pi(p_{n})$.

On the other hand, the Pliss Lemma [27] shows that the set of
$(\lambda,T_{0})$-backward hyperbolic times on the orbit of $p_{n}$ have
density $a>0$. Thus for $n$ large enough, there must be hyperbolic times on
the subsegment ${\rm Orb}(p_{n})\cap B_{r}(\sigma^{\pm})$. We consider the
following cases:

  1. (1)

If there exists a backward hyperbolic time $p_{n}^{\prime}$ contained in the
orbit segment ${\rm Orb}(p_{n})\cap B_{r}(\sigma^{+})$, then by Lemma 3.9
$p_{n}^{\prime}\notin D^{u}_{\alpha}(\sigma^{+})$; on the other hand, the
transition time from $D^{cs}_{\alpha}(\sigma^{+})$ to
$D^{u}_{\alpha}(\sigma^{+})$ is bounded (Lemma 3.2); this shows that
$p_{n}^{\prime}\in D_{\alpha}^{cs}(\sigma^{+})$;

  2. (2)

if there exists a backward hyperbolic times $p^{\prime}_{n}\in{\rm
Orb}(p_{n})\cap B_{r}(\sigma^{-})$, then Lemma 3.8 (applied to $-X$) shows
that the orbit segment from $p^{\prime}_{n}$ to $p^{\sigma^{-}}_{n}$ is
backward contracted by the scaled linear Poincaré flow. By Lemma 2.6,
$p_{n}^{\sigma^{-}}$ itself must be a backward hyperbolic time.

Next, by Lemma 3.9, the orbit segment from $p^{\sigma^{+}}_{n}$ to $p_{n}$ is
backward contracting, and the orbit segment from $p_{n}^{\sigma^{-}}$ to
$p^{\sigma^{+}}_{n}$ has very small length comparing to the former. As a
result, $p_{n}$ is a backward hyperbolic time.

It then follows from both cases, that there exists a backward hyperbolic time
inside ${\rm Cl}(D_{\alpha}^{cs}(\sigma^{+}))$. As a result of Lemma 3.11,
$W^{u}({\rm Orb}(p_{n}))$ must intersect transversally with
$W^{cs}(\sigma^{+})$.

The same argument applied to the flow $-X$ shows that the stable manifold of
${\rm Orb}(p_{n})$ intersects transversally with the unstable manifold of
$\sigma^{-}$. We conclude that ${\rm Orb}(p_{n})$ is contained in the chain
recurrent class of $\sigma^{\pm}$, finishing the proof of this proposition.

∎

######  Proposition 4.3.

There exists a residual set $\tilde{\mathcal{R}}\subset\mathscr{X}_{*}^{1}(M)$
such that for every $X\in\tilde{{\mathcal{R}}}$, if $C$ is a chain recurrent
class of $X$ with singularity and positive topological entropy then $C$
contains a periodic point $p$.

###### Proof.

In view of the discussion in Section 2.4, we consider the following two cases
for $X\in{\mathcal{R}}\cap{\mathcal{R}}_{0}$:

Case 1. All singularities in $C$ have the same index. Then they must be of the
same type: either they are all Lorenz-like or all reverse Lorenz-like. By [28,
Theorem 3.7], $C$ is sectional hyperbolic for $X$ or $-X$. We may assume that
$C$ is sectional hyperbolic for $X$. When $h_{top}(\phi_{t}|_{C})>0$, [26,
Theorem B] guarantees that there are periodic orbits $p$ contained in $C$.

Case 2. $C$ contains singularities with different indices. Then the
proposition follows from Proposition 4.2. ∎

This finishes the proof of Step (1).

###  4.2. All periodic orbits close to $C$ must be contained in $C$

Now we turn our attention to Step (2). The main result of this subsection is
Lemma 4.6 which shows that whenever a periodic orbit ${\rm Orb}(p)$ gets
sufficiently close to a chain recurrent class containing a periodic orbit, we
must have $W^{u}(p)\pitchfork W^{s}(x)$ for some $x\in C$.

The proof of this lemma requires a careful analysis on how the backward
hyperbolic times along the orbit of $p$ approach points in $C$. There are
essentially three cases:

  1. (1)

backward hyperbolic times near a regular point; this is the easy case;

  2. (2)

backward hyperbolic times near a Lorenz-like singularity $\sigma$; this has
been taken careful of by Lemma 3.12;

  3. (3)

backward hyperbolic times near a reverse Lorenz-like singularity $\sigma$; in
this case we consider $-X$, under which we have a sequence of forward
hyperbolic times near a Lorenz-like singularity $\sigma$; this is taken care
of by the following lemma, as well as Lemma 3.13.

######  Lemma 4.4.

There exists a residual set $\tilde{\mathcal{R}}\subset\mathscr{X}_{*}^{1}(M)$
such that for every $X\in\tilde{{\mathcal{R}}}$, let $C$ be a chain recurrent
class of $X$ that contains some Lorenz-like singularity $\sigma$ and a
hyperbolic periodic point $q$. Let $\\{p_{n}\\}$ be a sequence of periodic
points with $\lim_{H}{\rm Orb}(p_{n})\subset C$. Furthermore, assume that
there exists $x_{n}\in{\rm Orb}(p_{n})$ that are $(\lambda,T_{0})$-forward
hyperbolic times for $\lambda\in(0,1),T_{0}>0$ independent of $n$, with
$x_{n}\to\sigma$. Then for $n$ large enough,

$W^{s}({\rm Orb}(p_{n}))\pitchfork W^{u}({\rm Orb}(q))\neq\emptyset.$

###### Proof.

Recall that $E^{ss}_{\sigma}$ is the subspace spanned by the (generalized)
eigenspaces corresponding to the eigenvalues
$\lambda_{1},\ldots,\lambda_{s-1}$; furthermore we have $E^{ss}_{\sigma}\oplus
E_{\sigma}^{c}=E_{\sigma}^{cs}$.

Consider the strong stable manifold of $\sigma$, $W^{ss}(\sigma)$ that is
tangent to $E^{ss}_{\sigma}$ at $\sigma$. It exists because of the dominated
splitting $T_{\sigma}M=E^{ss}_{\sigma}\oplus E_{\sigma}^{cu}$ with
$E_{\sigma}^{cu}=E^{c}_{\sigma}\oplus E^{u}_{\sigma}$. Moreover,
$W^{ss}(\sigma)$ locally divides the stable manifold $W^{s}(\sigma)$ into two
branches, which we denote by $W^{s,\pm}(\sigma)$.

We may assume that ${p_{n}}$ themselves are forward hyperbolic times with
$p_{n}\to\sigma$. Fix some $r>0$ small enough, we consider $t_{n}<0$ the
largest real number such that $\phi_{t_{n}}(p_{n})\in\partial B_{r}(\sigma)$.
In particular, $z_{n}:=\phi_{t_{n}}(p_{n})\to z\in W^{s,+}(\sigma)\cap
C\cap\partial B_{r}(\sigma)$. Since the orbit of $p_{n}$ can only approach
$\sigma$ along the center direction $E^{c}_{\sigma}$, we must have
$z_{n}\in{\rm Cl}(D_{\beta}^{c}(\sigma))$, for some $\beta=\beta(r)>0$. By
Lemma 3.8, the orbit segment $\phi_{[t_{n},0]}(p_{n})$ is forward contracting.
It follows from Lemma 2.6 that $z_{n}$ itself is a $(\lambda,T_{0})$-forward
hyperbolic time, and possesses stable manifold with size proportional to the
flow speed at $z_{n}$.

On the other hand, we have $W^{s,+}(\sigma)\cap W^{u}({\rm
Orb}(q))\neq\emptyset$, a courtesy of the connecting lemma [2] (see also [26,
Proposition 4.9]). Let $a$ be a point of intersection between
$W^{s,+}(\sigma)$ and $W^{u}({\rm Orb}(q))$, and $D$ a disk in $W^{u}({\rm
Orb}(q))$ with $a\in D\subset W^{u}({\rm Orb}(q))$. By the inclination lemma,
$\phi_{t}(D)$ approximates $W^{u}(\sigma)$ as $t\to\infty$.

Note that $\tilde{D}:=\\{\phi_{t}(D):t>0\\}$ is tangent to the cone
$C_{\beta}(E^{cu})$ and has dimension $\dim E^{u}_{\sigma}+1$. See figure 3.
Shrinking $r$ such that $\beta(r)<\alpha_{3}$ and applying Lemma 3.13,444
$\tilde{D}$ defined in this way dos not contain $\sigma$. However it can be
extended to a disk $\overline{D}$ that contains $\sigma$ in its interior. Note
that $\overline{D}$ and $\tilde{D}$ coincide within the upper half of the cone
$D^{c}_{\alpha}(\sigma)$, which contains the point of the transverse
intersection with $W^{s}(z_{n})$. we obtain
$W^{s}(z_{n})\pitchfork\\{\phi_{t}(D):t>0\\}\neq\emptyset$ for all $n$ large
enough, with which we conclude the proof.

∎

######  Remark 4.5.

The proof of the lemma is reminiscent of [26, Corollary E] with two major
differences. Firstly, in [26], $C$ is Lyapunov stable. This guarantees not
only the existence of a periodic orbit $q\in C$, but the disk $D$ is contained
in the class $C$, which immediately allows one to conclude that $p_{n}\in C$
for $n$ large enough. Without Lyapunov stability, we only obtain the chain
attainability from $C$ to $p_{n}$. Secondly, $C$ is sectional hyperbolic in
[26]. Therefore nearby periodic orbits have dominated splitting $E^{s}\oplus
F^{cu}$ and, consequently, have stable manifolds with uniform size.

Now we are ready to state the main lemma of this subsection, which allows us
to establish the chain attainability from nearby periodic orbits to the class
$C$. Note that the lemma does not impose any condition on the type of
singularities contained in $C$, therefore can be applied to $-X$.

######  Lemma 4.6.

There exists a residual set $\tilde{\mathcal{R}}\subset\mathscr{X}_{*}^{1}(M)$
such that for every $X\in\tilde{{\mathcal{R}}}$, let $C$ be a chain recurrent
class of $X$ with singularities and a hyperbolic periodic point. Then there
exists a neighborhood $U$ of $C$ such that every periodic orbit ${\rm
Orb}(p)\subset U$ satisfies

$W^{u}(p)\pitchfork W^{s}(x)\neq\emptyset$

for some $x\in C$.

###### Proof.

We prove by contradiction. Assume that there is a sequence of periodic orbits
${\rm Orb}(p_{n})\subset U_{n}$ with $\cap_{n}U_{n}=C$, such that
$W^{u}(p)\pitchfork W^{s}(x)=\emptyset$ for every $x\in C$. It is easy to see
that the period of $p_{n}$ must tend to infinity; otherwise, we could take a
subsequence of $\\{p_{n}\\}$ that converges to a periodic orbit $p_{0}\in C$;
by the star assumption, $p_{0}$ is hyperbolic, and must be homoclinically
related with $p_{n}$ for $n$ large enough, a contradiction.

By [28, Theorem 5.7], the index of $p_{n}$ coincides with ${\rm Ind}_{C}$ for
all $n$ large enough. By [28, Lemma 2.1], there exists
$\lambda\in(0,1),T_{0}>0$ such that ${\rm Orb}(p_{n})$ contains
$(\lambda,T_{0})$-backward hyperbolic times $x_{n}$. Moreover, the collection
of such points $\Lambda_{n}=\\{x_{n}\in{\rm Orb}(p_{n}):x_{n}\mbox{ is a
backward hyperbolic time}\\}$ have positive density (independent of $n$) in
${\rm Orb}(p_{n})$ with respect to the empirical measure on ${\rm
Orb}(p_{n})$, thanks to the Pliss lemma [27].

Taking subsequence if necessary, we write $\tilde{C}\subset C$ for the
Hausdorff limit of ${\rm Orb}(p_{n})$, and
$\Lambda=\limsup_{n\to\infty}\Lambda_{n}\subset\tilde{C}$ (note that it is not
invariant). We may also assume that the empirical measure $\mu_{n}$ on ${\rm
Orb}(p_{n})$ converges to an invariant measure $\mu$ supported on $\tilde{C}$.
The backward hyperbolic times having uniform positive density implies that
$\mu(\Lambda)>0$.

We consider the following two cases:

Case 1. There is an ergodic component of $\mu$, denoted by $\mu_{1}$, which
satisfies $\mu_{1}(\Lambda)>0$ and $\mu_{1}({\rm Sing})=0$.

Then $\mu_{1}$ must be a non-trivial hyperbolic measure, thanks to [28,
Theorem 5.6]. The same argument used in Lemma 2.8 (f) shows that for $n$ large
enough, $W^{u}({\rm Orb}(p_{n}))$ has transverse intersection with the stable
manifold of some point in $\operatorname{supp}\mu_{1}$. Roughly speaking,
every regular point in $\Lambda\cap\,\operatorname{supp}\mu_{1}$ have stable
manifold (with uniform size if we choose a subset of $\Lambda$ away from all
singularities; note that such subset still has positive $\mu_{1}$ measure),
which must intersect transversally with the unstable manifold of the backward
hyperbolic times $x_{n}\in\Lambda_{n}$ (recall that such points have unstable
manifold up to the flow speed; if we take them uniformly away from all
singularities, then their unstable manifolds also have uniform size), a
contradiction.

Case 2. Every ergodic component $\mu_{1}$ of $\mu$ with $\mu_{1}(\Lambda)>0$
is supported on some singularities $\\{\sigma\\}$.

As per our discussion in Section 2.4, singularities in $C$ must be either
Lorenz-like or reverse Lorenz-like. We further consider two subcases:

Subcase 1. One of those $\sigma$ is reverse Lorenz-like555Note that this case
does not happen when $C$ is sectional hyperbolic..

In this case we consider the vector field $-X$. Note that backward hyperbolic
times for $X$ are forward hyperbolic times for $-X$. Therefore we have a
sequence of forward hyperbolic times on the orbit of $p_{n}$, approaching a
Lorenz-like singularity $\sigma$. We are in a position to apply Lemma 4.4,
which shows that

$W^{s,-X}({\rm Orb}(p_{n}))\pitchfork W^{u,-X}(q)\neq\emptyset$

for a hyperbolic periodic point $q\in C$. Reverse back to $X$, we see that
$W^{u}({\rm Orb}(p_{n}))\pitchfork W^{s}(x)\neq\emptyset$, which
contradictions with our assumption on $C$.

Subcase 2. One of those $\sigma$ is Lorenz-like.

In this case we have backward hyperbolic times in a neighborhood of a Lorenz-
like singularity with positive density. By Lemma 3.12, $W^{u}({\rm
Orb}(p_{n}))$ intersect transversely with $W^{cs}(\sigma^{+})$, which is a
contradiction.

The proof is now complete. ∎

In the proof of Lemma 4.6, the assumption that $C$ contains a periodic orbit
is only used in Case 2, Subcase 1. Since this case is impossible when $C$ is
sectional hyperbolic, we obtain the following proposition which has intrinsic
interest.

######  Proposition 4.7.

Let $C$ be a sectional hyperbolic chain recurrent class for $C^{1}$ generic
star flow $X$. Assume that $C$ contains a singularity $\sigma$. Then there
exists a neighborhood $U$ of $C$, such that $C$ is chain attainable from all
periodic orbits in $U$.

###  4.3. Proof of Theorem A

Now we are ready to prove Theorem A. By Proposition 4.3, every singular chain
recurrent class with positive entropy contains a periodic point. We then apply
Lemma 4.6 to $C$ for $X$ and $-X$. This shows that there exists a neighborhood
$U$ of $C$, such that all periodic orbits in $U$ are homoclinically related
with $C$, therefore, are contained in $C$. Finally we use Lemma 4.1 to
conclude that $C$ is isolated.

The only remaining case is chain recurrent classes with zero entropy. By
Proposition 4.2, singularities in such classes must have the same stable
index. Then [28, Theorem 3.7] shows that $C$ is sectional hyperbolic for $X$
or $-X$. $C$ cannot contain any periodic orbit; otherwise it must coincide
with the homoclinic class of said periodic orbit, resulting in positive
topological entropy.

To conclude the proof, we need to show that the only ergodic invariant
measures supported on $C$ are point masses of singularities.

For this purpose, let $\mu$ be an ergodic invariant measure with
$\operatorname{supp}\mu\subset C$. We prove by contradiction and assume that
$\mu\neq\delta_{\sigma}$ for any $\sigma\in{\rm Sing}(X)$. By [28], $\mu$ is a
non-trivial hyperbolic measure whose support contains regular orbits of $X$.
By Lemma 2.7 and 2.8 (f), there exists a periodic orbit in $C$, a
contradiction.

###  4.4. Proof of the corollary

We conclude this paper with the proof of the Corollary.

###### Proof of Corollary B.

In [20] it is proven that $C^{1}$ generic star flows have only finitely many
periodic sinks. As a result, if $X$ has infinitely many distinct Lyapunov
stable chain recurrent classes $\\{C_{n}\\}_{n=1}^{\infty}$, then we may
assume that $C_{n}$’s are non-trivial and approach, under Hausdorff topology,
to a chain recurrent class $C$. Note that $C$ cannot be trivial since trivial
chain recurrent classes, i.e., periodic orbits and singularities, are all
hyperbolic and isolated due to the star assumption. Therefore $C$ is non-
trivial and sectional hyperbolic with zero topological entropy due to Theorem
A.

Denote by $\lambda_{C}>0$ the sectional volume expanding rate on $F^{cu}$,
then by the continuity of the dominated splitting (see [4, Appendix B] and
[1]), nearby classes $C_{n}$ must be sectional hyperbolic and have sectional
volume expanding rate $\lambda_{C_{n}}>\lambda_{C}/2$. Since $C_{n}$ are
Lyapunov stable, we are in a position to apply [26, Theorem C] on $C_{n}$, and
see that $h_{top}(X|_{C_{n}})>\lambda_{C}/2$. By the variational principle, we
can take $\mu_{n}$ an ergodic invariant measure supported on $C_{n}$, such
that $h_{\mu_{n}}(X)\geq\lambda_{C}/2$.

On the other hand, [26, Theorem A] shows that $X$ is robustly entropy
expansive in s small neighborhood of $C$; in particular, this together with
[5] shows that the metric entropy must be upper semi-continuous in a small
neighborhood of $C$. Let $\mu$ be any limit point of $\mu_{n}$ in weak-*
topology, then we see that $\operatorname{supp}\mu\subset C$ and
$h_{\mu}(X)\geq\lambda_{C}/2$. It then follows from the variational principle
that $h_{top}(X|_{C})\geq\lambda_{C}/2$, a contradiction.

∎

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