# Cherry flow: physical measures and perturbation theory

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

In this article we consider Cherry flows on torus which have two
singularities: a source and a saddle, and no periodic orbits. We show that
every Cherry flow admits a unique physical measure, whose basin has full
volume. This proves a conjecture given by R. Saghin and E. Vargas in [21]. We
also show that the perturbation of Cherry flow depends on the divergence at
the saddle: when the divergence is negative, this flow admits a neighborhood,
such that any flow in this neighborhood belongs to the following three cases:
(a) has a saddle connection; (b) a Cherry flow; (c) a Morse-Smale flow whose
non-wandering set consists two singularities and one periodic sink. In
contrary, when the divergence is non-negative, this flow can be approximated
by non-hyperbolic flow with arbitrarily larger number of periodic sinks.

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

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
  3. 3 Proof of Lemma 2.15
  4. 4 Proof of Theorem B
  5. 5 Proof of Theorems C and D
  6. A Proof of Lemma 2.7

##  1\. Introduction

In this article we consider Cherry flows on torus which have two
singularities: a source and a saddle, and no periodic orbits and saddle
connections. Such an example was given first by Cherry in [4] (see also [18]),
and the classification of Cherry flows was treated in [15]. In this article we
are interested in the physical measures and perturbation theory for Cherry
flows.

An invariant probability $\mu$ of a flow $\phi$ is a physical measure if its
basin

$B(\mu)=\\{x\in\mathbb{T}^{2}:\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}f(\phi_{s}(x))ds=\int
fd\mu\\\ \;\;\text{for every continuous function}\;\;f\\}$

has positive volume.

We prove the following conjecture given by Saghin and Vargas ( [21]):

######  Theorem A.

Every $\operatorname{C}^{\infty}$ Cherry flow admits a unique physical
measure, whose basin has full volume.

Throughout the paper we consider a Cherry flow $\phi^{X}$ on torus
$\mathbb{T}^{2}$ generated by a vector field $X$, with a source
$\tilde{\sigma}^{X}$ and a quasi-minimal attractor $\Lambda^{X}$, where
$\Lambda^{X}$ is called quasi-minimal if it is transitive, contains a saddle
$\sigma^{X}$, and has neither periodic orbits nor saddle connections. We
denote by ${\mathcal{D}}_{X}(\sigma^{X})$ the divergence at $\sigma^{X}$. It
was observed by [16, 17, 21] that the behavior of physical measures for Cherry
flows depends on the divergence at the saddle. And the flows in general were
assumed to be $\operatorname{C}^{\infty}$, this is because the
$\operatorname{C}^{\infty}$ regularity implies that this flow is
$\operatorname{C}^{1+\text{bounded variation}}$ linearizable at a neighborhood
of $\sigma^{X}$ (see [15][Appendix]).

The existence of physical measures for $\operatorname{C}^{\infty}$ Cherry
flows with non-positive divergence at the saddle was proved in [21][Theorem
1.1]. Theorem A is a corollary of the following result about the case of
positive divergence at the saddle, where the regularity assumption on the flow
is removed.

######  Theorem B.

Suppose $\phi^{X}$ is $\operatorname{C}^{1}$ and
${\mathcal{D}}_{X}(\sigma^{X})>0$. Then $\phi^{X}$ admits a non-trivial
physical measure supported on $\Lambda^{X}$, this measure is ergodic, non-
hyperbolic, and whose basin has full volume.

An ergodic measure is non-trivial means that it is not supported on critical
elements, i.e., on singularities or on periodic orbits. Theorem B generalizes
the results of [21, 20]. Different to most of the dynamical systems with
physical measures, where the physical measures in general are either
hyperbolic (see [3, 1]) or atomic (see [9, 22]), the physical measures in
Theorem B are non-hyperbolic and non-trivial. An interesting and important
fact is that, the proof of Theorem B depends on the non-hyperbolicity of this
measure.

In [16, 17], a different kind of Cherry flows was considered, where
$\tilde{\sigma}_{0}$ is a sink. This sink supports automatically a physical
measure, which is not the case we are interested.

We also consider the perturbation of Cherry flow. It turns out that the
perturbation depends also on the divergence at the singularity. For a flow $Y$
sufficiently close to $X$, we denote by $\sigma^{Y}$ the analytic continuation
of the hyperbolic saddle $\sigma^{X}$. A flow is called star flow if the
critical elements of any $\operatorname{C}^{1}$ small perturbation are all
hyperbolic.

######  Theorem C.

Suppose $\phi^{X}$ is $\operatorname{C}^{\infty}$ and
${\mathcal{D}}_{X}(\sigma^{X})<0$. Then $\phi^{X}$ is a star flow. Moreover,
there is a $\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of $\phi^{X}$,
such that every flow $Y\in{\mathcal{U}}$ belongs to one of the following three
cases:

  * •

has a saddle connection;

  * •

or is a Morse-Smale flow, whose non-wandering set consists two singularities
and one periodic sink,

  * •

or is still a Cherry flow, which admits a unique physical measure
$\delta_{\sigma^{Y}}$ with basin
$B(\delta_{\sigma^{Y}})=\mathbb{T}^{2}\setminus\tilde{\sigma}^{Y}$.

######  Theorem D.

Suppose $\phi^{X}$ is $\operatorname{C}^{1}$ and
${\mathcal{D}}_{X}(\sigma^{X})\geq 0$. Then $\phi^{X}$ is
$\operatorname{C}^{1}$ approximated by non-hyperbolic flows with arbitrarily
large number of periodic sinks.

The techniques of Liao in [13] is important for us to remove the regularity
assumption on the flows, which is mainly used in the proof of Lemma 2.15. This
author would like to thank Shaobo Gan for his explanation on Liao Theory.

##  2\. Preliminary

###  2.1. Ergodic measures for Cherry flow

One can always take a circle ${\mathcal{S}}_{0}$ which does not bound a disk
and is everywhere transverse to $X$ e.g., see [15] [Proposition 7.1]. The
inverse of the Cherry flow $\phi^{X}$ is a suspension of a continuous circle
map $g^{X}:{\mathcal{S}}_{0}\to{\mathcal{S}}_{0}$, where $g^{X}$ is a monotone
map and constant on an interval. Such circle maps were well studied in [15, 8,
7, 19]. Note that $g^{X}$ has no periodic point, which implies that $g^{X}$
has irrational rotation number $\rho(g^{X})$ and is semi-conjugate to the
rotation $R_{\rho(g^{X})}:x\to x+\rho(g^{X})\mod\;1$. The semi-conjugacy
$\pi^{X}$ is continuous, monotone, has degree one and maps orbits of $g^{X}$
to orbits of $R_{\rho(g^{X})}$. Because the $\pi_{X}$ pre-image of every point
in the circle is either a singular point or a non-trivial connected interval,
and by the fact that a circle contains at most countable many disjoint open
intervals, there are at most countable many points have non-trivial $\pi^{X}$
pre-images. In particular, this set has vanishing Lebesgue measure. Then
$\nu^{X}=(\pi^{X})^{*}\operatorname{vol}|_{S^{1}}$ is well defined, and is the
unique ergodic measure of $g^{X}$. For every $\theta\in{\mathcal{S}}^{1}_{0}$,
denote by $\tau_{X}(\theta)$ the first return time of $\theta$ for $\phi^{X}$.
It is well known that $\int\tau_{X}(\theta)d\nu^{X}(\theta)<\infty$ if and
only if

$\mu^{X}=\int_{{\mathcal{S}}^{1}}\int_{t\in[0,\tau_{X}(\theta))}\delta_{\phi^{X}_{t}(\theta)}d\nu^{X}(\theta)$

is an ergodic measure of $\phi^{X}$. As a summary, we have that:

######  Proposition 2.1.

$\phi^{X}$ has three ergodic measures $\delta_{\sigma^{X}}$,
$\delta_{\tilde{\sigma}^{X}}$ and $\mu^{X}$ if and only if
$\int\tau_{X}(\theta)d\nu^{X}(\theta)<\infty$.

######  Theorem 2.2.

[ [21][Theorems 1.1, 1.2]]

  * (a)

Suppose $\phi^{X}$ is $\operatorname{C}^{\infty}$ and
${\mathcal{D}}_{X}(\sigma^{X})\leq 0$. Then $\delta_{\tilde{\sigma}^{X}}$ and
$\delta_{\sigma^{X}}$ are the only two ergodic measures for $\phi^{X}$.
Consequently, $\delta_{\sigma^{X}}$ is a physical measure with basin equal to
$\mathbb{T}^{2}\setminus\tilde{\sigma}^{X}$.

  * (b)

Suppose $\phi^{X}$ is $\operatorname{C}^{1}$ and
${\mathcal{D}}_{X}(\sigma^{X})>0$. Then there exist exactly three ergodic
measures for $\phi^{X}$: $\delta_{\tilde{\sigma}^{X}}$, $\delta_{\sigma^{X}}$
and a third invariant probability measure $\mu^{X}$ supported on the quasi-
minimal set $\Lambda^{X}$.

###  2.2. Ergodic theory for flows

Throughout this subsection, we consider $\phi^{Y}$ a flow over surface $M$
which is generated by vector field $Y$, and it admits a non-trivial ergodic
measure $\mu$. Denote by ${\rm Sing}(Y)$ the set of singularities.

The linear Poincaré flow $\psi_{t}$ is defined as following. Denote the normal
bundle of $\phi^{Y}$ over $M$ by

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

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

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

Denote the orthogonal projection of $T_{x}M$ to ${\mathcal{N}}_{x}$ by
$\pi_{x}$. Write the tangent flow by $\Phi_{t}=d\phi^{Y}_{t}:TM\to TM$. Given
$v\in{\mathcal{N}}_{x},x\in\Lambda\setminus\sigma$, $\psi_{t}(v)$ is the
orthogonal projection of $\Phi_{t}(v)$ on ${\mathcal{N}}_{\phi^{Y}_{t}(x)}$
along the flow direction, i.e.,

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

where $<.,.>$ is the inner product on $T_{x}M$ given by the Riemannian metric.

####  2.2.1. Oseledets Theorem for flows

The following is the flow version of Oseledets theorem.

######  Proposition 2.3.

There exists a real number $\lambda$ such that for $\mu$ almost every $x$:

$\underset{t\to\pm\infty}{\lim}\frac{1}{t}\log\|\psi_{t}(v)\|=\lambda\;\;\text{for
every non-zero}\;\;v\in{\mathcal{N}}_{x}\setminus\\{0\\}.$

When the time $t_{0}>0$ is fixed, $\phi^{Y}_{t_{0}}$ is a diffeomorphism and
$\mu$ is still an invariant measure of this diffeomorphism. Suppose
$\hat{\mu}$ is an ergodic decomposition of $\mu$ for $\phi^{Y}_{t_{0}}$, then

$\mu=\int_{t\in[0,t_{0})}d(\phi^{Y}_{t})_{*}\hat{\mu}.$

We may always choose $t_{0}$ such that $\mu$ is ergodic for
$\phi^{Y}_{t_{0}}$. Taking such a time $t_{0}$ and denote
$f=\phi^{Y}_{t_{0}}$. Now let us state the following relation between Lyapunov
exponent of $\mu$ for $\phi^{Y}_{t}$ and for $f$, whose proof is simple and we
will not provide.

######  Theorem 2.4.

The Lyapunov exponents of $\mu$ for $f$ coincide to
$\\{t_{0}\lambda\\}\cup\\{0\\}$, and for $\mu$ almost every $x\in
M\setminus{\rm Sing}(Y)$, the flow direction $<Y(x)>$ is contained in the
subbundle of the Oseledets splitting of $f$ corresponding to zero exponent.
Moreover, we have $\lim\frac{1}{n}\log\det(Df^{n}(x))=t_{0}\lambda$.

######  Remark 2.5.

The above theorem implies that, for $\mu$ almost every $x$,

$\lim_{t\to\infty}\frac{1}{t}\det\Phi_{t}|_{T_{x}M}=\lambda.$

######  Definition 2.6.

A non-trivial ergodic measure $\mu$ for flow $\phi^{Y}$ is a hyperbolic
measure if its exponent is non-vanishing.

####  2.2.2. Divergence

We need the following version of Liouville Theorem.

######  Lemma 2.7.

For any $x\in M$ and $t>0$,

$\operatorname{ln}\det\Phi_{t}|_{T_{x}M}=\int_{0}^{t}div_{Y}(\phi^{Y}_{s}(x))ds.$

The definition of divergence depends on the Riemmanian metric. Because in this
paper we only consider two dimensional torus, for simplicity, in Appendix we
provide a short proof of the above lemma with the assumption that
$M=\mathbb{T}^{2}$ and the divergence is defined by the flat metric on torus.
More precisely, every small open set of $\mathbb{T}^{2}$ can be looked as a
subset of $\mathbb{R}^{2}$, choosing such a local coordinate $(x,y)$, in this
coordinate we may write $Y=(Y_{1},Y_{2})$ and
$\phi^{Y}_{t}=(f_{t}(x,y),g_{t}(x,y))$. Then for any $x\in\mathbb{T}^{2}$, the
divergence of $Y$ at $x$ is defined by $div_{Y}(x)=\frac{\partial
Y_{1}}{\partial x}+\frac{\partial Y_{2}}{\partial y}$.

######  Definition 2.8.

Let ${\rm Orb}(p)$ be a periodic orbit of $\phi^{Y}$ with period
$\tau_{Y}(p)$:

  * •

the divergence of ${\rm Orb}(p)$ is defined by

${\mathcal{D}}_{Y}({\rm
Orb}(p))=\frac{1}{\tau_{Y}(p)}\int_{0}^{\tau_{Y}(p)}div_{Y}(\phi^{Y}_{s}(p))ds;$

  * •

the divergence of $\mu$ is defined by ${\mathcal{D}}_{Y}(\mu)=\int
div_{Y}(x)d\mu(x)$.

######  Remark 2.9.

For an atomic measure $\delta_{\sigma}$ supported on a singularity
$\sigma\in{\rm Sing}(Y)$, ${\mathcal{D}}_{Y}(\delta_{\sigma})$ coincides to
the classical definition of divergence at $\sigma$,
${\mathcal{D}}_{Y}(\sigma)$. And when $\mu$ is supported on a periodic orbit
${\rm Orb}(p)$, the above two definitions coincide, and by Lemma 2.7, which
equal to
$\operatorname{ln}\det\Phi_{\tau_{Y}(p)}|_{T_{p}M}=\operatorname{ln}\|\psi_{\tau_{Y}(p)}|_{{\mathcal{N}}_{p}}\|$.

As a corollary of Remark 2.5 and Lemma 2.7, by Birkhoff Ergodic Theorem, we
obtain that:

######  Corollary 2.10.

${\mathcal{D}}_{Y}(\mu)$ coincides to the Lyapunov exponent of $\mu$ for the
flow $\phi^{Y}$.

The following lemma implies [21][Theorem 1.2]:

######  Lemma 2.11.

There exists an ergodic measure $\mu$ of $Y$ such that
${\mathcal{D}}_{Y}(\mu)\leq 0$.

###### Proof.

Suppose by contradiction that the divergence of every ergodic measure of
$\phi^{Y}$ is positive. Denote by ${\mathcal{M}}_{Y}$ the space of invariant
measures of $\phi^{Y}$. Then by Ergodic Decomposition Theorem, there is $a>0$
such that ${\mathcal{D}}_{Y}|_{{\mathcal{M}}_{Y}}>2a>0$. Take ${\mathcal{V}}$
a small neighborhood of ${\mathcal{M}}_{Y}$ in the probability space over $M$,
such that for any $\tilde{\mu}\in{\mathcal{V}}$,

$\int div_{Y}(x)d\tilde{\mu}(x)>a.$

Note that the measures contained in ${\mathcal{V}}$ are not necessary to be
invariant. It is well known that for any $x\in M$, there is $t_{x}>0$ such
that $\frac{1}{t}\int_{0}^{t}\delta_{\phi^{Y}_{s}(x)}ds\in{\mathcal{V}}$ for
any $t\geq t_{x}$. A short proof is following: Suppose this is false, then
there are $t_{n}\to\infty$ such that
$\mu_{n}=\frac{1}{t_{n}}\int_{0}^{t_{n}}\delta_{\phi^{Y}_{s}(x)}ds\notin{\mathcal{V}}$.
Because the probability space over any compact manifold is compact, we can
take a converging point $\mu_{0}$ of $\\{\mu_{n}\\}_{n\in\mathbb{N}}$. Then
$\mu_{0}$ is an invariant measure but not contained in ${\mathcal{V}}$, a
contradiction.

Let us continue the proof. For $n>0$, denote by $M_{n}=\\{x;t_{x}\leq n\\}$.
Because $\bigcup_{n}M_{n}=M$, we can choose $N$ sufficiently large, such that
$\operatorname{vol}(M_{N})>0$. Observe that for any $x\in M_{N}$ and $n>N$,

$\int
div_{Y}(y)d\big{(}\frac{1}{n}\int_{0}^{n}\delta_{\phi^{Y}_{s}(x)}ds\big{)}(y)=\frac{1}{n}\int_{0}^{n}div_{Y}(\delta_{\phi^{Y}_{s}(x)})ds>a.$

Then by Lemma 2.7,

$\det\Phi_{n}|_{T_{x}M}=\exp^{\int_{0}^{n}div_{Y}(\delta_{\phi^{Y}_{s}(x)})ds}>\exp^{na},$

which implies that $\lim_{n}\operatorname{vol}(\phi_{n}^{Y}(M_{N}))=\infty$, a
contradiction to the fact that the volume of $M$ is bounded.

∎

####  2.2.3. Star flow

Now let us give a general criterion for a two-dimensional flow $Y$ to be a
star flow.

######  Theorem 2.12.

Suppose all the singularities of $Y$ are hyperbolic, and all the ergodic
measures of $\phi^{Y}$ have negative divergence, then $Y$ is a star flow.

######  Example 2.13.

Consider a 2-dimensional flow $Y_{0}$ on $\mathbb{R}^{2}$ with a saddle
$\sigma$ of negative divergence, and each branch of the unstable manifold of
$\sigma$ is connected to a branch of its stable manifold– locally this set
looks like a figure ‘8’. Denote by $\Gamma$ the set consisting of $\sigma$ and
its two saddle connections. It is easy to show that $\Gamma$ is an attractor:
for each branch of unstable manifold, we take a transverse section near to
$\sigma$ and analyze the Poincaré return maps, which are uniformly
contracting. Taking a small contracting neighborhood $U$, $\delta_{\sigma}$ is
the unique invariant measure supported in $U$. Applying Theorem 2.12, $Y_{0}$
is a star flow in $U$.

###### Proof of Theorem 2.12:.

Because all the singularities of $Y$ are hyperbolic, $\phi^{Y}$ has only
finitely many singularities. By the continuation of hyperbolic singularities,
the singularities for flows in a small neighborhood are all hyperbolic. To
prove this theorem, it suffices to show that the periodic orbits of nearby
flows are all periodic sinks.

Write ${\mathcal{M}}_{Y}$ the space of invariant measures of $\phi^{Y}$. By
the assumption, for any ergodic measure $\mu\in{\mathcal{M}}_{Y}$,
${\mathcal{D}}_{Y}(\mu)<0$. Then by Ergodic Decomposition Theorem,
${\mathcal{D}}_{Y}(.)|_{{\mathcal{M}}_{Y}}<0$. Because ${\mathcal{M}}_{Y}$ is
a compact space and ${\mathcal{D}}_{Y}(.)$ is a continuous function, this
implies that ${\mathcal{D}}_{Y}(.)|_{{\mathcal{M}}_{Y}}<a<0$ for some $a<0$.

For any vector field $Z\in{\mathcal{U}}$, suppose ${\rm Orb}_{Z}(p)$ is a
periodic orbit of $Z$. Denote $\mu_{{\rm Orb}_{Z}(p)}$ the $\phi^{Z}$ ergodic
measure supported on ${\rm Orb}_{Z}(p)$. Then by the upper semi-continuation
of the invariant measures space, for ${\mathcal{U}}$ sufficiently small,
$\mu_{{\rm Orb}_{Z}(p)}$ is close to ${\mathcal{M}}_{Z}$ in the weak star
topology. From the definition of divergence for measures,

${\mathcal{D}}_{Z}({\rm Orb}_{Z}(p))={\mathcal{D}}_{Z}(\mu_{{\rm
Orb}_{Z}(p)})<\frac{a}{2}<0.$

By Remark 2.9, ${\rm Orb}_{Z}(p)$ is a periodic sink. The proof is complete.

∎

As an immediately corollary of Theorem 2.2 (a) and Theorem 2.12, we show that:

######  Lemma 2.14.

Suppose $\phi^{X}$ is a $\operatorname{C}^{\infty}$ Cherry flow and
${\mathcal{D}}_{X}(\sigma_{X})<0$. Then it is a star flow.

We also need the following general description on ergodic measures for flows,
whose proof depends on Liao Theory, and is postponed to Section 3. The main
difficulty in the proof arises from the existence of singularities. A similar
result for diffeomorphisms can be deduced by considering the corresponding
suspension flows or using the $\operatorname{C}^{1}$ version of Pesin Theory
for diffeomorphisms in [25].

######  Lemma 2.15.

Suppose $\mu$ is a non-trivial ergodic measure of a $\operatorname{C}^{1}$
flow. Then $\mu$ has at least one non-negative Lyapunov exponent.

By considering again the inverse of the flow, we show that:

######  Corollary 2.16.

Every non-trivial ergodic measure of a two-dimensional flow is non-hyperbolic,
i.e., the Lyapunov exponent and the divergence of this measure are both
vanishing.

###  2.3. Perturbation of flows

We need the following flow version of Franks Lemma of [2][Theorem A.1] to
modify the linear Poincaré flow along a periodic orbit. Similar statement was
obtained by Liao in [11][Proposition 3.4].

######  Theorem 2.17.

Given a $\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of $Y$, there is a
neighborhood ${\mathcal{V}}\subset{\mathcal{U}}$ of $Y$ and $\varepsilon>0$
such that for any $Z\in{\mathcal{V}}$, for any periodic orbit ${\rm
Orb}_{Z}(p)$ of $Z$ with period $\tau(p)\geq 1$, any neighborhood $U$ of ${\rm
Orb}_{Z}(p)$ and any partition of $[0,\tau_{Z}(p)]$:

$0=t_{0}<t_{1}<\dots<t_{l}=\tau_{Z}(p),\;\;\;1\leq t_{i+1}-t_{i}\leq
2,i=0,1,\dots,l-1,$

and any linear isomorphisms
$L_{i}:{\mathcal{N}}_{\phi^{Z}_{t_{i}}(p)}\to{\mathcal{N}}_{\phi^{Z}_{t_{i+1}}(p)},i=0,1,\dots,l-1$
satisfying
$\|L_{i}-\psi^{Z}_{t_{i+1}-t_{i}}|_{{\mathcal{N}}_{\phi_{t_{i}}(p)}}\|\leq\varepsilon$,
there exists $\tilde{Z}\in{\mathcal{U}}$ such that
$\psi^{\tilde{Z}}_{t_{i+1}-t_{i}}|_{{\mathcal{N}}_{\phi_{t_{i}}(p)}}=L_{i}$
and $\tilde{Z}=Z$ on $(M\setminus U)\cup{\rm Orb}_{Z}(p)$.

######  Remark 2.18.

In fact, for any fixed $k\in\mathbb{N}$, one can choose ${\mathcal{V}}$
smaller, such that for any $Z\in{\mathcal{V}}$, and $p_{1},p_{2},\dots,p_{k}$
periodic orbits of $Z$, we can perturb this flow in the above manner near to
all the $k$ periodic orbits simultaneously.

We also need the Closing Lemma:

######  Theorem 2.19.

Suppose $x\in M$ is regular point of flow $Y$ and $\omega(x)$ contains regular
points. Then for any $\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of
$Y$ and any neighborhood $U$ of $\omega(x)$, there is a flow
$Z\in{\mathcal{U}}$ such that $Z=Y|_{M\setminus U}$ and $Z$ has a periodic
orbit contained in $U$.

A deep result of Liao [13] shows that the number of periodic sinks is
uniformly bounded in a small neighborhood of a star flow. The following
version of Liao’s result which can be obtained directly from the original
proof, is the motivation of our Theorem C. Since it will not be used in this
article, we do not provide a proof.

For a flow $Y$, denote by ${\rm Sing}_{1}(Y)$ the set of hyperbolic
singularities which are contained in non-trivial chain recurrent classes, and
whose tangent bundle admits a codimension-1 dominated splitting $E^{cs}\oplus
E^{u}_{1}$ where $E^{u}_{1}$ is a one-dimensional unstable bundle. Note that
every $\sigma\in{\rm Sing}_{1}(Y)$ admits a one-dimensional strong unstable
manifold. We also write $S(Y)$ the number of periodic sinks of $\phi^{Y}$ and
$K(Y)=\\#({\rm Sing}_{1}(Y))$.

######  Theorem 2.20.

Suppose $Y$ is a star flow. Then $S(Y)$ is finite, and there is a
$\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}$ of $Y$, such that for any
$Z\in{\mathcal{U}}$,

$S(Z)\leq S(Y)+2K(Y).$

Moreover, the basin of every ‘new’ periodic sink of $Z$ intersects one branch
of the one-dimensional strong unstable manifold of a singularity of $Z$ which
is the analytic continuation of a singularity $\sigma\in{\rm Sing}_{1}(Y)$.

######  Example 2.21.

Applying the above theorem on the star flow $Y_{0}$ in Example 2.13, there is
an open neighborhood ${\mathcal{U}}$ of $Y_{0}$, such that any flow
$Z\in{\mathcal{U}}$ has at most two periodic orbits/sinks, and a saddle
connection is broken when one periodic sink appears.

##  3\. Proof of Lemma 2.15

Throughout this section we suppose $\phi^{Y}$ is a flow generated by vector
field $Y$ over manifold $M^{d}$, $\mu$ is a non-trivial ergodic measure of
$\phi^{Y}$ with all Lyapunov exponents negative. We prove Lemma 2.15 by
showing the contradiction. In fact, we make use of scaled linear Poincaré flow
in Liao Theory to guarantee that there are infinite number of distinct
periodic sinks, and each basin contains a uniform size of ball, which is a
contradiction.

We need the following version of [25][Theorem 5.1], where the ‘$F$ bundle’ of
the original statement is taken to be empty here. For completeness, we give a
proof in Subsection 3.1.

######  Theorem 3.1.

There is a compact, positive $\mu$ measure subset
$\Lambda\subset\operatorname{supp}(\mu)\setminus{\rm Sing}(Y)$ satisfying: for
any $\varepsilon>0$, there are $L,L^{{}^{\prime}},\delta>0$ such that for any
$x$ and $\phi^{Y}_{T}(x)\in\Lambda$ with $T>L^{{}^{\prime}}$ and
$d(x,\phi^{Y}_{T}(x))<\delta$, there exists a point $p\in M$ 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 sink of $\phi^{Y}$ with period $\tau(p)=\theta(T)$;

  * (c)

$d(\phi^{Y}(x),\phi^{Y}_{\theta(t)}(p))\leq\varepsilon\|Y(\phi^{Y}_{t}(x))\|,t\in[0,T]$;

  * (d)

${\rm Orb}_{Y}(p)$ has uniform size of stable manifold at $p$.

###### Proof of Lemma 2.15:.

Fix $\Lambda$, $\varepsilon,\delta,L,L^{{}^{\prime}}>0$ as in Theorem 3.1. By
Birkhoff Ergodic Theorem, there is $x\in\Lambda$ and
$L^{{}^{\prime}}<t_{1}<t_{2}<\dots$ such that

  * (i)

$\phi^{Y}_{t_{i}}(x)\in\Lambda$ for any $i>0$ and $\phi^{Y}_{t_{i}}(x)\to x$;

  * (ii)

$t_{i+1}(1-\varepsilon)>t_{i}(1+\varepsilon)$.

Applying Theorem 3.1, each pseudo orbit
$\big{(}\phi_{t}^{Y}(x)\big{)}_{t\in[0,t_{i}]}$ is shadowed by a periodic sink
$p_{i}$ with period $(1-\varepsilon)t_{i}<\tau(p_{i})<(1+\varepsilon)t_{i}$.

By (ii) above, all the periods are different, which implies that all these
periodic sinks are distinct. By (d) of Theorem 3.1, each periodic sink ${\rm
Orb}(p_{i})$ has uniform size of stable manifold at $p_{i}$ for every $i$,
which is a contradiction.

∎

###  3.1. Proof of Theorem 3.1

In this subsection, we provide a proof of Theorem 3.1. In the proof we need
another flow $\psi^{*}_{t}:{\mathcal{N}}_{M}\rightarrow{\mathcal{N}}_{M}$,
which is called scaled linear Poincaré flow:

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

######  Lemma 3.2.

$\psi^{*}_{t}$ is a bounded cocycle on ${\mathcal{N}}_{M}$ 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, the two cocycles $\psi_{t}$ and $\psi^{*}_{t}$ have the same
Lyapunov exponent.

###### Proof.

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

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

∎

####  3.1.1. Pesin block

######  Definition 3.3.

The orbit arc $\phi_{[0,T]}(x)$ is called $(\lambda,T_{0})^{*}$-quasi
contracting with respect to 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 $k=0,1,\dots,l-1$, we have

$\prod_{i=0}^{k-1}\|\psi^{*}_{t_{i+1}-t_{i}}|_{{\mathcal{N}}_{\phi_{t_{i}}(x)}}\|\leq\lambda^{k}$

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

######  Lemma 3.4.

There are $L^{{}^{\prime}},\eta,T_{0}>0$ and a positive $\mu$ measure subset
$\Lambda\setminus{\rm Sing}(X)$, such that for any
$x,\phi^{Y}_{T}(x)\in\Lambda$ with $T>L^{{}^{\prime}}$, $\phi^{Y}_{[0,T]}(x)$
is $(\eta,T_{0})^{*}$-quasi contracting with respect to the scaled linear
Poincaré flow $\psi^{*}_{t}$.

###### Proof.

Taking $1\leq t_{0}<2$ such that $\mu$ is an ergodic measure for
$f=\phi^{Y}_{t_{0}}$. For simplicity, we assume $t_{0}=1$.

By subadditive ergodic theorem, there is $a<0$ such that

$\lim_{t\to\infty}\frac{1}{t}\int\log\|\psi_{t}^{*}\|d\mu<a.$

Take $N_{0}$ sufficiently large, such that
$\frac{1}{N_{0}}\int\log\|\psi_{N_{0}}^{*}\|d\mu<a.$ Consider the ergodic
decomposition of $\mu$ for $f^{N_{0}}=\phi^{Y}_{N_{0}}$:

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

By changing the order, we suppose that:

$\frac{1}{N_{0}}\int\log\|\psi_{N_{0}}^{*}\|d\mu_{1}<a.$

Note that $\mu_{1}$ is an $f^{N_{0}}$ ergodic measure. Applying Birkhoff
ergodic theorem, for $\mu_{1}$ almost every $x$,

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

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

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

Choose $N_{1}$ such that the set $\Lambda^{{}^{\prime}}=\\{x;n(x)<N_{1}\\}$
has positive $\mu_{1}$ measure. Let $\Lambda\subset\Lambda^{{}^{\prime}}$ be a
compact subset with positive $\mu_{1}$ measure. It follows immediately that
$\mu(\Lambda)>0$. By Lemma 3.2, let

$K=\sup_{|t|\leq
N_{0},y\in\Lambda}\\{\|\psi_{t}^{*}|_{{\mathcal{N}}_{y}}\|\\}.$

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 $0\leq i\leq l-1$ and $x\in\Lambda$:

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

Let continue the proof. Choose $L^{{}^{\prime}}>0$ 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

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

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

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

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

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

∎

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

In this subsection we introduce the Liao’s shadowing lemma for scaled linear
Poincaré flow of [13]. The idea of the proof is explained:

######  Theorem 3.5.

Given a compact set $\Lambda\cap{\rm Sing}(Y)=\emptyset$, 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 contracting orbit arc
$\phi_{[0,T]}(x)$ with respect to the scaled linear Poincaré flow
$\psi_{t}^{*}$ which satisfies $x,\phi^{Y}_{T}(x)\in\Lambda$ and
$d(x,\phi_{T}(x))\leq\delta$, there exists a point $p\in M$ 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 sink with $\tau(p)=\theta(T)$;

  * (c)

$d(\phi^{Y}_{t}(x),\phi^{Y}_{\theta(t)}(p))\leq\varepsilon\|Y(\phi^{Y}_{t}(x))\|,t\in[0,T]$;

  * (d)

${\rm Orb}(p)$ has uniform size of stable manifold at $p$.

###### 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. For $\beta>0$, we denote by ${\mathcal{N}}_{x}(\beta)$ the ball
contained in ${\mathcal{N}}_{x}$ with radius $\beta$.

The first step is to translate the problem for flow into the shadowing lemma
for a sequence of maps: by [6][Lemmas 2.2, 2.3], there is $\beta$ depending on
$T_{0}$ such that the holonomy map induced by $\phi^{Y}$ is well defined
between

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

which is conjugate to the following map:

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

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

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

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

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

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

Then, by the definition of quasi contracting, there is $0<\lambda<1$ such that
for $k=1,\dots,l$, we have

$\prod_{i=1}^{k}\|{\mathcal{T}}_{i}\|\leq\lambda^{k}.$

Now we apply the version of Liao’s shadowing lemma on discrete quasi
hyperbolic maps ([5][Theorem 1.1]), see also [10, 12, 14] and the explanation
below. On the $d(x,\phi^{Y}_{T}(x))$-pseudo orbit
$\\{0_{x},0_{\phi^{Y}_{t_{1}}(x)},\dots,0_{\phi^{Y}_{t_{l-1}}(x)}\\}$, there
is $L>0$ and a periodic point $p\in{\mathcal{N}}_{x}$ for the maps
$\\{{\mathcal{T}}_{0},\cdots,{\mathcal{T}}_{l-1}\\}$, whose orbit
$Ld(x,\phi^{Y}_{T}(x))$-shadows the pseudo orbit.

Using the version of flow tubular theorem [6][Lemmas 2.2], one can prove (a),
(b) and (c) above. Now let us focus on the proof of (d). We follow the proof
of [5]. By Lemma 3.1 of [5], 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 $\tilde{{\mathcal{T}}}_{j}(a)=g_{j}^{-1}{\mathcal{T}}_{j}(g_{j-1}a)$
for $a\in\mathbb{R}^{d-1}$, then

$\|T\tilde{{\mathcal{T}}}_{j}(0)|_{E_{j}}\|\leq\lambda.$

It was shown in [5] (p. 631) that the well adapted sequence is uniformly
bounded from above and from zero, and the sequence of contracting 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

(1)
$\begin{split}\Psi_{k}=g_{k}\tilde{\Psi}_{k}\;\;&\text{and}\;\;\Psi_{l}=\tilde{\Psi}_{l}.\\\
\end{split}$

Observe that
$\\{0_{x},0_{\phi^{Y}_{t_{1}}}(x),\cdots,0_{\phi^{Y}_{t_{l-1}}(x)}\\}$ is
still a $d(x,\phi^{Y}_{T}(x))$-pseudo orbit of sequence of uniformly
contracting diffeomorphisms $\\{\tilde{{\mathcal{T}}}_{j}\\}_{j=0}^{l-1}$. By
the standard shadowing lemma for hyperbolic diffeomorphisms (see [5][Lemma
2.1]), there is $L>0$ such that the pseudo orbit is
$Ld(x,\phi^{Y}_{T}(x))$-shadowed by periodic orbit
$\\{\tilde{p},\tilde{p}_{1},\dots,\tilde{p}_{l-1}\\}$, and this periodic orbit
has uniform size of stable manifold. By (1),
$\\{g_{k}\tilde{p}_{k}\\}_{k=1}^{l}$ is a pseudo orbit for the original
sequence of maps $\\{{\mathcal{T}}_{k}\\}_{k=0}^{l-1}$. Observe that
$g_{l}=1$, which implies that $p_{0}$ and $\tilde{p}_{0}$ coincide and have
the same stable manifold. Hence, $p_{0}$ has uniform size of stable manifold.
We conclude the proof of (d).

∎

####  3.1.3. Proof of Theorem 3.1

It is easy to see that Theorem 3.1 is a direct corollary of Lemma 3.4 and
Theorem 3.5.

##  4\. Proof of Theorem B

###### Proof of Theorem B.

: By Theorem 2.2(b), $\Lambda^{X}$ admits two ergodic measures:
$\delta_{\sigma^{X}}$ and $\mu^{X}$. By Corollaries 2.16 and 2.10, $\mu^{X}$
is non-hyperbolic and ${\mathcal{D}}_{X}(\mu^{X})=0$.

For every $x\in M\setminus\tilde{\sigma}^{X}$, its forward orbit converges to
the quasi-minimal set. In particular, any accumulated point of
$\frac{1}{t}\int_{0}^{t}\delta_{\phi^{X}_{s}(x)}ds$ can be written as
$a\delta_{\sigma^{X}}+(1-a)\mu^{X}$ for some $0\leq a\leq 1$.

For $0\leq b\leq 1$, denote $K_{b}=\\{a\delta_{\sigma^{X}}+(1-a)\mu^{X};b\leq
a\leq 1\\}$ and

$\Gamma_{b}=\\{x\in\mathbb{T}^{2};\;\text{there
is}\;t_{i}\to\infty\;\text{such
that}\;\lim_{i}\frac{1}{t_{i}}\int_{0}^{t_{i}}\delta_{\phi^{X}_{s}(x)}ds\in
K_{b}\\}.$

We claim that $\operatorname{vol}(\Gamma_{b})=0$ for every $b>0$. It is easy
to see that Theorem B follows from this claim immediately. Now let us prove
this claim.

###### Proof.

Because ${\mathcal{D}}_{X}(\sigma^{X})>0$ and ${\mathcal{D}}_{X}(\mu^{X})=0$,
there is $c>0$ such that ${\mathcal{D}}_{X}|_{K_{b}}>2c$. Taking
${\mathcal{V}}$ a small neighborhood of $K_{b}$ in the probability space of
$\mathbb{T}^{2}$, such that for any measure $\mu\in{\mathcal{V}}$ which is
possibly not invariant, we have $\int div_{X}(x)d\mu(x)>c$.

Denote by

$\Gamma_{b,n}=\\{x;\;\text{there is}\;t_{x}\in[n,n+1)\;\text{such
that}\;\frac{1}{t_{x}}\int_{0}^{t_{x}}\delta_{\phi^{X}_{s}(x)}ds\in{\mathcal{V}}\\}.$

Then for every $x\in\Gamma_{b,n}$, we have

$\int
div_{X}(y)d\big{(}\frac{1}{t_{x}}\int_{0}^{t_{x}}\delta_{\phi^{X}_{s}(x)}ds\big{)}(y)=\frac{1}{t_{x}}\int_{0}^{t_{x}}div(\phi^{X}_{s}(x))ds>c,$

which implies by Lemma 2.7 that

$\det\Phi_{t_{x}}|_{T_{x}\mathbb{T}^{2}}=\exp^{t_{x}\int_{0}^{t_{x}}div_{X}(\phi^{X}_{s}(x))ds}>\exp^{ct_{x}}.$

Denote by
$C=\max_{x\in\mathbb{T}^{2},t\in[0,1)}\\{\det\Phi_{t}|_{T_{x}\mathbb{T}^{2}}\\}$.
Then for any $x\in\Gamma_{b,n}$,

$\det\Phi_{n}|_{T_{x}\mathbb{T}^{2}}>\frac{1}{C}\exp^{cn},$

which implies that $\operatorname{vol}(\Gamma_{b,n})\leq C\exp^{-cn}$.

Because $\Gamma_{b}\subset\bigcap_{n}\bigcup_{i>n}\Gamma_{b,n}$, the above
argument shows that $\operatorname{vol}(\Gamma_{b})=0$. The proof of the claim
is complete. ∎

∎

##  5\. Proof of Theorems C and D

###  5.1. Perturbation of Cherry flow

Recall that ${\mathcal{S}}_{0}$ is a circle transverse to $X$. There is a
$\operatorname{C}^{1}$ neighborhood ${\mathcal{U}}_{0}$ of $X$ such that for
any flow $Y\in{\mathcal{U}}_{0}$, ${\mathcal{S}}_{0}$ is still transverse to
$Y$. Moreover, the inverse of $\phi^{Y}$ can be looked as a suspension of a
continuous circle map $g^{Y}$, where $g^{Y}$ is a monotone circle map which is
constant on an interval $I^{Y}$. The intersection between a periodic orbit of
$\phi^{Y}$ with ${\mathcal{S}}_{0}$ corresponds to a periodic orbit of
$g^{Y}$. In contrary, a periodic orbit of $g^{Y}$ which does not intersect
with $I^{Y}$ corresponds to a periodic orbit of $\phi^{Y}$. Denote by
$\operatorname{Per}(Y)$ and $\operatorname{Per}(g^{Y})$ the set of periodic
orbits for $Y$ and $g^{Y}$ respectively. A segment
$I=[p,q]\subset{\mathcal{S}}_{0}$ is called a $g^{Y}$-periodic segment if
$p,q$ both are periodic points of $g^{Y}$. The proof of the following lemma is
the same as for homeomorphisms over circle.

######  Lemma 5.1.

Suppose $g^{Y}$ has a periodic point with period $k$. Then its rotation number
$\rho(g^{Y})$ is rational, all the periodic points have the same period, and
every periodic segment is fixed by $(g^{Y})^{k}$. Moreover, for any
$\theta\in{\mathcal{S}}_{0}$, $\omega(\theta)$ is a periodic orbit of $g^{Y}$.

######  Corollary 5.2.

$\\#\operatorname{Per}(Y)\geq\\#\operatorname{Per}(g^{Y})-1$. More precisely,
suppose $Y$ contains $n$ periodic sources, then it has at least $n$ periodic
orbits which are not periodic sources, and $\\#\operatorname{Per}(Y)\geq 2n$;
in contrary, when $Y$ contains $n$ periodic sinks, then it has at least $n-1$
periodic orbits which are not periodic sinks, and
$\\#\operatorname{Per}(Y)\geq 2n-1$.

###### Proof.

Observe that $g^{Y}(I^{Y})$ is a point, $I^{Y}$ intersects at most one
periodic orbit, and in this case, $I^{Y}$ belongs to the stable set of this
periodic orbit. Hence, the periodic orbit which intersects with $I^{Y}$ is not
a source for $g^{Y}$.

When $Y$ has $n$ periodic sinks: ${\rm Orb}_{Y}(p_{1}),\dots,{\rm
Orb}_{Y}(p_{n})$, suppose $p_{1},\dots,p_{n}\in{\mathcal{S}}_{0}$. Then for
every $1\leq i\leq n$, $p_{i}$ is an isolated source for $g^{Y}$. For each
$1\leq i\leq n$, take $q_{i}$ the nearest periodic point of $g^{Y}$ to $p_{i}$
in the counterclockwise direction, then $q_{i}$ has a half neighborhood which
is topologically contracting by $g^{Y}$. There are at least
$\\#\operatorname{Per}(g^{Y})-1$ distinct such periodic orbits of $g^{Y}$
which do not intersect $I^{Y}$, these periodic orbits correspond to
intersections of periodic orbits of $Y$ with ${\mathcal{S}}_{0}$. Therefore,
there are at least $n-1$ $g^{Y}$-periodic orbits of $\\{{\rm
Orb}_{g^{Y}}(q_{1}),\dots,{\rm Orb}_{g^{Y}}(q_{n})\\}$ can be lifted to
periodic orbits of $Y$ which are not periodic sinks. As a conclusion,
$\phi^{Y}$ has at least $2n-1$ periodic orbits.

When $Y$ has $n$ periodic sources: ${\rm Orb}_{Y}(p_{1}),\dots,{\rm
Orb}_{Y}(p_{n})$, suppose $p_{1},\dots,p_{n}\in{\mathcal{S}}_{0}$. Then for
every $1\leq i\leq n$, $p_{i}$ is an isolated sink for $g^{Y}$. We first
consider the case that $I^{Y}$ does not intersect a periodic orbit of $g^{Y}$.
Taking $q_{i}$ the nearest periodic point of $g^{Y}$ to $p_{i}$ in the
counterclockwise direction, then $q_{i}$ has at least a half neighborhood
which is expanding by $g^{Y}$. As observed above, $\\{{\rm
Orb}_{g^{Y}}(q_{1}),\dots,{\rm Orb}_{g^{Y}}(q_{n})\\}$ can be lifted to
periodic orbits of $Y$ which are not periodic sources. Hence, $Y$ has at least
$2n$ periodic orbits.

In the second case, we suppose $p_{0}\in I^{Y}$ is a periodic point of
$g^{Y}$. We may further suppose that $I\setminus p_{0}$ contains a segment in
the counterclockwise direction, the proof of other case is similar. Taking
$q_{i}$ $i=0,1,\dots,n$ the nearest periodic point of $g^{Y}$ in the
counterclockwise direction, Then each $q_{i}$ has at least a half neighborhood
is expanding by $(g^{Y})^{k}$. There are at least $n$ $g^{Y}$-periodic orbits
of $\\{q_{0},\dots,q_{n}\\}$ can be lift to periodic orbits of $Y$, which are
not periodic sources. This implies that $Y$ has at least $2n$ periodic orbits.

The proof of this corollary is finished. ∎

###  5.2. Proof of Theorems C and D

###### Proof of Theorem C:.

Fix ${\mathcal{U}}_{0}$ the neighborhood of $X$ given in subsection 5.1. By
Lemma 2.14, $\phi_{t}$ is a star flow. There is a $\operatorname{C}^{1}$
neighborhood ${\mathcal{U}}\subset{\mathcal{U}}_{0}$ of $X$ such that for any
flow $Y\in{\mathcal{U}}$, $Y$ belongs to the following three cases:

  * (a)

has a saddle connection;

  * (b)

has no periodic orbits,

  * (c)

has periodic orbits, and all the periodic orbits are periodic sinks.

From now on, we assume that $Y$ has no saddle connection.

Recall the general definition of Cherry flow in [15]:

######  Definition 5.3.

A flow on $\mathbb{T}^{2}$ is called a Cherry flow if it has:

  * (i)

a finite number of singularities all of which are hyperbolic;

  * (ii)

no periodic orbits;

  * (iii)

no saddle-connections;

  * (iv)

sinks.

When $\phi^{Y}$ contains no periodic orbit, by the above definition of Cherry
flow, $Y$ is automatically a $\operatorname{C}^{1}$ Cherry flow. Denote by
$\Lambda^{Y}$ the quasi-minimal attractor and $\sigma^{Y}$ the singularity
contained in $\Lambda^{Y}$. By Proposition 2.1, $\Lambda^{Y}$ admits at most
two ergodic measures. We claim that $\Lambda^{Y}$ admits a unique ergodic
measure $\delta_{\sigma^{Y}}$.

###### Proof.

Suppose that $\Lambda^{Y}$ admits another non-trivial ergodic measure
$\mu^{Y}$ of $\phi^{Y}$. Because $\Lambda^{X}$ only supports a unique
invariant measure $\delta_{\sigma_{0}}$ of $\phi^{X}$ with negative
divergence, by the upper-semi continuation of the invariant measures spaces,
we have that $\int div_{Y}(x)d\mu^{Y}(x)<0$. This contradicts to Corollary
2.16.

∎

The above claim implies that $\delta_{\sigma^{Y}}$ is a physical measure,
whose basin coincides to $\mathbb{T}^{2}\setminus\tilde{\sigma}^{Y}$ where
$\tilde{\sigma}^{Y}$ denotes the unique source of $Y$.

Now suppose $\phi^{Y}$ contains a periodic sink $p$. Let us show that ${\rm
Orb}_{Y}(p)$ is the unique periodic orbit of $\phi^{Y}$. Suppose there is a
different periodic sink ${\rm Orb}_{Y}(q)$. By Corollary 5.2, $\phi^{Y}$ has a
periodic orbit which is not a periodic sink. This contradicts to the fact that
every periodic orbit of $\phi^{Y}$ is a periodic sink. Hence, $\phi^{Y}$
contains only one periodic sink.

It remains to show that $\phi^{Y}$ is a Morse-Smale flow. We are going to
prove that every
$x\in\mathbb{T}^{2}\setminus(\tilde{\sigma}^{Y}\cup\sigma^{Y}\cup
W^{s}(\sigma^{Y}))$ belongs to the attracting basin of the periodic sink ${\rm
Orb}_{Y}(p)$. Suppose this is false, then there is
$x\in\mathbb{T}^{2}\setminus(\tilde{\sigma}^{Y}\cup\sigma^{Y}\cup
W^{s}(\sigma^{Y}))$ such that $\omega(x)$ is not a critical element. Take a
smaller neighborhood $V$ of $\omega(x)$ such that $V\cap{\rm
Orb}_{Y}(p)=\emptyset$. Then by Closing Lemma, there is another flow
$Z\in{\mathcal{U}}$ such that $Z|_{V^{c}}=Y$ and $\phi^{Z}$ has a periodic
orbit ${\rm Orb}_{Z}(q)\subset V$. Then $\phi^{Z}$ has two periodic orbits, a
contradiction to the above discussion that $\phi^{Z}$ has at most one periodic
orbit. The proof is complete.

∎

###### Proof of Theorem D:.

We first prove that $\phi_{t}$ can be approximated by flows with arbitrarily
large number of periodic sinks. Suppose by contradiction that there exist
$l>0$ and ${\mathcal{U}}\subset{\mathcal{U}}_{0}$ a neighborhood of $X$ such
that every flow $Y\in{\mathcal{U}}$ has at most $l$ periodic sinks. We claim
that $l\geq 1$.

###### Proof.

Suppose by contradiction that $l=0$. By Closing lemma, there is a flow $Y$
arbitrarily close to $X$ such that $Y$ has at least one periodic orbit ${\rm
Orb}_{Y}(p)$ which is not a periodic sink. After a small perturbation at the
singularity $\sigma^{Y}$, we may always assume that
${\mathcal{D}}_{Y}(\sigma^{Y})>0$. When ${\mathcal{D}}_{Y}({\rm
Orb}_{Y}(p))=0$, which means this periodic orbit is non-hyperbolic, we can
apply the Franks Lemma (Theorem 2.17) to show that, there is flow $Z$
arbitrarily close to $Y$ such that ${\rm Orb}_{Z}(p)$ is a periodic sink. This
contradicts our previous assumption.

From now on, we assume that all the periodic orbits of $Y$ are sources. By
Corollary 5.2, $Y$ has at least another periodic orbit ${\rm Orb}_{Y}(q)$
which is not a periodic source. This contradicts to the above assumption that
all the periodic orbits of $Y$ should be periodic sources.

∎

Take a flow $Y$ which is arbitrarily close to $X$ such that $\phi^{Y}$
contains $l$ periodic sinks ${\rm Orb}_{Y}(p_{1}),\dots,{\rm Orb}_{Y}(p_{l})$.
By Corollary 5.2, $Y$ contains at least $l-1$ periodic orbits which are not
periodic sinks. Replacing by arbitrary small perturbation, we suppose the
$l-1$ periodic orbits ${\rm Orb}_{Y}(q_{1}),\dots,{\rm Orb}_{Y}(q_{l-1})$ are
all periodic sources. Because all the invariant measures of $X$ have non-
negative divergence, by the upper semi-continuation of the invariant measures
space, there is $\varepsilon$ close to zero such that for any $1\leq i\leq l$,
${\mathcal{D}}_{Y}({\rm Orb}_{Y}(p_{i}))\geq-\varepsilon$. Applying Franks
Lemma as observed in Remark 2.18, there is a flow $Z$ which is close to $Y$
such that

  * •

$\\{{\rm Orb}_{Z}(p_{i});i=1,\dots,l\\}$ are all periodic sources with
divergence smaller than $\varepsilon$,

  * •

$Z=Y$ outside a small neighborhood of $\cup_{i=1,\dots,l}{\rm
Orb}_{Y}(p_{i})$.

In particular, $Z$ has $2l-1$ periodic sources. By Corollary 5.2, $Z$ has at
least another $2l-1$ periodic orbits ${\rm Orb}_{Z}(a_{1}),\dots,{\rm
Orb}_{Z}(a_{2l-1})$ which are not periodic sources. Because $\\{{\rm
Orb}_{Z}(p_{i})\\}_{i=1}^{l}$ all have small divergence, using again the
Franks Lemma, we obtain a flow $\hat{Z}$ such that

${\rm Orb}_{\hat{Z}}(p_{1}),\dots,{\rm Orb}_{\hat{Z}}(p_{l})),{\rm
Orb}_{\hat{Z}}(a_{1}),\dots,{\rm Orb}_{\hat{Z}}(a_{2l-1})$

are all periodic sinks. Hence $\hat{Z}$ has at least $3l-1>l$ periodic sinks,
a contradiction to the assumption that $\hat{Z}$ has at most $l$ periodic
sinks. This contradiction shows that $X$ can be approached by a flow $Y$ with
arbitrarily large number of periodic sinks.

For any $l>0$, take $Y$ sufficiently close to $X$ such that $Y$ has $l$
periodic sinks $\\{{\rm Orb}_{Y}(p_{1}),\dots,{\rm Orb}_{Y}(p_{l})\\}$. As
explained above, all these periodic sinks have negative divergence close to
zero. Applying Franks Lemma on ${\rm Orb}_{Y}(p_{1})$, there is a flow $Z$
arbitrarily close to $Y$ such that

  * •

${\rm Orb}_{Z}(p_{1})$ is not-hyperbolic,

  * •

${\rm Orb}_{Z}(p_{2}),\dots,{\rm Orb}_{Z}(p_{l})$ are still periodic sinks of
$Z$.

The proof is complete. ∎

##  Appendix A Proof of Lemma 2.7

###### Proof of Lemma 2.7:.

For $s\in[0,t]$, denote by $x_{s}=\phi^{Y}_{s}(x)$. Because

$\displaystyle\frac{d}{ds}\big{|}_{s_{0}}(\operatorname{ln}\det\Phi_{s}|_{T_{x}M})$
$\displaystyle=$ $\displaystyle\lim_{\Delta s\to 0}\frac{1}{\Delta
s}(\operatorname{ln}\det\Phi_{s_{0}+\Delta
s}|_{T_{x}M}-\operatorname{ln}\det\Phi_{s_{0}}|_{T_{x}M})$ $\displaystyle=$
$\displaystyle\lim_{\Delta s\to 0}\frac{1}{\Delta
s}(\operatorname{ln}\det\Phi_{\Delta s}|_{T_{x_{s_{0}}}M})$ $\displaystyle=$
$\displaystyle\lim_{\Delta s\to 0}\frac{1}{\Delta
s}\operatorname{ln}\det\left|\begin{array}[]{cc}\frac{\partial f_{\Delta
s}}{\partial x}|_{x_{s_{0}}}&\frac{\partial g_{\Delta s}}{\partial
x}|_{x_{s_{0}}}\\\ \frac{\partial f_{\Delta s}}{\partial
y}|_{x_{s_{0}}}&\frac{\partial g_{\Delta s}}{\partial y}|_{x_{s_{0}}}\\\
\end{array}\right|$ $\displaystyle=$ $\displaystyle\frac{d}{d\Delta
s}|_{\Delta s=0}\operatorname{ln}\big{(}\frac{\partial f_{\Delta s}}{\partial
x}\frac{\partial g_{\Delta s}}{\partial y}-\frac{\partial g_{\Delta
s}}{\partial x}\frac{\partial f_{\Delta s}}{\partial
y}\big{)}\big{|}_{x_{s_{0}}}$ $\displaystyle=$
$\displaystyle\frac{\frac{d}{d\Delta s}|_{\Delta s=0}\big{(}\frac{\partial
f_{\Delta s}(x_{s_{0}})}{\partial x}\frac{\partial g_{\Delta
s}(x_{s_{0}})}{\partial y}-\frac{\partial g_{\Delta s}(x_{s_{0}})}{\partial
x}\frac{\partial f_{\Delta s}(x_{s_{0}})}{\partial
y}\big{)}}{\det\Phi_{0}|_{T_{x_{s_{0}}}M}}$

Note that

$\frac{d}{d\Delta s}|_{\Delta s=0}(\frac{\partial f_{\Delta
s}(x_{s_{0}})}{\partial x})=\frac{\partial}{\partial x}(\frac{d}{d\Delta
x}|_{\Delta s=0}(f_{\Delta s}(x_{s_{0}})))=\frac{\partial}{\partial
x}(Y_{1}(x_{s_{0}})).$

Similarity, we have that

$\frac{d}{d\Delta s}|_{\Delta s=0}(\frac{\partial g_{\Delta
s}(x_{s_{0}})}{\partial y})=\frac{\partial}{\partial x}(Y_{2}(x_{s_{0}})),$
$\frac{d}{d\Delta s}|_{\Delta s=0}(\frac{\partial g_{\Delta
s}(x_{s_{0}})}{\partial x})=\frac{\partial}{\partial y}(Y_{1}(x_{s_{0}})),$
$\frac{d}{d\Delta s}|_{\Delta s=0}(\frac{\partial f_{\Delta
s}(x_{s_{0}})}{\partial y})=\frac{\partial}{\partial y}(Y_{2}(x_{s_{0}})).$

Because $\Phi_{0}|_{T_{x_{s_{0}}}M}=\operatorname{ID}$, we have

$\left(\begin{array}[]{cc}\frac{\partial f_{\Delta s}(x_{s_{0}})}{\partial
x}&\frac{\partial g_{\Delta s}(x_{s_{0}})}{\partial x}\\\ \frac{\partial
f_{\Delta s}(x_{s_{0}})}{\partial y}&\frac{\partial g_{\Delta
s}(x_{s_{0}})}{\partial y}\\\ \end{array}\right)_{\Delta
s=0}=\operatorname{ID}.$

Combining the above argument, we show that

$\frac{d}{ds}\big{|}_{s_{0}}(\operatorname{ln}\det\Phi_{s}|_{T_{x}M})=\frac{\partial{Y_{1}}}{\partial{x}}+\frac{\partial{Y_{2}}}{\partial{y}}=div_{Y}(x_{s_{0}})$

Hence,

$\operatorname{ln}\det\Phi_{t}|_{T_{x}M}=\int_{0}^{t}\frac{d}{ds}(\operatorname{ln}\det\Phi_{s}|_{T_{x}M})ds=\int_{0}^{t}div_{Y}(x_{s})ds.$

The proof is complete.

∎

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