# Ergodic optimization theory for Axiom A flows

Wen Huang  Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of
Sciences and Department of Mathematics
University of Science and Technology of China,
Hefei, Anhui, China<EMAIL_ADDRESS>,  Zeng Lian  College of
Mathematical Sciences
Sichuan University
Chengdu, Sichuan, 610016, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Xiao Ma  Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of
Sciences and Department of Mathematics
University of Science and Technology of China
Hefei, Anhui, China<EMAIL_ADDRESS>,  Leiye Xu  Wu Wen-Tsun Key
Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of
Mathematics
University of Science and Technology of China
Hefei, Anhui, China<EMAIL_ADDRESS>and  Yiwei Zhang  School of
Mathematics and Statistics, Center for Mathematical Sciences, Hubei Key
Laboratory of Engineering Modeling and Scientific Computing, Hua-Zhong
University of Sciences and Technology
Wuhan 430074, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we consider the weighted ergodic optimization problem Axiom A
attractors of a $C^{2}$ flow on a compact smooth manifold. The main result
obtained in this paper is that for a generic observable from function space
$\mathcal{C}^{0,\alpha}$ ($\alpha\in(0,1]$) or $\mathcal{C}^{1}$ the
minimizing measure is unique and is supported on a periodic orbit.

Huang is partially supported by NSF of China (11431012,11731003). Lian is
partially supported by NSF of China (11725105,11671279). Xu is partially
supported by NSF of China (11801538, 11871188). Zhang is partially supported
by NSF of China (117010200,11871262).

##  1\. Introduction

Context and motivation. Ergodic optimization theory focuses on the ergodic
measures on which a given observable taking an extreme ergodic average
(maximum or minimum), which has strong connection with other fields, such as
Anbry-Mather theory [Co2, Mane, CIPP] in Lagrangian Mechanics; ground state
theory [BLL] in thermodynamics formalism and multifractal analysis; and
controlling chaos [HO1, OGY90, SGOY] in control theory.

In this paper, we study the typical optimization problem in weighted ergodic
optimization theory for Axiom A attractors of a $C^{2}$ flow on a compact
smooth manifold. For discrete time case, ergodic optimization theory has been
developed broadly. Among them, Yuan and Hunt proposed an open problem in [YH,
Conjecture 1.1] on 1999, which provides a mathematical mechanism on Hunt and
Ott’s experimental and heuristic results in [HO2, HO3] and becomes one of the
fundamental questions raised in the field of ergodic optimization theory. A
more general form of Yuan and Hunt’s conjecture is now called the Typical
Periodic Optimization Conjecture, and has attracted sustained attentions and
yielded considerable results, for instances [BZ, Bousch_poisson,
Bousch_Walters, Bo4, Contreras_EO, CLT, Morris_entropy, QS]. For a more
comprehensive survey for the classical ergodic optimization theory, we refer
the reader to Jenkinson [Jenkinson_survey, Je2], to Bochi [B], to Baraviera,
Leplaideur, Lopes [BLL], and to Garibaldi [G] for a historical perspective of
the development in this area. In our recent paper [HLMXZ], we extend the
applicability of the theory both to a broader class of systems and to a
broader class of observables, which leads to a positive answer to Yuan and
Hunt’s conjecture for $C^{1}$-observable case. To our knowledge, because of
difficulties appears on both conceptual level and technical level, there is no
existing result of ergodic optimization theory for flows so far, which make
the results obtained in the present paper the first achievement on flows
towards ergodic optimization theory.

On the other hand, as mentioned in [HLMXZ], the reason of adding the non-
constant weight $\psi$ mainly lies in the studies on the zero temperature
limit (or ground state) of the $(u,\psi)$-weighted equilibrium state for
thermodynamics formalism (for more details, we refer readers to works [BF,
BCW, FH]).

Summary of the results. To avoid unnecessary complexity, we only introduce the
result in the framework of standard ergodic optimization theory. Consider a
$C^{2}$ flow $\Phi$ on a compact smooth manifold $M$. Let $\Lambda$ be an
Axiom A attractor of $\Phi$ (detailed definition is given by Definition 2.1).
For a given observable $u:M\to\mathbb{R}$, the ergodic averages of $u$ on
$\Lambda$ is defined by the integration of $u$ with respect to
$\Phi|_{\Lambda}$-ergodic measures, and the $u$-maximizing (or minimizing)
measure is the measure with respect to which the ergodic average of $u$ takes
maximum (or minimum) value. As a consequence of the main result (Theorem 2.2)
of the present paper, we have the following result:

Theorem A: Let $(\Lambda,\Phi)$ be an Axiom A attractor on a compact smooth
Riemannian manifold $M$, then for a generic observable $u$ in function space
$\mathcal{C}^{0,\alpha}(M)$ or $\mathcal{C}^{1}(M)$, the $u$-maximizing (or
$u$-minimizing) measure is unique and is supported on a periodic orbit.

Remarks on techniques of the proof. It seems that the proof given in [HLMXZ]
provides a more general mechanism in the study on ergodic optimization
problems, which also shed a light on the case of flows for sure. However, the
results of the present paper depend crucially on the continuous time nature of
the system; that is to say, they do not follow from the properties of their
time-1 maps. Therefore, we must build certain theoretical base and create
certain new techniques to address issues raised in the case of flows.

We mention three differences of note between our setting and the existing
literatures at both conceptual level and technical level which pervade the
arguments in this paper: (1) At conceptual level, the most significant issue
is that the gap function of a discrete time periodic orbit, i.e. the minimum
separation of finite isolated points, is not well defined for continuous
periodic orbit. Such a gap function plays a key role in the proof of [HLMXZ].
(2) The presence of shear, i.e. the sliding of some orbits past other nearby
orbits due to the slightly different speed at which they travel, is a typical
phenomenon of continuous time systems, which causes tremendous amount of ”tail
estimates” throughout this paper. (3) Several main fundamental theoretical
tools are not existing and need to be rebuilt from the base, such as Anosov
Closing Lemma, Mañé-Conze-Guivarc’h-Bousch’s Lemma and Periodic Approximation
Lemma.

Structure of this paper. In Section 2, we set up the theoretic model and state
the main results; In Section 3, we state (without proofs) some well known
properties of Axiom A attractors, and some theoretical tools including Anosov
Closing Lemma, Mañé-Conze-Guivarc’h-Bousch’s Lemma and Periodic Approximation
Lemma preparing for the proof the main results; In Section 4, we give the
proof of Theorem 2.2, of which proving Part I) of the Theorem costs most
efforts; As follows, we leave the proofs of all the technical lemmas to
Section 5. On one hand, readers may go through the main proof by assuming the
validity of these technical lemmas without extra interruptions; on the other
hand, these technical lemmas with their proofs may be of independent interest.
Finally, we discuss the case when observables have higher regularity in
Section 6 in which only some partial results are presented.

##  2\. main setting and results

Let $M$ be a compact smooth Riemannian manifold with Riemannian metric $d$ and
$\Phi=\\{\phi_{t}:M\to M\\}_{t\in\mathbb{R}}$ be a $C^{2}$ flow on $M$.

######  Definition 2.1.

For $\Lambda\subset M$, $(\Lambda,\Phi)$ is called an Axiom A attractor if the
following conditions hold:

  * A1)

$\Lambda$ is a nonempty $\Phi$-invariant compact set.

  * A2)

There exists an $\epsilon_{0}>0$ such that for any $x\in M$ with
$d(x,\Lambda)<\epsilon_{0}$

$\lim_{t\to\infty}d\left(\phi_{t}(x),\Lambda\right)=0.$

  * A3)

There exist $\lambda_{0}>0$, $C_{0}>1$ and a continuous splitting of
tangential spaces of $M$ restricted on $\Lambda$, $T_{x}M=E^{u}_{x}\oplus
E^{c}_{x}\oplus E^{s}_{x}\ \forall\ x\in\Lambda$, such that the following hold

$\displaystyle(D_{M}\phi_{t})_{x}(E^{\tau}_{x})=E^{\tau}_{\phi_{t}(x)},\
\tau=u,c,s,\ \forall t\in\mathbb{R}\text{ and }x\in\Lambda,$
$\displaystyle\max\left\\{\|(D_{M}\phi_{-t})_{x}|_{E^{u}_{x}}\|,\|(D_{M}\phi_{t})_{x}|_{E^{s}_{x}}\|\right\\}\leq
C_{0}e^{-t\lambda_{0}},\ \forall t\in\mathbb{R}^{+},$

where $(D_{M}\phi_{t})_{x}$ is the derivative of the time-$t$ map $\phi_{t}$
on $x$ with respect to space variables.

  * A4)

$\inf_{x\in\Lambda}\left\|\frac{d\phi_{t}(x)}{dt}\right\|>0\text{ and
}E^{c}_{x}=span\left\\{\frac{d}{dt}\phi_{t}(x)\right\\},\ \forall
x\in\Lambda$.

Denote by $\mathcal{M}(\Lambda,\Phi)$ the set of all $\Phi$-invariant Borel
probability measures on $\Lambda$, which is a non-empty convex and compact
topological space with respect to weak∗ topology. Denote by
$\mathcal{M}^{e}(\Lambda,\Phi)\subset\mathcal{M}(\Lambda,\Phi)$ the set of
ergodic measures, which is the set of the extremal points of
$\mathcal{M}(\Lambda,\Phi)$. Let $u:M\to\mathbb{R}$ and
$\psi:M\to\mathbb{R}^{+}$ be continuous functions. The quantity
$\beta(u;\psi,\Lambda,\Phi)$ being defined by

$\beta(u;\psi,\Lambda,\Phi):=\min_{\nu\in\mathcal{M}(\Lambda,\Phi)}\frac{\int
ud\nu}{\int\psi d\nu},$ (2.1)

is called the _ratio minimum ergodic average_ , and any
$\nu\in\mathcal{M}(\Lambda,\Phi)$ satisfying

$\frac{\int ud\nu}{\int\psi d\nu}=\beta(u;\psi,\Lambda,\Phi)$

is called a _$(u,\psi)$ -minimizing measure_. Denote that

$\mathcal{M}_{\min}(u;\psi,\Lambda,\Phi):=\left\\{\nu\in\mathcal{M}(\Lambda,\Phi):\frac{\int
ud\nu}{\int\psi d\nu}=\beta(u;\psi,\Lambda,\Phi)\right\\}.$

By compactness of $\mathcal{M}(\Lambda,\Phi)$, and the continuity of the
operator $\frac{\int ud(\cdot)}{\int\psi d(\cdot)}$, it directly follows that
$\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)\neq\emptyset$, which contains at least
one ergodic $(u,\psi)$-minimizing measure by ergodic decomposition.

For $\alpha\in(0,1]$, let $\mathcal{C}^{0,\alpha}(M)$ be the space of
$\alpha$-Hölder continuous real-valued functions on $M$ endowed with the
$\alpha$-Hölder norm $\|u\|_{\alpha}:=\|u\|_{0}+[u]_{\alpha}$, where
$\|u\|_{0}:=\sup_{x\in M}|u(x)|$ is the super norm, and
$[u]_{\alpha}:=\sup_{x\neq y}\frac{|u(x)-u(y)|}{(d(x,y))^{\alpha}}$. Also note
that when $\alpha=1$, $C^{0,1}(M)$ becomes the collection of all real-valued
Lipschitz continuous functions, and $[u]_{1}$ becomes the minimum Lipschitz
constant of $u$. Additionally, denote by $\mathcal{C}^{1,0}(M)$ the Banach
space of continuous differentiable functions on $M$ endowed with the standard
$C^{1}$-norm.

In this paper, we consider the weighted ergodic optimization problem and
derive the following result.

######  Theorem 2.2.

Let $M$ be a compact smooth Riemannian manifold with Riemannian metric $d$ and
$\Phi$ be a $C^{2}$ flow on $M$. Suppose that $(\Lambda,\Phi)$ is an Axiom A
attractor, then the following hold:

  * I)

For $\alpha\in(0,1]$, given a $\psi\in\mathcal{C}^{0,\alpha}(M)$ with
$\inf_{x\in M}\psi(x)>0$, then there exists an open and dense set
$\mathfrak{P}\subset\mathcal{C}^{0,\alpha}(M)$ such that for any
$u\in\mathfrak{P}$, the $(u|_{\Lambda},\psi|_{\Lambda})$-minimizing measure of
$(\Lambda,\Phi)$ is unique and is supported on a periodic orbit of $\Phi$.

  * II)

For $\psi\in\mathcal{C}^{0,1}(M)$ with $\inf_{x\in M}\psi(x)>0$, there exists
an open and dense set $\mathfrak{P}\subset\mathcal{C}^{1,0}(M)$ such that for
any $u\in\mathfrak{P}$, the $(u|_{\Lambda},\psi|_{\Lambda})$-minimizing
measure of $(\Lambda,\Phi)$ is unique and is supported on a periodic orbit of
$\Phi$.

We remark here that $M,\Lambda,\Phi$ are assumed to satisfy conditions in
Theorem 2.2 throughout the rest of this paper.

##  3\. Properties of Axiom A attractors

This section devotes to building theoretic tools as preparations for the proof
of Theorem 2.2.

###  3.1. Invariant Manifolds

For a point $x\in\Lambda$ and $\epsilon>0$ the local stable and unstable sets
are defined by

$\displaystyle W_{\epsilon}^{s}(x)=$ $\displaystyle\\{y\in
M:d(\phi_{t}(x),\phi_{t}(y))\leq\epsilon\ \forall t\geq 0,\
d(\phi_{t}(x),\phi_{t}(y))\to 0\text{ as }t\to+\infty\\},$ $\displaystyle
W_{\epsilon}^{u}(x)=$ $\displaystyle\\{y\in
M:d(\phi_{-t}(x),\phi_{-t}(y))\leq\epsilon\ \forall t\geq 0,\
d(\phi_{-t}(x),\phi_{-t}(y))\to 0\text{ as }t\to+\infty\\}.$

The following Lemma is a standard result of invariant manifolds in existing
literature, of which the proof is omitted.

######  Lemma 3.1.

For any $\lambda_{1}\in(0,\lambda_{0})$, there exists $\epsilon_{1}>0$ and
$C_{1}\geq 1$ such that for any $\epsilon\in(0,\epsilon_{1}]$, the following
hold:

  * i)

$W^{s}_{\epsilon}(x),W^{u}_{\epsilon}(x)$ are $C^{2}$ embedded discs for all
$x\in\Lambda$ with $T_{x}W_{\epsilon}^{\tau}(x)=E^{\tau}_{x}$, $\tau=u,s$;

  * ii)

$d(\phi^{t}(x),\phi^{t}(y))\leq C_{1}e^{-t\lambda_{1}}d(x,y)$ for $y\in
W^{s}_{\epsilon}(x)$, $t\geq 0$, and
$d(\phi^{-t}(x),\phi^{-t}(y))\leq C_{1}e^{-t\lambda}d(x,y)$ for $y\in
W^{u}_{\epsilon}(x)$, $t\geq 0$;

  * iii)

$W^{s}_{\epsilon}(x),W^{u}_{\epsilon}(x)$ vary continuously with respect to
$x$ (in $C^{1}$ topology).

By choosing the Riemannian metric, the Axiom A flow in Theorem 2.2 meets the
following basic canonical setting : There are positive constants
$\delta,\epsilon,\beta,\lambda,C$ with $C\geq 1$ and
$\delta\ll\epsilon\ll\min\\{\epsilon_{0},\epsilon_{1}\\}$, where
$\epsilon_{0}$ is as in A2) of the definition of Axiom A attractors and
$\epsilon_{1}$ is as in Lemma 3.1, such that:

  * (1)

For $x,y\in M$ with $d(x,y)\leq\delta$, there is a unique time $v=v(x,y)$ with
$|v|\leq Cd(x,y)$ such that

    * (a)

$W^{s}_{\epsilon}(\phi_{v}(x))\cap W^{u}_{\epsilon}(y)$ is not empty and
contains only one element which is noted by $w=w(x,y)$.

    * (b)

$d(x,y)\geq C^{-1}\max\\{d(\phi_{v}(x),w),d(y,w),d(\phi_{v}(x),x),d(w,x)\\}.$

  * (2)

For $x\in M$ , $y\in W_{\epsilon}^{u}(x)$ and $t\geq 0$,
$d(\phi_{-t}x,\phi_{-t}y)\leq Ce^{-\lambda t}d(x,y)$,
For $x\in M$ , $y\in W_{\epsilon}^{s}(x)$ and $t\geq 0$,
$d(\phi_{t}x,\phi_{t}y)\leq Ce^{-\lambda t}d(x,y)$.

  * (3)

For $x,y\in M$, $d(\phi_{t}x,\phi_{t}y)\leq Ce^{\beta|t|}d(x,y)$ for all
$t\in\mathbb{R}$.

######  Remark 3.2.

In our following text, $\delta,\epsilon,\lambda,\beta,C$ are the positive
constants as above. Additionally, for convenience, we assume $C\gg
1,0<\delta\ll\epsilon\ll 1$. Otherwise, we set a positive constant
$\epsilon^{\prime}$ such that
$\epsilon^{\prime}\ll\frac{\epsilon}{C^{10}e^{10\beta}}$. We set another
positive constant $\delta^{\prime}$ with $\delta^{\prime}\ll\delta$ such that
for any $x,y\in M$ with
$d(x,y)\leq\frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta^{\prime}$,
there is an unique time $v=v(x,y)$ with $|v|\leq Cd(x,y)$ such that
$W^{s}_{\epsilon^{\prime}}(\phi_{v}(x))\cap W^{u}_{\epsilon^{\prime}}(y)$ is
not empty and contains only one element.

######  Remark 3.3.

For proofs and more details of Lemma 3.1 and the basic canonical setting, we
refer readers to [PSh], [Bowen], and [BR]. The only property which is not
appearing in the above references is the following inequality

$|v(x,y)|\leq Cd(x,y)$ (3.1)

appearing in (1) of basic canonical setting. We remark here that this
inequality holds when $\Phi$ is $C^{2}$. When $\Phi$ is $C^{1+\alpha}$ for
some $\alpha\in(0,1]$, the above inequality will be replaced by $|v(x,y)|\leq
Cd^{\alpha}(x,y)$ which is still sufficient for the proof of this paper
(although necessary modifications are required). This concludes that the
Theorem 2.2 still holds for $C^{1+\alpha}$ flows.

Finally, to our knowledge, there is no explicit statement equivalent to (3.1)
in existing literature. Nevertheless, (3.1) can be proved by combining Lemma
6, Proposition 8, Proposition 9 and Lemma 13 from [LY]. Since (3.1) is
intuitively natural but at the same time the proof involves considerable
technical complexity, we decide not to put the detailed proof in this paper
for the sake of simplicity.

###  3.2. Anosov Closing Lemma

Let $\delta^{\prime},\epsilon^{\prime},\delta,\epsilon$, $\lambda,\beta,C$ be
the constants as in Remark 3.2. Then we have the following Lemma.

######  Lemma 3.4.

Given
$\eta\leq\frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta^{\prime}$ and
$T>0$, if $x,y\in\Lambda$ and continuous function $s:\mathbb{R}\to\mathbb{R}$
with $s(0)=0$ satisfy

$d(\phi_{t+s(t)}(y),\phi_{t}(x))\leq\eta\text{ for }t\in[0,T],$

then for all $t\in[0,T]$, the following hold:

  * ASh1)

$|s(t)|\leq 2C\eta$;

  * ASh2)

$d(\phi_{t}\phi_{v(y,x)}(y),\phi_{t}(x))\leq
C^{2}e^{-\lambda\min(t,T-t)}\left(d(y,x)+d\left(\phi_{T}(y),\phi_{T}(x)\right)\right)$,
where $v(y,x)$ is as in Remark 3.2 satisfying $|v(y,x)|\leq Cd(x,y)$.

Especially, one has that

  * (1)

If $d(\phi_{t+s(t)}(y),\phi_{t}(x))\leq\eta$ for all $t\geq 0,$ then

$d(\phi_{t}\phi_{v(y,x)}(y),\phi_{t}(x))\to 0\text{ as }t\to+\infty.$

  * (2)

If $d(\phi_{t+s(t)}(y),\phi_{t}(x))\leq\eta$ for all $t\in\mathbb{R},$ then
$\phi_{v(y,x)}(y)=x.$

A segment of $\Phi$ is a curve $\mathcal{S}:[a,b]\to M:t\to\phi_{t}(x)$ for
some $x\in M$ and real numbers $a\leq b$. We denote the left endpoint of
$\mathcal{S}$ by $\mathcal{S}^{L}=\phi_{a}(x)$, the right endpoint of
$\mathcal{S}$ by $\mathcal{S}^{R}=\phi_{b}(x)$ and the length of $\mathcal{S}$
by $|\mathcal{S}|=b-a$. By a segment $\mathcal{S}$, if
$\mathcal{S}^{L}=\mathcal{S}^{R}$, we say $\mathcal{S}$ is a periodic segment.
We have the following version of Anosov Closing Lemma.

######  Lemma 3.5 (Anosov Closing Lemma).

There are positive constants $L$ and $K$ depending on the system constants
only such that if segment $\mathcal{S}$ of $\Phi|_{\Lambda}$ satisfy

  * (a)

$|\mathcal{S}|\geq K$;

  * (b)

$d(\mathcal{S}^{L},\mathcal{S}^{R})\leq\delta^{\prime}$.

Then, there is a periodic segment $\mathcal{O}$ such that

$\left||\mathcal{S}|-|\mathcal{O}|\right|\leq
Ld(\mathcal{S}^{L},\mathcal{S}^{R})$

and

$d(\phi_{t}(\mathcal{O}^{L}),\phi_{t}(\mathcal{S}^{L}))\leq
Ld(\mathcal{S}^{L},\mathcal{S}^{R})\text{ for all }0\leq
t\leq\max(|\mathcal{S}|,|\mathcal{O}|).$

######  Remark 3.6.

In the following text, we also use $\mathcal{S}$ (so is $\mathcal{O}$ and
$\mathcal{Q}$) to represnet the collection of points
$\phi_{t}(\mathcal{S}^{L}),0\leq t\leq|\mathcal{S}|$ as no confusion being
caused. By Lemma 3.4 and the choices of $\epsilon$ and $\delta$, $\mathcal{O}$
clearly belongs to $\Lambda$.

###  3.3. Mañé-Conze-Guivarc’h-Bousch’s Lemma

For $\gamma\in\mathbb{R}\setminus\\{0\\}$ and continuous function
$u:M\to\mathbb{R}$, define that

$u_{\gamma}(x):=\frac{1}{\gamma}\int_{0}^{\gamma}u(\phi_{t}(x))dt.$ (3.2)

######  Lemma 3.7 (Mañé-Conze-Guivarc’h-Bousch’s Lemma).

For $0<\alpha\leq 1$ and $N_{0}>0$, there exists a positive constant
$\gamma=\gamma(\alpha)>N_{0}$ such that if $u\in\mathcal{C}^{0,\alpha}(M)$
satisfies $\beta(u;1,\Lambda,\Phi|_{\Lambda})\geq 0$, then there is a
$v\in\mathcal{C}^{0,\alpha}(\Lambda)$ such that $u_{\gamma}|_{\Lambda}\geq
v\circ\phi_{\gamma}|_{\Lambda}-v.$

######  Remark 3.8.

We remark that the key point of Lemma 3.7 lies in the fact that $v$ is chosen
with the same Hölder exponent as $u$. Indeed, there were a number of weak
versions of Lemma 3.7 in the setting of smooth Anosov flows without fixed
points, or certain expansive non-Anosov geodesic flows, where $v$ is still
Hölder, but the Hölder exponent is less than $\alpha$ (for details, see [LRR,
LT, PR]).

By using Lemma 3.7, we have the following Lemma.

######  Lemma 3.9.

For $0<\alpha\leq 1$, there exists large $\gamma=\gamma(\alpha)$ such that,
for $u\in\mathcal{C}^{0,\alpha}(M)$ and strictly positive
$\psi\in\mathcal{C}^{0,\alpha}(M)$, there is a
$v\in\mathcal{C}^{0,\alpha}(\Lambda)$ such that

  * (1)

$u_{\gamma}|_{\Lambda}-v\circ\phi_{\gamma}|_{\Lambda}+v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}|_{\Lambda}\geq
0;$

  * (2)

$Z_{u,\psi}\subset\left\\{x\in\Lambda:\left(u_{\gamma}|_{\Lambda}+v\circ\phi_{\gamma}|_{\Lambda}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}|_{\Lambda}\right)(x)=0\right\\},$

where
$Z_{u,\psi}=\overline{\cup_{\mu\in\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)}supp(\mu)}.$

######  Remark 3.10.

For convenience, in the following text, if we need to use Lemma 3.9, we use
$\bar{u}$ to represent
$u_{\gamma}|_{\Lambda}+v\circ\phi_{\gamma}|_{\Lambda}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}|_{\Lambda}$
for short. Then, $\bar{u}\geq 0$ and
$Z_{u,\psi}\subset\\{x\in{\Lambda}:\bar{u}(x)=0\\}.$

###  3.4. Periodic Approximation

For $\alpha\in(0,1]$, $Z\subset M$ and a segment $\mathcal{S}$ of $\Phi$, we
define the $\alpha$-deviation  of $\mathcal{S}$ with respect to $Z$ by

$d_{\alpha,Z}(\mathcal{S})=\int_{0}^{|\mathcal{S}|}d^{\alpha}\left(\phi_{t}\left(\mathcal{S}^{L}\right),Z\right)dt.$

For $P\geq 0$, using $\mathcal{O}^{P}_{\Lambda}$ denote the collection of
periodic segments in $\Lambda$ with length not larger than $P$. Now we have
the following version of Quas and Bressaud’s periodic approximation Lemma.

######  Lemma 3.11.

Let $Z\subset\Lambda$ be a $\Phi$-invariant compact subset of $\Lambda$. Then,
for all $\alpha\in(0,1],k\geq 0$,

$\lim_{P\to+\infty}P^{k}\min_{\mathcal{S}\in\mathcal{O}^{P}_{\Lambda}}d_{\alpha,Z}(\mathcal{S})=0.$

##  4\. Proof of Theorem 2.2.

This section contains two main subsections 4.1 and 4.2 corresponding to the
proofs of Part I and Part II of Theorem 2.2 respectively. Indeed, Proposition
4.7 in Subsection 4.1.3 plays the key role in the proof of Theorem 2.2, based
on which the Part I result follows immediately and the Part II result follows
in a straightforward way with the help of an approximation lemma. We also note
that throughout the whole section
$\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ are the
fixed constants as in Remark 3.2.

###  4.1. Proof of Part I) of Theorem 2.2

This section mainly contains three parts: 4.1.1, 4.1.2 and 4.1.3. As mentioned
above, Proposition 4.7 in Subsection 4.1.3 is the key to prove the main
theorem of this paper, proving which is the main aim of Section 4.1. While
Section 4.1.1 and 4.1.2 are devoted to building notions, tools and results as
preparations for the proof of Proposition 4.7, of which Section 4.1.1
investigate the basic properties of periodic orbits and Section 4.1.2
constructs periodic orbits with ”good shapes”.

####  4.1.1. Locking Property of Periodic Segments

In this subsection, we show that periodic segments have locking property in
some sense.

For $0<\eta\leq\delta$, the $\eta$-disk of $x$ is defined by

$\mathcal{D}(x,\eta)=\\{y\in\Lambda:d(x,y)\leq\eta,W^{s}_{\epsilon}(x)\cap
W^{u}_{\epsilon}(y)\neq\emptyset\\}.$ (4.1)

$\mathcal{D}(x,\eta)$ has the following properties:

  * (a)

$W^{s}_{\eta}(x)\subset\mathcal{D}(x,\eta)$ and
$\phi_{t}(W^{s}_{\eta}(x))\subset\mathcal{D}(\phi_{t}(x),Ce^{-\lambda t}\eta)$
for $t\geq 0$;

  * (b)

$W^{u}_{\eta}(x)\subset\mathcal{D}(x,\eta)$ and
$\phi_{t}(W^{u}_{\eta}(x))\subset\mathcal{D}(\phi_{t}(x),Ce^{\lambda t}\eta)$
for $t\leq 0$;

  * (c)

$\phi_{t}(\mathcal{D}(x,\eta))\subset\mathcal{D}\left(\phi_{t}(x),Ce^{\beta|t|}\eta\right)$
for $t\in\mathbb{R}$ satisfying $Ce^{\beta|t|}\eta<\delta$.

  * (d)

for $\eta\leq\frac{\delta}{C}$ and $x,y\in\Lambda$ with $d(x,y)\leq\eta$,
there exists a unique time $v=v(x,y)$ with $|v|\leq Cd(x,y)$ such that
$y\in\mathcal{D}(\phi_{v}(x),\delta)$. In fact, $v$ is the one given by the
basic canonical setting.

Now we define $D:\Lambda\times\Lambda\to[0,+\infty)$ by

$D(x,y)=\left\\{\begin{array}[]{ll}\delta^{\prime},&\text{ if
}y\notin\mathcal{D}(x,\delta^{\prime}),\\\ d(x,y),&\text{ if
}y\in\mathcal{D}(x,\delta^{\prime}).\end{array}\right.$ (4.2)

By a periodic segment $\mathcal{O}$ of $\Phi|_{\Lambda}$, we define the gap
of $\mathcal{O}$ by

$D(\mathcal{O})=\min_{x\in\mathcal{O},0<t<|\mathcal{O}|_{\min}}D(x,\phi_{t}(x)),$
(4.3)

where $|\mathcal{O}|_{\min}=\min\\{t>0|\phi_{t}(x)=x,x\in\mathcal{O}\\}$. In
[HLMXZ], the gap function of periodic orbits of a discrete time system plays
an irreplaceable role in identifying the optimal periodic measures. Here, the
gap of a periodic segment is an analogue of the gap function for discrete
periodic orbits, which reflects the geometric characteristic of the periodic
segment. Indeed, the definition of the gap of periodic segment is an
conceptual enhancement, which originates the study of the ergodic optimization
theory for the flow case.

Note that A4) of Definition 2.1 implies that $D(\mathcal{O})>0$ automatically,
therefore in the rest of this section we keep in mind that $D(\mathcal{O})>0$
without extra explanation.

Firstly, we present several technical Lemmas.

######  Lemma 4.1.

Let $\mathcal{O}$ be a periodic segment of $\Phi|_{\Lambda}$. If
$x,y\in\mathcal{O}$ satisfy $d(x,y)<\frac{D(\mathcal{O})}{C}$, then
$\phi_{v}(x)=y$ where $v=v(x,y)$.

###### Proof.

Let $\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ be the
constants as in Remark 3.2. Then,

$d(x,y)\leq\frac{D(\mathcal{O})}{C}<\delta^{\prime}\ll\delta.$

Hence, by the basic canonical setting, there is a constant $v=v(x,y)$ such
that

$y\in\mathcal{D}(\phi_{v}(x),Cd(x,y))\subset\mathcal{D}(\phi_{v}(x),\delta).$

If $\phi_{v}(x)\neq y$, then

$D(\mathcal{O})\leq d(\phi_{v}(x),y)\leq Cd(x,y)<D(\mathcal{O}),$

which is impossible. Thus, $\phi_{v}(x)=y$. This ends the proof. ∎

By a periodic segment $\mathcal{O}$ of $\Phi$, the periodic measure
$\mu_{\mathcal{O}}$ is defined by

$\mu_{\mathcal{O}}=\frac{1}{|\mathcal{O}|}\int_{0}^{|\mathcal{O}|}\delta_{\phi_{t}(\mathcal{O}^{L})}dt.$

By an ergodic measure $\mu\in\mathcal{M}^{e}(\Lambda,\Phi|_{\Lambda})$, a
point $x\in M$ is called a generic point of $\mu$ if the following holds

$\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}f(\phi_{t}(x))dt=\int fd\mu\text{
for all }f\in\mathcal{C}(M).$

The following Lemma shows that periodic segments have locking property in some
sense.

######  Lemma 4.2.

Let $\mathcal{O}$ be a periodic segment of $\Phi|_{\Lambda}$ and $x\in M$. If

$d(\phi_{t}(x),\mathcal{O})\leq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}\text{
for all }0\leq t\leq T,$ (4.4)

then there is a $y\in\mathcal{O}$ such that

$d(\phi_{t}(x),\phi_{t}(y))\leq Cd(\phi_{t}(x),\mathcal{O})\text{ for all
}0\leq t\leq T.$

Especially, if
$d(\phi_{t}(x),\mathcal{O})\leq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}$ for all
$t\geq 0$, then $x$ is a generic point of $\mu_{\mathcal{O}}.$

###### Proof.

Let $\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ be the
constants as in Remark 3.2. Take a positive constant $\theta\ll\delta$ such
that $d(\phi_{t}(z),z)\leq\frac{D(\mathcal{O})}{4C}$ for all $|t|\leq\theta$
and $z\in M$. By assumption (4.4), there are $y_{t}^{\prime}\in\mathcal{O}$
such that

$d(\phi_{t}(x),y_{t}^{\prime})=d(\phi_{t}(x),\mathcal{O})\leq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}\ll\delta\text{
for all }0\leq t\leq T.$

Let $y_{t}=\phi_{v(y_{t}^{\prime},\phi_{t}(x))}(y_{t}^{\prime})$ for $0\leq
t\leq T$. Then

$\displaystyle\phi_{t}(x)\in\mathcal{D}\left(y_{t},Cd\left(\phi_{t}(x),\mathcal{O}\right)\right)\subset\mathcal{D}\left(y_{t},\frac{D(\mathcal{O})}{4Ce^{\beta}}\right)\text{
for all }0\leq t\leq T.$ (4.5)

Fix $t_{1},t_{2}\in[0,T]$ with $|t_{1}-t_{2}|\leq\theta$. Then, by (4.5),

$\displaystyle d(y_{t_{1}},y_{t_{2}})$ $\displaystyle\leq
d(y_{t_{1}},\phi_{t_{1}}(x))+d(\phi_{t_{1}}(x),\phi_{t_{2}}(x))+d(\phi_{t_{2}}(x),y_{t_{2}})$
$\displaystyle\leq\frac{D(\mathcal{O})}{4Ce^{\beta}}+\frac{D(\mathcal{O})}{4C}+\frac{D(\mathcal{O})}{4Ce^{\beta}}<\frac{D(\mathcal{O})}{C}.$

Thus, by Lemma 4.1, there is a constant $\tau$ satisfying $|\tau|\leq
Cd(y_{t_{1}},y_{t_{2}})\ll\delta$ such that

$\phi_{\tau}(y_{t_{1}})=y_{t_{2}}.$

By the uniqueness of $v$ given in the basic canonical setting and the
smallness of both $\theta$ and $|\tau|$, one has $\tau=t_{2}-t_{1}$. Hence,

$y_{t_{2}}=\phi_{t_{2}-t_{1}}(y_{t_{1}})\text{ for all
}t_{1},t_{2}\in[0,T]\text{ with }|t_{1}-t_{2}|\leq\theta.$

Therefore, $y_{t}=\phi_{t}(y_{0})$ for all $t\in[0,T]$ by induction, which
implies that

$d(\phi_{t}(y_{0}),\phi_{t}(x))=d(y_{t},\phi_{t}(x))\leq
Cd(y_{t}^{\prime},\phi_{t}(x))=Cd(\phi_{t}(x),\mathcal{O})\text{ for all
}0\leq t\leq T.$

Thus, $y=y_{0}$ is the point as required.

Now we assume that
$d(\phi_{t}(x),\mathcal{O})\leq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}$ for all
$t\geq 0$. Then by the arguments above, there is $y\in\mathcal{O}$ such that

$d(\phi_{t}(x),\phi_{t}(y))\leq\frac{D(\mathcal{O})}{4Ce^{\beta}}\leq\delta^{\prime}\text{
for all }t\geq 0.$

Then by Lemma 3.4, we have

$d(\phi_{t}\phi_{v}(x),\phi_{t}(y))\to 0\text{ as }t\to+\infty,$

where $v=v(x,y)$. Then $x$ must be a generic point of $\mu_{\mathcal{O}}$.
This ends the proof. ∎

######  Lemma 4.3.

There is positive constant $\tau_{0}$ such that for any $x\in\Lambda$

$\phi_{t}(x)\notin\mathcal{D}(x,\delta)\text{ for all }0<|t|\leq\tau_{0}.$

###### Proof.

Let $\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ be the
constants as in Remark 3.2. Take $\tau_{0}>0$ small enough such that
$d(\phi_{t}(z),z)\leq\frac{\delta^{\prime}}{C^{2}}$ for all $|t|\leq\tau_{0}$
and $z\in M$. Suppose that there is an $x\in\Lambda$ and
$0<|\tau|\leq\tau_{0}$ such that $\phi_{\tau}(x)\in\mathcal{D}(x,\delta)$.
Note

$w=W^{s}_{\epsilon}(x)\cap W^{u}_{\epsilon}(\phi_{\tau}(x)).$

Then, by (1)(b) of the basic canonical setting, one has that

$\max\left\\{d(w,x),d(w,\phi_{\tau}(x))\right\\}\leq
Cd(x,\phi_{\tau}(x))\leq\frac{\delta^{\prime}}{C}.$

Then for $t\geq 0$,

$d(\phi_{t}(w),\phi_{t}(x))\leq Ce^{-\lambda t}d(w,x)\leq
Cd(w,x)\leq\frac{\delta^{\prime}}{C},$

and for $t<0$,

$d(\phi_{t}(w),\phi_{t}(x))\leq
d(\phi_{t}(w),\phi_{t}\phi_{\tau}(x))+d(\phi_{\tau}\phi_{t}(x),\phi_{t}(x))\leq\frac{\delta^{\prime}}{C}+\frac{\delta^{\prime}}{C^{2}}\leq\delta^{\prime},$

where we used $w=W^{s}_{\epsilon}(x)\cap W^{u}_{\epsilon}(\phi_{\tau}(x))$ and
the selection of $\tau$. Then by Lemma 3.4, there is a constant $l$ with
$|l|\ll\delta$ such that $w=\phi_{l}(x)=\phi_{l-\tau}\phi_{\tau}(x)$. It is
clear that at least one of $l$ and $l-\tau$ is not zero since $\tau\neq 0$.
Without loss of any generality, we assume that $l\neq 0$, then
$\\{x,\phi_{l}(x)\\}\subset W^{s}_{\epsilon}(x)$ (otherwise
$\\{\phi_{l}(x),\phi_{\tau}(x)\\}\subset W^{u}_{\epsilon}(\phi_{\tau}(x))$).
Thus

$W_{\epsilon}^{s}(x)\cap W^{u}_{\epsilon}(\phi_{l}(x))\neq\emptyset\text{ and
}W_{\epsilon}^{s}(x)\cap W^{u}_{\epsilon}(x)\neq\emptyset,$

which is impossible by the uniqueness of function $v$ given in the basic
canonical setting and A4) of Definition 2.1. This ends the proof. ∎

######  Remark 4.4.

Lemma 4.3 provides a lower bound $\tau_{0}$ of the periods of periodic
segments.

######  Remark 4.5.

We say a periodic segment $\mathcal{O}$ is pure if $\phi_{t}(y)\neq y$ for all
$y\in\mathcal{O}$ and $0<t<|\mathcal{O}|$. By Lemma 4.3, a periodic segment
$\mathcal{O}$ is pure if and only if $|\mathcal{O}|=|\mathcal{O}|_{\min}$.

####  4.1.2. Good periodic orbits

In this subsection, we mainly demonstrate that for a given compact invariant
set, there exist periodic segments being closed enough as well as with
reasonable large gap, which are the candidates to support certain minimizing
measures.

######  Proposition 4.6.

For any $\alpha\in(0,1]$, a given $\tilde{L}>0$ and $\Phi$-forward-invariant
non-empty subset $Z\subset\Lambda$ (i.e. $\phi_{t}(Z)\subset Z,\ \forall t\geq
0$), there exists a periodic segment $\mathcal{O}$ of $\Phi_{\Lambda}$ such
that

$\displaystyle\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z}(\mathcal{O})}>\tilde{L}.$
(4.6)

###### Proof.

Fix $0<\alpha\leq 1$, and recall that
$\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ are as in
Remark 3.2, and $K$, $L$ are as in Lemma 3.5. For the sake of convenience, we
assume that $K\geq L$ additionally. By Lemma 3.11, for any $k\in\mathbb{N}$,
there exists a periodic segment $\mathcal{O}_{0}$ of $\Phi|_{\Lambda}$ with
period $P_{0}$ large enough such that

$d_{\alpha,Z}(\mathcal{O}_{0})<P_{0}^{-k}<<\delta^{\prime}.$ (4.7)

We remark here that the period of a periodic segment is always assumed to be
the MINIMUM period, which will avoid unnecessary complexity without harming
the argument.

If $D^{\alpha}(\mathcal{O}_{0})>\tilde{L}d_{\alpha,Z}(\mathcal{O}_{0})$, the
proof is done. Otherwise, one has that

$D^{\alpha}(\mathcal{O}_{0})\leq\tilde{L}d_{\alpha,Z}(\mathcal{O}_{0})<\tilde{L}P_{0}^{-k}.$
(4.8)

Since $P_{0},k$ can be chosen as large as needed, one can request that
$D(\mathcal{O}_{0})<\delta^{\prime}$. Therefore, by definition of
$D(\mathcal{O}_{0})$ (see (4.3)), there exist $y\in\mathcal{O}_{0}$ and
$t_{0}\in(0,P_{0})$ such that

$\phi_{t_{0}}(y)\in\mathcal{D}(y,D(\mathcal{O}_{0})).$

Split the periodic segment $\mathcal{O}_{0}$ into two segments which are noted
by

$\displaystyle\mathcal{Q}^{1}_{0}:[0,t_{0}]\to\Lambda:t\to\phi_{t}(y);$
$\displaystyle\mathcal{Q}^{2}_{0}:[t_{0},P_{0}]\to\Lambda:t\to\phi_{t}(y).$

We choose the segment with smaller length and note it by $\mathcal{Q}_{0}$.
Then either
$\mathcal{Q}_{0}^{L}\in\mathcal{D}(\mathcal{Q}^{R},\delta^{\prime})$ or
$\mathcal{Q}_{0}^{R}\in\mathcal{D}(\mathcal{Q}^{L},\delta^{\prime})$. It is
clear that in either case

$d(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R})\leq\delta^{\prime}\text{ and
}d_{\alpha,Z}(\mathcal{Q}_{0})\leq d_{\alpha,Z}(\mathcal{O}_{0}).$

Next, we will estimate the increment of orbit deviation after orbit splitting.
We will employ different discussions for two different situations according to
the length of the segment for which we set $3K$ as a landmark.

Case 1. If the following condition holds

$|\mathcal{Q}_{0}|>3K,$ (4.9)

also note that
$d(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R})\leq\delta^{\prime}$, then Lemma
3.5 is applicable here, by which one has that there exists a periodic segment
$\mathcal{O}_{1}$ such that

$\left|\left|\mathcal{Q}_{0}\right|-\left|\mathcal{O}_{1}\right|\right|\leq
Ld(\mathcal{Q}_{1}^{L},\mathcal{Q}_{1}^{R})\leq
LD(\mathcal{O}_{0})<L\tilde{L}P_{0}^{-k},$ (4.10)
$d\left(\phi_{t}\left(\mathcal{Q}_{0}^{L}\right),\phi_{t}\left(\mathcal{O}_{1}^{L}\right)\right)\leq
Ld(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R})\ \forall
0<t<\max\\{|\mathcal{Q}_{0}|,|\mathcal{O}_{1}|\\}.$ (4.11)

Since $K\geq L$ and $\delta^{\prime}<<1$, (4.10) together with the assumption
$|\mathcal{Q}_{0}|>3K$ implies that

$|\mathcal{O}_{1}|\leq\frac{4}{3}|\mathcal{Q}_{0}|\leq\frac{2}{3}|\mathcal{O}_{0}|.$
(4.12)

$\displaystyle\begin{split}d_{\alpha,Z}(\mathcal{O}_{1})=&\int_{0}^{|\mathcal{O}_{1}|}d^{\alpha}\left(\phi_{t}(\mathcal{O}_{1}^{L}),Z\right)dt\\\
=&\int_{0}^{|\mathcal{O}_{1}|}d^{\alpha}\left(\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),Z\right)dt\\\
\leq&\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),Z\right)dt\\\
&+\int_{\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}^{|\mathcal{Q}_{0}|+\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}d^{\alpha}\left(\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),Z\right)dt\\\
\leq&\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),\phi_{t}(\mathcal{Q}_{0}^{L})\right)dt\quad\quad\quad\quad\quad{\bf\left(Int(a)\right)}\\\
&+\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{t}(\mathcal{Q}^{L}_{0}),Z\right)dt\quad\quad\quad\quad\quad\quad\quad\quad\quad\
\quad{\bf\left(Int(b)\right)}\\\
&+\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),Z\right)dt\quad\quad{\bf\left(Int(c)\right)}.\end{split}$

By applying Ash2) of Lemma 3.4, one has that

${\bf
Int(a)}\leq\int_{0}^{|\mathcal{Q}_{0}|}2C^{2}e^{-\lambda\min(t,|\mathcal{Q}_{0}|-t)}Ld^{\alpha}\left(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R}\right)dt\leq\frac{2(2C^{2}L)^{\alpha}}{\lambda\alpha}d^{\alpha}(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R}).$
(4.13)

By definition, one has that

${\bf Int(b)}=d_{\alpha,Z}(\mathcal{Q}_{0}).$ (4.14)

By (3) of the basic canonical setting, one has that

$\displaystyle\begin{split}{\bf
Int(c)}=&\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),\phi_{\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}Z\right)dt\\\
\leq&\left(Ce^{\beta\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}\right)^{\alpha}\int_{0}^{|\mathcal{Q}_{0}|}d^{\alpha}\left(\phi_{t}\phi_{v(\mathcal{O}^{L}_{1},\mathcal{Q}^{L}_{0})}(\mathcal{O}_{1}^{L}),Z\right)dt\\\
\leq&\left(Ce^{\beta\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|}\right)^{\alpha}\left({\bf
Int(a)+Int(b)}\right).\end{split}$ (4.15)

By taking $P_{0}$ and $k$ large, one is able to make
$\left||\mathcal{Q}_{0}|-|\mathcal{O}_{1}|\right|<1$. Therefore, one has the
following simplified estimate

$d_{\alpha,Z}(\mathcal{O}_{1})\leq
L_{1}d^{\alpha}(\mathcal{Q}_{0}^{L},\mathcal{Q}_{0}^{R})+L_{2}d_{\alpha,Z}(\mathcal{Q}_{0})\leq\hat{L}d_{\alpha,Z}(\mathcal{O}_{0}),$
(4.16)

where

$\displaystyle L_{1}$
$\displaystyle=\left(C^{\alpha}e^{\alpha\beta}+1\right)\frac{2(2C^{2}L)^{\alpha}}{\lambda\alpha};$
$\displaystyle L_{2}$ $\displaystyle=C^{\alpha}e^{\alpha\beta}+1;$
$\displaystyle\hat{L}$
$\displaystyle=L_{1}\tilde{L}+L_{2}=\left(C^{\alpha}e^{\alpha\beta}+1\right)\left(\frac{2(2C^{2}L)^{\alpha}}{\lambda\alpha}\tilde{L}+1\right).$

Once $d_{\alpha,Z}(\mathcal{O}_{1})>\tilde{L}d_{\alpha,Z}(\mathcal{O}_{1})$,
$\mathcal{O}_{1}$ is the periodic segment as required, the splitting process
stops. Otherwise, repeat the operation above as long as it is doable. Note
that such a process will stop at a finite time, since the operation above will
reduce the period of periodic segment at least $\frac{1}{3}$ by (4.12).
Therefore, under the assumption that the operation above is always doable, the
process will end on an periodic segment $\mathcal{O}_{m}$ for some
$m\in\mathbb{N}\cup\\{0\\}$, which either satisfies the requirement of this
Proposition or $|\mathcal{O}_{m}|\geq 3K$ while $|\mathcal{Q}_{m}|<3K$. In
either cases, one has that

$m\leq\frac{\log P_{0}-\log(3K)}{\log 1.5}+1,$

and

$d_{\alpha,Z}(\mathcal{O}_{i})\leq\hat{L}^{i}d_{\alpha,Z}(\mathcal{O}_{0}),\
\forall 1\leq i\leq m.$

In order to make each operation above doable, one needs that

$D(\mathcal{O}_{i})<\delta^{\prime},\ \forall 1\leq i\leq m-1,$

which can be done by assuming the largeness of $P_{0}$ and $k$ in advance. To
be precise, one can take

$k>\frac{\log\hat{L}}{\log 1.5}\text{ and }P_{0}^{\frac{\log\hat{L}}{\log
1.5}-k}<\frac{(\delta^{\prime})^{\alpha}}{\tilde{L}\hat{L}},$ (4.17)

where the second inequality above implies that for all $0\leq i\leq m-1$

$\tilde{L}d_{\alpha,Z}(\mathcal{O}_{i})\leq
P_{0}^{-k}\tilde{L}\hat{L}^{m}<(\delta^{\prime})^{\alpha},$

which ensures the existence of $\mathcal{O}_{i+1}$ and
$\tilde{L}d_{\alpha,Z}(\mathcal{O}_{m})<(\delta^{\prime})^{\alpha}$.

Case 2. We will deal with the case that $|\mathcal{Q}_{m}|\leq 3K$, which is
the counterpart of the case when (4.9) holds. We will show that by rearranging
extra largeness of $P_{0}$ and $k$, one can make $\mathcal{O}_{m}$ satisfy the
requirement of Proposition 4.6. We will prove this by contradiction.

Before going to further discussion, we should note first that the union of all
periodic orbits of $\Phi|_{\Lambda}$ with period $\leq 4K$ is a nonempty
compact subset of $\Lambda$, which is denoted by $Per_{4K}$. Once $Z\cap
Per_{4K}\neq\emptyset$ Proposition 4.6 holds automatically; otherwise, there
exists $\sigma>0$ such that

$d(x,Z)>\sigma\ \forall x\in Per_{4K}.$ (4.18)

Suppose that

$D^{\alpha}(\mathcal{O}_{m})\leq\tilde{L}d_{\alpha,Z}(\mathcal{O}_{m})<(\delta^{\prime})^{\alpha}.$
(4.19)

When

$K\leq|\mathcal{Q}_{m}|\leq 3K,$ (4.20)

by the exactly same argument as on $\mathcal{Q}_{0}$, one has that there
exists a periodic segment $\mathcal{O}_{m+1}$ such that

$|\mathcal{O}_{m+1}|\leq 4K\text{ and
}d_{\alpha,Z}(\mathcal{O}_{m+1})\leq\hat{L}d_{\alpha,Z}(\mathcal{O}_{m})\leq\hat{L}^{m+1}d_{\alpha,Z}(\mathcal{O}_{0})<\hat{L}^{m+1}P_{0}^{-k}.$

By choosing $P_{0}$ and $k$ large enough one can make
$d_{\alpha,Z}(\mathcal{O}_{m+1})<\sigma$ which implies an contradiction with
(4.18). Therefore (4.19) and (4.20) can not hold simultaneously for $P_{0}$
and $k$ large enough.

When

$|\mathcal{Q}_{m}|<K,$ (4.21)

Lemma 3.5 is not applicable directly. For sake of convenience, note
$l=|\mathcal{Q}_{m}|$. By (4.19), $\mathcal{Q}_{m}^{L}\in
D(\mathcal{Q}_{m}^{R},\delta^{\prime})$ or $\mathcal{Q}_{m}^{R}\in
D(\mathcal{Q}_{m}^{L},\delta^{\prime})$. Then by Lemma 4.3, $l>\tau_{0}$. Let
$q$ be the integer such that

$K\leq ql\leq 2K\text{ and thus }2\leq q\leq\left[\frac{2K}{\tau_{0}}\right].$

Note

$\mathcal{S}_{i}:[0,l]\to M:t\to\phi_{il+t}(\mathcal{Q}_{m}^{L})\text{ for
}i=0,1,2,\cdots,q-1,$

and

$\mathcal{S}:[0,ql]\to M:t\to\phi_{t}(\mathcal{Q}_{m}^{L}).$

Then,

$d(\mathcal{S}_{0}^{L},\mathcal{S}_{0}^{R})=d(\mathcal{Q}_{m}^{L},\mathcal{Q}_{m}^{R})$
$d(\mathcal{S}_{1}^{L},\mathcal{S}_{1}^{R})=d(\phi_{l}(\mathcal{S}_{0}^{L}),\phi_{l}(\mathcal{S}_{0}^{R}))\leq
Ce^{\beta l}d(\mathcal{S}_{0}^{L},\mathcal{S}_{0}^{R})\leq Ce^{\beta
K}d(\mathcal{Q}_{m}^{L},\mathcal{Q}_{m}^{R}),$ $\dots$
$d(\mathcal{S}_{q-1}^{L},\mathcal{S}_{q-1}^{R})\leq(Ce^{\beta
K})^{q-1}d(\mathcal{Q}_{m}^{L},\mathcal{Q}_{m}^{R}).$

Therefore,

$d(\mathcal{S}^{L},\mathcal{S}^{R})\leq\sum_{i=0}^{q-1}(Ce^{\beta
K})^{i}d(\mathcal{Q}_{m}^{L},\mathcal{Q}_{m}^{R})\leq\frac{(Ce^{\beta
K})^{\left[\frac{2K}{\tau_{0}}\right]}-1}{Ce^{\beta K}-1}D(\mathcal{O}_{m}),$

which together with (4.19) implies that

$d(\mathcal{S}^{L},\mathcal{S}^{R})\leq\frac{(Ce^{\beta
K})^{\left[\frac{2K}{\tau_{0}}\right]}-1}{Ce^{\beta
K}-1}\left(\tilde{L}d_{\alpha,Z}(\mathcal{O}_{m})\right)^{\frac{1}{\alpha}}\leq\frac{(Ce^{\beta
K})^{\left[\frac{2K}{\tau_{0}}\right]}-1}{Ce^{\beta
K}-1}\left(\tilde{L}\hat{L}^{m}P_{0}^{-k}\right)^{\frac{1}{\alpha}}.$ (4.22)

By taking $P_{0}$ and $k$ large enough, one can make
$d(\mathcal{S}^{L},\mathcal{S}^{R})<\delta^{\prime}$. Also note that
$|\mathcal{S}|\geq K$, then Lemma 3.5 is applicable to $\mathcal{S}$.
Therefore, there exists a periodic segment $\mathcal{O}_{*}$ such that
$|\mathcal{O}_{*}|\leq|\mathcal{S}|+Ld(\mathcal{S}^{L},\mathcal{S}^{R})\leq
3K$ and

$\displaystyle
d\left(\phi_{t}\phi_{v(\mathcal{O}_{*}^{L},\mathcal{S}^{L})}(\mathcal{O}_{*}^{L}),\phi_{t}(\mathcal{S}^{L})\right)\leq
L\frac{(Ce^{\beta K})^{\left[\frac{2K}{\tau_{0}}\right]}-1}{Ce^{\beta
K}-1}\left(\tilde{L}\hat{L}^{m}P_{0}^{-k}\right)^{\frac{1}{\alpha}}\ \forall
0\leq t\leq|\mathcal{S}|,$ (4.23)

where the right hand side of the above inequality can be make smaller than
$\frac{1}{3}\sigma$ by taking $P_{0}$ and $k$ large enough. On the other hand,

$d_{\alpha,Z}(\mathcal{S}_{0})=d_{\alpha,Z}(\mathcal{Q}_{m})$
$d_{\alpha,Z}(\mathcal{S}_{1})=\int_{0}^{l}d^{\alpha}(\phi_{l+t}(\mathcal{S}_{0}^{L}),Z)\leq(Ce^{\beta
l})^{\alpha}d_{\alpha,Z}(\mathcal{S}_{0})\leq(Ce^{\beta
K})^{\alpha}d_{\alpha,Z}(\mathcal{Q}_{m}),$ $\dots$
$d_{\alpha,Z}(\mathcal{S}_{q-1})\leq(Ce^{\beta
K})^{(q-1)\alpha}d_{\alpha,Z}(\mathcal{Q}_{m}).$

Thus,

$\displaystyle\begin{split}d_{\alpha,Z}(\mathcal{S})&=\sum_{i=0}^{q-1}d_{\alpha,Z}(\mathcal{S}_{i})\leq\sum_{i=0}^{q-1}(Ce^{\beta
K})^{i\alpha}d_{\alpha,Z}(\mathcal{Q}_{m})\\\
&\leq\sum_{i=0}^{[\frac{2K}{\tau_{0}}]-1}C^{i\alpha}e^{i\beta
K\alpha}d_{\alpha,Z}(\mathcal{O}_{m})\\\
&\leq\sum_{i=0}^{[\frac{2K}{\tau_{0}}]-1}C^{i\alpha}e^{i\beta
K\alpha}\hat{L}^{m}d_{\alpha,Z_{u}}(\mathcal{O}_{0})\\\ &\leq\frac{(Ce^{\beta
K})^{\left[\frac{2K}{\tau_{0}}\right]\alpha}-1}{\left(Ce^{\beta
K}\right)^{\alpha}-1}\hat{L}^{m}P_{0}^{-k},\end{split}$ (4.24)

which can be make smaller than $\frac{1}{3}\sigma$ by taking $P_{0}$ and $k$
large enough. Since $|\mathcal{S}|\geq K>1$, there is a point
$t_{*}\in[0,|\mathcal{S}|]$ such that

$d(\phi_{t_{*}}(\mathcal{S}^{L}),Z)\leq\frac{\sigma}{3}.$

Therefore, by (4.23), by taking $P_{0}$ and $k$ large enough, we have

$d(\phi_{t_{*}}\phi_{v_{*}}(\mathcal{O}_{*}^{L}),Z)\leq
d\left(\phi_{t_{*}}\phi_{v(\mathcal{O}_{*}^{L},\mathcal{S}^{L})}(\mathcal{O}_{*}^{L}),\phi_{t_{*}}(\mathcal{S}^{L}))+d(\phi_{t_{*}}(\mathcal{S}^{L}),Z\right)\leq\frac{2}{3}\sigma<\sigma$

which contradicts with (4.18) as $K\leq|\mathcal{S}|\leq 2K$.
Hence (4.19) can not hold for large enough $P_{0}$ and $k$. This ends the
proof. ∎

Here, we remark that there is no fixed point under the setting of this paper
by A4) of Definition 2.1, thus the orbit splitting process cannot stop at a
fixed point, which is the main difference comparing to the discrete time case
in [HLMXZ]. The Case 2. above is mainly taking care this issue.

####  4.1.3. Main Proposition

In this subsection, we state and prove our main proposition. For a continuous
function $u$ and a segment $\mathcal{S}$ of $\Phi$, define the integration of
$u$ along $\mathcal{S}$ with time interval $[a,b]$ and starting point $x$ by
the following

$\langle\mathcal{S},u\rangle:=\int_{a}^{b}u\left(\phi_{t}\left(\mathcal{S}^{L}\right)\right)dt,$
(4.25)

also recall that the definition of $u_{\gamma}$ for $\gamma>0$ by

$u_{\gamma}(x)=\frac{1}{\gamma}\int_{0}^{\gamma}u\left(\phi_{t}(x)\right)dt.$

Now we have the following proposition.

######  Proposition 4.7.

Given $0<\varepsilon\leq 1$, $0<\alpha\leq 1$, a strictly positive function
$\psi\in\mathcal{C}^{0,\alpha}(M)$ and $u\in\mathcal{C}^{0,\alpha}(M)$, if a
periodic segment $\mathcal{O}$ of $\Phi|_{\Lambda}$ satisfies the following
comparison condition

$\displaystyle\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z_{u,\psi}}(\mathcal{O})}>\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon\tau_{0}}+1+\frac{1}{\tau_{0}}\right)\frac{\|\bar{u}\|_{\alpha}\|\psi\|_{0}}{\psi_{min}}\cdot\frac{100(4C^{3}e^{2\beta})^{\alpha}}{\varepsilon},$
(4.26)

where $\bar{u}$ is defined in Remark 3.9 and $\tau_{0}$ is the constant in
Lemma 4.3, then the periodic measure

$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h;\psi,\Lambda,\Phi),$

where $h\in\mathcal{C}^{0,\alpha}(M)$ satisfying
$\|h\|_{\alpha}<10\varepsilon$ and

$\|h\|_{0}<\min\left\\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)^{\alpha}}{\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right)\frac{\|\psi\|_{0}+\psi_{min}}{\psi_{min}}},1\right\\}.$

###### Proof.

Fix $\varepsilon,\alpha,\mathcal{O},\psi,u,h$ as in the Proposition,
$\delta^{\prime},\epsilon^{\prime},\delta,\epsilon,\lambda,\beta,C$ as in
Remark 3.2, $\bar{u},Z_{u,\psi},\gamma,\psi_{\gamma}$ as in Remark 3.10,
$\tau_{0}$ as in Lemma 4.3. Note

$G=\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h-a_{\mathcal{O}}\psi_{\gamma}$

where

$\displaystyle\begin{split}a_{\mathcal{O}}&:=\frac{\left\langle\mathcal{O},\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h\right\rangle}{\left\langle\mathcal{O},\psi_{\gamma}\right\rangle}=\frac{\left\langle\mathcal{O},\bar{u}+h\right\rangle}{\left\langle\mathcal{O},\psi_{\gamma}\right\rangle}.\end{split}$

By straightforward computation, one has that

$\displaystyle\begin{split}\left|a_{\mathcal{O}}\right|\leq\frac{\left\langle\mathcal{O},\|\bar{u}\|_{\alpha}d^{\alpha}(\cdot,Z_{u,\psi})+h\right\rangle}{\left\langle\mathcal{O},\psi_{min}\right\rangle}\leq\frac{\|\bar{u}\|_{\alpha}d_{\alpha,Z_{u,\psi}}(\mathcal{O})}{|\mathcal{O}|\psi_{min}}+\frac{\|h\|_{0}}{\psi_{min}},\end{split}$
(4.27)

where we used $\bar{u}|_{Z_{u,\psi}}=0$. Notice that for all
$\mu\in\mathcal{M}(\Lambda,\Phi)$

$\displaystyle\frac{\int\left(u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+hd\right)\mu}{\int\psi d\mu}$
$\displaystyle=\frac{\int\left(u_{\gamma}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h\right)d\mu}{\int\psi_{\gamma}d\mu}$
$\displaystyle=\frac{\int\left(\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h\right)d\mu}{\int\psi_{\gamma}d\mu}+\beta(u;\psi,\Lambda,\Phi)$
$\displaystyle=\frac{\int Gd\mu}{\int\psi
d\mu}+a_{\mathcal{O}}+\beta(u;\psi,\Lambda,\Phi).$

Then, in order to show that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h;\psi,\Lambda,\Phi)$, it is enough to show that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(G;\psi_{\gamma},\Lambda,\Phi)$. Since
$\psi$ is strictly positive and $\int Gd\mu_{\mathcal{O}}=0$, it is enough to
show that

$\int Gd\mu\geq 0\text{ for all }\mu\in\mathcal{M}^{e}(\Lambda,\Phi).$ (4.28)

Define a compact set $\mathcal{R}\subset M$ by

$\mathcal{R}=\left\\{y\in
M:d(y,\mathcal{O})\leq\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}\right\\}.$

We have the following Claim.

Claim 1. $\mathcal{R}$ contains all $x\in M$ with $G(x)\leq 0$.

###### Proof of Claim 1..

Given $x\in M\setminus\mathcal{R}$, we are to show that $G(x)>0$. Note that

$\displaystyle\begin{split}\bar{u}+h-a_{\mathcal{O}}\psi_{\gamma}&\geq-|a_{\mathcal{O}}|\|\psi_{\gamma}\|_{0}-\|h\|_{0}.\end{split}$
(4.29)

where we used $\bar{u}\geq 0$ and $\|\psi_{\gamma}\|_{0}\leq\|\psi\|_{0}$.
Then

$\displaystyle G(x)$ $\displaystyle=\bar{u}(x)+\varepsilon
d^{\alpha}(x,\mathcal{O})+h(x)-a_{\mathcal{O}}\psi_{\gamma}$
$\displaystyle\geq\varepsilon
d^{\alpha}(x,\mathcal{O})-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}$
$\displaystyle>\varepsilon\cdot\left(\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}\right)^{\alpha}-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}$
$\displaystyle=0.$

This ends the proof of Claim 1. ∎

Define a compact set $\mathcal{R}^{\prime}\subset M$ by

$\mathcal{R}^{\prime}=\left\\{y\in
M:d(y,\mathcal{O})\leq\left(\frac{2(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0})}{\varepsilon}\right)^{\frac{1}{\alpha}}\right\\}.$

It is easy to see that $\mathcal{R}$ is in the interior of
$\mathcal{R}^{\prime}$ and the following holds because of (4.26), (4.27) and
the range of $\|h\|_{0}$

$d(y,\mathcal{O})\leq\left(\frac{2(a_{\mathcal{O}}\|\psi\|_{0}+\|h\|_{0})}{\varepsilon}\right)^{\frac{1}{\alpha}}\leq\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}},\
\forall y\in\mathcal{R}^{\prime}.$ (4.30)

By Claim 1, there is a constant $\tau$ with $0<\tau<1$ such that
$G(\phi_{t}(x))>0$ for all $x\in M\setminus\mathcal{R}^{\prime}$ and
$|t|\leq\tau$.

Now we claim the following assertion:
Claim 2. If $z\in M$ is not a generic point of $\mu_{\mathcal{O}}$, then there
is $m\geq\tau$ such that

$\int_{0}^{m}G(\phi_{t}(z))dt>0.$

Next we prove the Proposition by assuming the validity of Claim 2, while the
proof of Claim 2. is left to the end of this section. For a given ergodic
measure $\mu\in\mathcal{M}^{e}(\Lambda,\Phi)$, if $\mu=\mu_{\mathcal{O}}$,
(4.28) obviously holds. Otherwise, let $z$ be a generic point of $\mu$, thus
$z$ is not a generic point of $\mu_{\mathcal{O}}$. Therefore, by Claim 2,
there is a $t_{1}\geq\tau$ such that

$\int_{0}^{t_{1}}G(\phi_{t}(z))dt>0.$

Since $\phi_{t_{1}}(z)$ is still not a generic point of $\mu_{\mathcal{O}}$,
by using claim 2 agian, one has a $t_{2}\geq t_{1}+\tau$ such that

$\int_{t_{1}}^{t_{2}}G(\phi_{t}(z))dt>0.$

By repeating the above process, one has $0\leq t_{1}<t_{2}<t_{3}<\cdots$ with
all gaps not less than $\tau$ such that

$\int_{t_{i}}^{t_{i+1}}G(\phi_{t}(z))dt>0\text{ for }i=0,1,2,3,\cdots,$

where we assign $t_{0}=0$. Therefore

$\displaystyle\int Gd\mu$
$\displaystyle=\lim_{l\to+\infty}\frac{1}{l}\int_{0}^{l}G(\phi_{t}(z))dt$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{t_{i}}\left(\int_{t_{0}}^{t_{1}}G(\phi_{t}(z))dt+\int_{t_{1}}^{t_{2}}G(\phi_{t}(z))dt+\cdots+\int_{t_{i-1}}^{t_{i}}G(\phi_{t}(z))dt\right)$
$\displaystyle\geq 0.$

Thus, $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h;\psi,\Lambda,\Phi)$. This ends the proof. ∎

######  Remark 4.8.

It is not difficult to see that for any $\varepsilon^{\prime}>\varepsilon$,
$\mu_{\mathcal{O}}$ is the unique measure in
$\mathcal{M}_{min}(u+\varepsilon^{\prime}d^{\alpha}(\cdot,\mathcal{O})+h;\psi,\Lambda,\Phi)$
whenever $\|h\|_{\alpha}<10\varepsilon$ and $\|h\|_{0}$ is sufficiently small.
The Proposition shows that there is an open set of $\mathcal{C}^{0,\alpha}(M)$
near $u$ such that these $\alpha$-Hölder functions in the open set has the
same unique minimizing measure with respect to $\psi$ being supported on a
periodic orbit.

###### Proof of Claim 2..

If $z\notin\mathcal{R}^{\prime}$, just take $m=\tau$, we have nothing to prove
since $G(\phi_{t}(z))>0$ for all $|t|\leq\tau$. Therefore, one needs only to
consider the case that $z\in\mathcal{R}^{\prime}$. Also note that, since $z$
is not a generic point of $\mu_{\mathcal{O}}$, Lemma 4.2 implies that the
following inequality

$d(\phi_{t}(z),\mathcal{O})\leq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}$

CANNOT hold for all $t\geq 0$. Thus there is an $m_{1}>0$ such that

$d(\phi_{m_{1}}(z),\mathcal{O})>\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}.$

Let $m_{2}>0$ be the smallest time such that

$\displaystyle
d(\phi_{m_{2}}(z),\mathcal{O})=\frac{D(\mathcal{O})}{4C^{2}e^{\beta}},$ (4.31)

where the existence of such $m_{2}$ is ensured by (4.30) and the continuity of
the flow. Then, by (3) of the basic canonical setting, one has the following

$d(\phi_{m_{2}-t}(z),\mathcal{O})>\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}},\
\forall 0<t\leq 1,$

which together with (4.30) implies that

$\displaystyle\phi_{m_{2}-t}(z)\notin\mathcal{R}^{\prime}\text{ for all
}0<t\leq 1.$ (4.32)

Thus

$\displaystyle\begin{split}\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt&=\int_{m_{2}-1}^{m_{2}}\bar{u}(\phi_{t}(z))+\varepsilon
d^{\alpha}(\phi_{t}(z),\mathcal{O})+h(\phi_{t}(z))-a_{\mathcal{O}}\psi_{\gamma}(\phi_{t}(z))dt\\\
&\geq\int_{m_{2}-1}^{m_{2}}\varepsilon
d^{\alpha}(\phi_{t}(z),\mathcal{O})-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}dt\\\
&\geq\varepsilon\cdot\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)^{\alpha}-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0},\\\
\end{split}$ (4.33)

where we used (4.29).

Now since $\mathcal{R}^{\prime}$ is compact, there is an $m_{3}$ which is the
largest time such that $0\leq m_{3}\leq m_{2}$ and
$\phi_{t}(z)\in\mathcal{R}^{\prime}$. By (4.32), it is clear that $m_{3}\leq
m_{2}-1$. Then by Claim 1, for all $m_{3}<t<m_{2}-1$

$\displaystyle G(\phi_{t}(z))>0,$ (4.34)

where we used the fact $\mathcal{R}\subset\mathcal{R}^{\prime}$. On the other
hand, since $m_{3}<m_{2}$, by the choice of $m_{2}$ (see (4.31)), one has that

$d(\phi_{t}(z),\mathcal{O})\leq\left(\frac{2(a_{\mathcal{O}}\|\psi\|_{0}+\|h\|_{0})}{\varepsilon}\right)^{\frac{1}{\alpha}}<\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}\text{
for all }0\leq t\leq m_{3}.$

Therefore, by Lemma 4.2, there is $y_{0}\in\mathcal{O}$ such that

$d(\phi_{t}(z),\phi_{t}(y_{0}))\leq
C\left(\frac{2(a_{\mathcal{O}}\|\psi\|_{0}+\|h\|_{0})}{\varepsilon}\right)^{\frac{1}{\alpha}}\leq\delta^{\prime}\text{
for all }t\in[0,m_{3}].$

By using Lemma 3.4, we have for all $0\leq t\leq m_{3}$,

$\displaystyle d(\phi_{t}\phi_{v(y_{0},z)}(y_{0}),\phi_{t}(z))\leq
2C^{2}e^{-\lambda\min(t,m_{3}-t)}C\left(\frac{2(a_{\mathcal{O}}\|\psi\|_{0}+\|h\|_{0})}{\varepsilon}\right)^{\frac{1}{\alpha}}.$

Hence,

$\displaystyle\begin{split}&\int_{0}^{m_{3}}d^{\alpha}(\phi_{t}(z),\phi_{t}\phi_{v(y_{0},z)}(y_{0}))dt\\\
\leq&\int_{0}^{m_{3}}\left(2C^{3}\left(\frac{2\left(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}\right)}{\varepsilon}\right)^{\frac{1}{\alpha}}\left(e^{-\lambda
t}+e^{-\lambda(m_{3}-t)}\right)\right)^{\alpha}dt\\\
\leq&\frac{4C^{3\alpha}}{\lambda\alpha}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}.\end{split}$

Therefore,

$\displaystyle\begin{split}&\int_{0}^{m_{3}}G(\phi_{t}(z))-G(\phi_{t}\phi_{v}(y_{0}))dt\\\
=&\int_{0}^{m_{3}}\bar{u}(\phi_{t}(z))+\varepsilon
d^{\alpha}(\phi_{t}(z),\mathcal{O})+h(\phi_{t}(z))-\bar{u}(\phi_{t+v}(y_{0}))-h(\phi_{t+v}(y_{0}))\\\
&-a_{\mathcal{O}}\big{(}\psi_{\gamma}(\phi_{t}(z))-\psi_{\gamma}(\phi_{t+v}(y_{0}))\big{)}dt\\\
\geq&\int_{0}^{m_{3}}\bar{u}(\phi_{t}(z))-\bar{u}(\phi_{t+v}(y_{0}))+h(\phi_{t}(z))-h(\phi_{t+v}(y_{0}))\\\
&-a_{\mathcal{O}}\big{(}\psi_{\gamma}(\phi_{t}(z))-\psi_{\gamma}(\phi_{t+v}(y_{0}))\big{)}dt\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+|a_{\mathcal{O}}|\|\psi_{\gamma}\|_{\alpha})\int_{0}^{m_{3}}d^{\alpha}(\phi_{t}(z),\phi_{t+v}(y_{0}))dt\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+|a_{\mathcal{O}}|\|\psi_{\gamma}\|_{\alpha})\cdot\frac{4C^{3\alpha}}{\lambda\alpha}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon},\end{split}$

where we write $v$ short for $v(y_{0},z)$. Also note that

$\displaystyle|a_{\mathcal{O}}|=\left|\frac{\left\langle\mathcal{O},\bar{u}+h\right\rangle}{\left\langle\mathcal{O},\psi_{\gamma}\right\rangle}\right|\leq\frac{\|\bar{u}\|_{0}+\|h\|_{0}}{\psi_{min}}\leq\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}.$

Thus, one has that

$\displaystyle\begin{split}&\int_{0}^{m_{3}}G(\phi_{t}(z))-G(\phi_{t}\phi_{v}(y_{0}))dt\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})\cdot\frac{4C^{3\alpha}}{\lambda\alpha}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon},\end{split}$
(4.35)

Rewrite $m_{3}=p|\mathcal{O}|+r$ for some nonnegative integer $p$ and real
number $0\leq r\leq|\mathcal{O}|$. By applying (4.29) and $\int
Gd\mu_{\mathcal{O}}=0$, one has that

$\displaystyle\begin{split}\int_{0}^{m_{3}}G(\phi_{t}\phi_{v}(y_{0}))dt&=\int_{m_{3}-r}^{m_{3}}G(\phi_{t}\phi_{v}(y_{0}))dt\geq-|\mathcal{O}|\cdot(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}).\\\
\end{split}$ (4.36)

Combining (4.27), (4.33), (4.34), (4.35) and (4.36), we have

$\displaystyle\int_{0}^{m_{2}}G(\phi_{t}(z))dt\geq\int_{0}^{m_{3}}G(\phi_{t}(z))dt+\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt$
$\displaystyle\text{by }(\ref{X-12})$ $\displaystyle=$
$\displaystyle\int_{0}^{m_{3}}\left(G(\phi_{t}(z))-G(\phi_{t+v}(y_{0}))\right)dt$
$\displaystyle+\int_{0}^{m_{3}}G(\phi_{t+v}(y_{0}))dt+\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt$
$\displaystyle\geq$
$\displaystyle-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})\cdot\frac{4C^{3\alpha}}{\lambda\alpha}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}$
$\displaystyle\text{by }(\ref{X-6})$
$\displaystyle-|\mathcal{O}|\cdot(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0})$
$\displaystyle\text{by }(\ref{X-7})$
$\displaystyle+\varepsilon\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)^{\alpha}-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}$
$\displaystyle\text{by }(\ref{X-5})$ $\displaystyle\geq$
$\displaystyle\varepsilon\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)^{\alpha}$
$\displaystyle-\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon|\mathcal{O}|}+1+\frac{1}{|\mathcal{O}|}\right)\frac{\|\bar{u}\|_{\alpha}\|\psi\|_{0}}{\psi_{min}}d_{\alpha,Z_{u,\psi}}(\mathcal{O})$
$\displaystyle-\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right)\frac{\|\psi\|_{0}+\psi_{min}}{\psi_{min}}\|h\|_{0}$
$\displaystyle\text{by }(\ref{X-1})$ $\displaystyle\geq$
$\displaystyle\varepsilon\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)^{\alpha}$
$\displaystyle-\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon\tau_{0}}+1+\frac{1}{\tau_{0}}\right)\frac{\|\bar{u}\|_{\alpha}\|\psi\|_{0}}{\psi_{min}}d_{\alpha,Z_{u,\psi}}(\mathcal{O})$
$\displaystyle-\left(\frac{4C^{3\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{\alpha})}{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right)\frac{\|\psi\|_{0}+\psi_{min}}{\psi_{min}}\|h\|_{0}$
$\displaystyle>$ $\displaystyle 0,$

where we used Remark 4.4 and condition (4.26). Therefore, $m=m_{2}$ is the
time as required since $m_{2}\geq 1\geq\tau$. This ends the proof of Claim 2.
∎

###  4.2. Proof of Part II) of Theorem 2.2

Firstly, we state a technical result on function approximation, which plays a
key role in proving Proposition 4.11. Proposition 4.11 can be viewed as a
$C^{1}$-version of Proposition 4.7 which implies the part II) of Theorem 2.2.

######  Theorem 4.9 ([GW]).

Let $M$ be a smooth compact manifold. Then
$\mathcal{C}^{\infty}(M)\cap\mathcal{C}^{0,1}(M)$ is Lip-dense in
$\mathcal{C}^{0,1}(M)$.

######  Remark 4.10.

In this theorem, $\mathcal{C}^{\infty}(M)\cap\mathcal{C}^{0,1}(M)$ is Lip-
dense in $\mathcal{C}^{0,1}(M)$ means that for any
$g_{1}\in\mathcal{C}^{0,1}(M)$ and $\varepsilon>0$ there is corresponding
$g_{2}\in\mathcal{C}^{\infty}(M)$ such that $\|g_{1}-g_{2}\|_{0}<\varepsilon$
and $\|g_{2}\|_{1}<\varepsilon+\|g_{1}\|_{1}$. Especially,
$\|D_{M}g_{2}\|_{0}<\varepsilon+\|g_{1}\|_{1}$, where $D_{M}g$ is the
derivative of function $g$ with respect to space variables.

######  Proposition 4.11.

Given $0<\varepsilon\leq 1$, a strictly positive $\psi\in\mathcal{C}^{1,0}(M)$
and $u\in\mathcal{C}^{1,0}(M)$, if a periodic segment $\mathcal{O}$ of
$\Phi|_{\Lambda}$ satisfies the following comparison condition

$\displaystyle
D(\mathcal{O})>\Big{(}\frac{4C^{3}(\|\bar{u}\|_{1}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{1})}{\lambda\varepsilon\tau_{0}}+1+\frac{1}{\tau_{0}}\Big{)}\frac{\|\bar{u}\|_{1}\|\psi\|_{0}}{\psi_{min}}\cdot\frac{400C^{3}e^{2\beta}}{\varepsilon}\cdot
d_{1,Z_{u,\psi}}(\mathcal{O}),$

where $\bar{u}$ is defined in Remark 3.9 and $\tau_{0}$ is the constant in
Lemma 4.3, then there is a function $w\in C^{\infty}(M)$ with
$\|w\|_{0}<2\varepsilon\cdot diam(M)$ and $\|D_{M}w\|_{0}<2\varepsilon$ such
that the probability measure

$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+w+h;\psi,\Lambda,\Phi),$

where $h$ is any $C^{1}$ function with $\|D_{M}h\|_{0}<5\varepsilon$ and

$\|h\|_{0}<\frac{1}{2}\cdot\min\left\\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)}{\left(\frac{4C^{3}(\|\bar{u}\|_{1}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{1})}{\lambda\varepsilon}+|\mathcal{O}|+1\right)\frac{\|\psi\|_{0}+\psi_{min}}{\psi_{min}}},1\right\\}.$

###### Proof.

By Theorem 4.9, there exists a function $w\in C^{\infty}$ such that

$\|D_{M}w\|_{0}<\|\varepsilon d(\cdot,\mathcal{O})\|_{1}+\varepsilon\leq
2\varepsilon$

and

$\|w-\varepsilon
d(\cdot,\mathcal{O})\|_{0}<\min\left(\frac{H}{2},\varepsilon\cdot
diam(M)\right),$

where

$H=\min\left\\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)}{\left(\frac{4C^{3}(\|\bar{u}\|_{1}+10\varepsilon+\frac{\|\bar{u}\|_{0}+1}{\psi_{min}}\|\psi_{\gamma}\|_{1})}{\lambda\varepsilon}+|\mathcal{O}|+1\right)\frac{\|\psi\|_{0}+\psi_{min}}{\psi_{min}}},1\right\\}.$

Next we show that $w$ is the function as required. Note that

$u+w+h=u+\varepsilon d(\cdot,\mathcal{O})+(w-\varepsilon
d(\cdot,\mathcal{O})+h).$

Notice that,

$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_{1}\leq\|D_{M}w\|_{0}+\|\varepsilon
d(\cdot,\mathcal{O})\|_{1}+\|h\|_{1}\leq
2\varepsilon+\varepsilon+5\varepsilon<10\varepsilon,$

and

$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_{0}\leq\|w-\varepsilon
d(\cdot,\mathcal{O})\|_{0}+\|h\|_{0}<\frac{H}{2}+\frac{H}{2}=H.$

Then by Proposition 4.7, we have that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+w+h;\psi,\Lambda,\Phi).$
Additionally,

$\|w\|_{0}<\|\varepsilon d(\cdot,\mathcal{O})\|_{0}+\varepsilon\cdot
diam(M)\leq 2\varepsilon\cdot diam(M).$

This ends the proof. ∎

######  Remark 4.12.

Let $\widetilde{w}\in\mathcal{C}^{1,0}(M)$ be such that
$\|\widetilde{w}\|_{1,0}<\varepsilon$, $\widetilde{w}|_{\mathcal{O}}=0$ and
$\widetilde{w}|_{M\setminus\mathcal{O}}>0$. Then $\mu_{\mathcal{O}}$ is the
unique measure in $\mathcal{M}_{min}(u+\widetilde{w}+w+h;\psi,\Lambda,\Phi)$
whenever $\|h\|_{1}<5\varepsilon$ and $\|h\|_{0}$ is sufficiently small. The
Proposition shows that there is an open set of $\mathcal{C}^{1,0}({M})$ near
$u$ such that functions in the open set have the same unique minimizing
measure with respect to $\psi$ and the measure supports on a periodic orbit.

##  5\. Proofs of Technical Lemmas

Note that throughout this section,
$\delta,\epsilon,\lambda,\beta,C,\epsilon^{\prime},\delta^{\prime}$ are same
as the ones in Remark 3.2.

###  5.1. Proof of Lemma 3.4

###### Proof.

We put a small positive constant $\tau$ with $\tau\ll 1$ such that
$|s(t_{1})-s(t_{2})|\leq\eta$ for all $|t_{1}-t_{2}|\leq\tau$ and
$t_{1},t_{2}\in[0,T].$ Since
$\eta\leq\frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta^{\prime}$,
for all $0\leq t\leq T,$ there exists $r(t)$ with $|r(t)|<C\eta$ such that

$\displaystyle
w(\phi_{t+s(t)+r(t)}(y),\phi_{t}(x))=W^{s}_{\epsilon^{\prime}}(\phi_{t+s(t)+r(t)}(y))\cap
W^{u}_{\epsilon^{\prime}}(\phi_{t}(x)).$ (5.1)

Then for $t^{\prime}\in[-\tau,\tau]$ and $t\in[\tau,T-\tau]$,

$\phi_{t^{\prime}}(w(\phi_{t+s(t)+r(t)}(y),\phi_{t}(x)))\in
W^{s}_{\epsilon^{\prime}}(\phi_{t+s(t)+r(t)+t^{\prime}}(y))\cap
W^{u}_{\epsilon^{\prime}}(\phi_{t+t^{\prime}}(x)).$

On the other hand, one has that

$w(\phi_{t+t^{\prime}+s(t+t^{\prime})+r(t+t^{\prime})}(y),\phi_{t+t^{\prime}}(x))=W^{s}_{\epsilon^{\prime}}(\phi_{t+t^{\prime}+s(t+t^{\prime})+r(t+t^{\prime})}(y))\cap
W^{u}_{\epsilon^{\prime}}(\phi_{t+t^{\prime}}(x)).$

Since

$|(t+t^{\prime}+s(t+t^{\prime})+r(t+t^{\prime}))-(t+s(t)+r(t)+t^{\prime})|\leq(2C+1)\eta\ll\delta,$

by the uniqueness of $v(\phi_{t+s(t)+r(t)+t^{\prime}}y,\phi_{t+t^{\prime}}x)$
given by (1) of the basic canonical setting, one has that

$t+t^{\prime}+s(t+t^{\prime})+r(t+t^{\prime})=t+s(t)+r(t)+t^{\prime},$

and

$w(\phi_{t+t^{\prime}+s(t+t^{\prime})+r(t+t^{\prime})}(y),\phi_{t+t^{\prime}}(x))=\phi_{t^{\prime}}(w(\phi_{t+s(t)+r(t)}(y),\phi_{t}(x))),$

for all $t^{\prime}\in[-\tau,\tau]$ and $t\in[\tau,T-\tau]$. Since $\tau$ can
be taken arbitrarily small, one has the following by induction

$\displaystyle s(t)+r(t)=s(\tau)+r(\tau)=s(0)+r(0)=r(0)=v(y,x),\ \forall
t\in[0,T].$ (5.2)

Thus

$|s(t)|\leq|r(t)|+|r(0)|\leq 2C\eta,$

and for all $t\in[\tau,T-\tau]$ and $t^{\prime}\in[-\tau,\tau]$

$w(\phi_{t+t^{\prime}+v(y,x)}(y),\phi_{t+t^{\prime}}(x))=\phi_{t^{\prime}}(w(\phi_{t+v(y,x)}(y),\phi_{t}(x))),$

which implies that

$\displaystyle
w(\phi_{t+v(y,x)}(y),\phi_{t}(x))=\phi_{t}(w(\phi_{v(y,x)}(y),x))\ \forall
t\in[0,T].$ (5.3)

Now, we prove Ash 2). Note
$w=w(\phi_{v(y,x)}(y),x)=W_{\epsilon^{\prime}}^{s}(\phi_{v(y,x)}(y))\cap
W_{\epsilon^{\prime}}^{u}(x)$. Then by (5.1), (5.2) and (5.3), one has

$\phi_{t}(w)=W^{s}_{\epsilon^{\prime}}(\phi_{t}\phi_{v(y,x)}(y))\cap
W^{u}_{\epsilon^{\prime}}(\phi_{t}(x))\text{ for all }t\in[0,T].$

Thus, for all $t\in[0,T]$, by (b) of the basic canonical setting,

$d(\phi_{t}(w),\phi_{t}\phi_{v(y,x)}(y))<Cd(\phi_{t}(x),\phi_{t}(y))\text{ and
}d(\phi_{t}(w),\phi_{t}(x))<Cd(\phi_{t}(x),\phi_{t}(y)).$

Therefore

$d(\phi_{t}(w),\phi_{t}(x))\leq
Ce^{-\lambda(T-t)}d(\phi_{T}(w),\phi_{T}(x))\leq
C^{2}e^{-\lambda(T-t)}d(\phi_{T}(x),\phi_{T}(y)),$

where we used $w\in W_{\epsilon^{\prime}}^{u}(x)$ and

$d(\phi_{t}(w),\phi_{t}\phi_{v(y,x)}(y))\leq
Ce^{-\lambda(t)}d(w,\phi_{v(y,x)}(y))\leq C^{2}e^{-\lambda(t)}d(x,y).$

where we used $w\in W_{\epsilon^{\prime}}^{s}(\phi_{(y,x)}v(y))$. By summing
up, we have

$d(\phi_{t}\phi_{v(y,x)}(y),\phi_{t}(x))\leq
C^{2}e^{-\lambda\min(t,T-t)}(d(x,y)+d(\phi_{T}(x),\phi_{T}(y)))\text{ for
}0\leq t\leq T.$

Now we assume that $d(\phi_{t+s(t)}(y),\phi_{t}(x))\leq\eta$ for all $t\geq
0,$ then by the arguments above. We have for all $t\geq 0$,

$\displaystyle d(\phi_{t}\phi_{v(y,x)}(y),\phi_{t}(x))$ $\displaystyle\leq
C^{2}e^{-\lambda\min(t,2t-t)}(d(x,y)+d(\phi_{2t}(x),\phi_{2t}(y)))$
$\displaystyle\leq 2C^{2}\eta e^{-\lambda\min(t,2t-t)}\to 0\text{ as
}t\to+\infty.$

This ends the proof. ∎

###  5.2. Proof of Lemma 3.5

###### Proof.

We partially follow Bowen’s arguments in [Bowen]. Firstly we fix a constant
$K\gg C$ with $2C^{2}e^{-\lambda K}\ll 1$ and a segment $\mathcal{S}$ as in
Lemma 3.5. We let $\tau=|\mathcal{S}|$ and
$\eta=d(\mathcal{S}^{L},\mathcal{S}^{R})$. Then $\eta<\delta^{\prime}$ and
$2Ce^{-\lambda\tau}\ll 1$. Therefore, we have the following claim.

Claim A.  For the segment $\mathcal{S}$ in Lemma 3.5, there is a $y\in\Lambda$
and a continuous function $\hat{s}:\mathbb{R}\to\mathbb{R}$ with
$\hat{s}(0)=0$ and $Lip(\hat{s})\leq\frac{2C\eta}{\tau}$ such that
$d(\phi_{i\tau+t_{1}+s(i\tau+t_{1})}(y),\phi_{t_{1}}(\mathcal{S}^{L}))\leq
L_{1}\eta$ for all $t_{1}\in[0,\tau]$ and $i\in\mathbb{Z}$ where
$L_{1}=2C^{2}(\frac{2}{e^{\lambda}-1}+e^{\beta}+2)$.

Since the proof of Claim A is long, we postpone the proof of Claim A to the
next subsection. Let $y\in M$ and $\hat{s}:\mathbb{R}\to\mathbb{R}$ be as in
Claim A. We divide the following proof into two steps.

Step 1.  At first, we show that $y$ is a periodic point.

By Claim A,

$\displaystyle d(\phi_{t+\hat{s}(t)}(y),\phi_{t+\tau+\hat{s}(t+\tau)}(y))\leq
2L_{1}\eta\text{ for }t\in\mathbb{R}.$

Since $Lip(\hat{s})\ll 1$ and $\hat{s}(0)=0$, $g(t)=t+\hat{s}(t)$ is a
homomorphism of $\mathbb{R}$ onto itself, the above inequality can be
rewritten as the following

$\displaystyle
d(\phi_{t}(y),\phi_{g^{-1}(t)+\tau+\hat{s}(g^{-1}(t)+\tau)}(y))\leq
2L_{1}\eta\text{ for }t\in\mathbb{R}.$

We note

$y^{\prime}=\phi_{g^{-1}(0)+\tau+\hat{s}(g^{-1}(0)+\tau)}(y)$

and

$s(t)=g^{-1}(t)+\hat{s}(g^{-1}(t)+\tau)-g^{-1}(0)-\hat{s}(g^{-1}(0)+\tau)-t.$

Then

$d(\phi_{t}(y),\phi_{t+s(t)}(y^{\prime}))\leq 2L_{1}\eta\text{ for
}t\in\mathbb{R}\text{ and }s(0)=0.$

Therefore, by Lemma 3.4, one has

$\phi_{v_{2}}(y^{\prime})=y\text{ and }|v_{2}|\leq 2CL_{1}\eta,$

where $v_{2}=v(y^{\prime},y)$. Thus

$\phi_{g^{-1}(0)+\tau+\hat{s}(g^{-1}(0)+\tau)+v_{2}}(y)=y.$

Notice that $g^{-1}(0)=0$ since $g(0)=0$. Thus,

$|g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_{2}|\leq|\hat{s}(\tau)|+|v_{2}|\leq(2C+2CL_{1})\eta\ll\tau.$

Therefore, $y$ is a periodic point.

Step 2. There is a periodic segment $\mathcal{O}$ such that

$||\mathcal{S}|-|\mathcal{O}||\leq Ld(\mathcal{S}^{L},\mathcal{S}^{R})$

and

$d(\phi_{t}(\mathcal{O}^{L}),\phi_{t}(\mathcal{S}^{L}))\leq
Ld(\mathcal{S}^{L},\mathcal{S}^{R})\text{ for all }0\leq
t\leq\max(|\mathcal{S}|,|\mathcal{O}|),$

where $L=2CL_{2}+L_{2}$ and $L_{2}=2C^{3}L_{1}+C^{4}L_{1}.$

By Claim A,

$\displaystyle d(\phi_{t+\hat{s}(t)}(y),\phi_{t}(\mathcal{S}^{L}))\leq
L_{1}\eta\text{ for }t\in[0,\tau].$

By Lemma 3.4, for $t\in[0,\tau],$ $|\hat{s}(t)|\leq 2CL_{1}\eta$ and

$\displaystyle\begin{split}d(\phi_{t}\phi_{v_{1}}(y),\phi_{t}(\mathcal{S}^{L}))&\leq
C^{2}e^{-\lambda\min(t,\tau-t)}(d(y,\mathcal{S}^{L})+d(\phi_{\tau}(y),\phi_{\tau}(\mathcal{S}^{L})))\\\
&\leq
C^{2}(d(y,\mathcal{S}^{L})+d(\phi_{\tau+\hat{s}(\tau)}(y),\phi_{\tau}(\mathcal{S}^{L}))+d(\phi_{\tau+\hat{s}(\tau)}(y),\phi_{\tau}(y)))\\\
&\leq L_{2}\eta,\end{split}$ (5.4)

where $v_{1}=v(y,\mathcal{S}^{L})$. Now we put $y^{*}=\phi_{v_{1}}y$ and we
have a periodic segment,

$\mathcal{O}:[0,\tau+g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_{2}]\to
M:t\to\phi_{t}(y^{*}),$

where $v_{2}$ is as in Step 1.

It is clear that

$||\mathcal{S}|-|\mathcal{O}||\leq|g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_{2}|\leq
L_{2}\eta.$

If $|\mathcal{O}|\leq|\mathcal{S}|$, by (5.4),

$d(\phi_{t}(y^{*}),\phi_{t}(\mathcal{S}^{L}))\leq L_{2}\eta,\text{ for
}t\in[0,\max(|\mathcal{S}|,|\mathcal{O}|)],$

where we used $\tau=\max(|\mathcal{S}|,|\mathcal{O}|).$
If $|\mathcal{O}|>|\mathcal{S}|$, by (5.4),

$d(\phi_{t}(y^{*}),\phi_{t}(\mathcal{S}^{L}))\leq L_{2}\eta,\text{ for
}t\in[0,|\mathcal{S}|],$

and for $t\in(|\mathcal{S}|,|\mathcal{O}|]$,

$\displaystyle d(\phi_{t}(y^{*}),\phi_{t}(\mathcal{S}^{L}))$
$\displaystyle\leq
d(\phi_{t}(y^{*}),\phi_{\tau}(y^{*}))+d(\phi_{\tau}(y^{*}),\phi_{\tau}(\mathcal{S}^{L}))+d(\phi_{\tau}(\mathcal{S}^{L}),\phi_{t}(\mathcal{S}^{L}))$
$\displaystyle\leq L\eta,$

where $L=2CL_{2}+L_{2}$. This ends the proof since $L_{2}\leq L$. ∎

####  5.2.1. Proof of Claim A

###### Proof.

Recall that $\mathcal{S}$ is a segment of $\Phi|_{\Lambda}$ with
$|\mathcal{S}|=\tau\geq K$ and
$d(\mathcal{S}^{L},\mathcal{S}^{R})=\eta<\delta^{\prime}$ where $K$ satisfies
$2C^{2}e^{-\lambda K}\ll 1$. We define $x_{-k},\zeta_{-k}$ recursively for
$k\geq 0$ by

$x_{0}=\mathcal{S}^{R},\zeta_{0}=0$

and

$\zeta_{-k-1}=v(\phi_{-\tau}(x_{-k}),\mathcal{S}^{R}),x_{-k-1}=W_{\epsilon^{\prime}}^{s}(\phi_{-\tau+\zeta_{-k-1}}(x_{-k}))\cap
W_{\epsilon^{\prime}}^{u}(\mathcal{S}^{R})\text{ for }k=1,2,\cdots.$

We have the following two assertions.

Assertion 1. $x_{-k}$ and $\zeta_{-k}$ are well defined and
$d(x_{-k},\mathcal{S}^{R})\leq 2C\eta$ for $k\geq 0.$

###### Proof.

In the case $k=0$, it is obviously true. Now assume that we have
$\zeta_{-k},x_{-k}$ and $d(x_{-k},\mathcal{S}^{R})\leq 2C\eta$. Then

$\displaystyle\begin{split}d(\phi_{-\tau}(x_{-k}),\mathcal{S}^{R})&\leq
d(\phi_{-\tau}(x_{-k}),\phi_{-\tau}(\mathcal{S}^{R}))+d(\mathcal{S}^{R},\phi_{-\tau}(\mathcal{S}^{R}))\\\
&\leq Ce^{-\lambda\tau}\cdot 2C\eta+\eta\\\ &\leq 2\eta,\end{split}$ (5.5)

where we used $x_{-k}\in W_{\epsilon^{\prime}}^{u}(\mathcal{S}^{R})$. Since
$2\eta\leq 2\delta^{\prime}$, $x_{-k-1}$ is well defined as well as
$\zeta_{-k-1}$, and moreover one has that

$\displaystyle d(x_{-k-1},\mathcal{S}^{R})\leq 2C\eta.$

This ends the proof. ∎

By (5.5), we have that

$\displaystyle|\zeta_{k}|\leq 2C\eta\ll 1.$ (5.6)

Next we denote $x^{(-k)}=\phi_{k\tau-\sum_{i=0}^{k}\zeta_{-i}}(x_{-k})$ for
$k\geq 0$. For $k\in\mathbb{N}$, we define
$s^{*}_{-k}:\mathbb{R}\to\mathbb{R}$ by

$s^{*}_{-k}(t)=\left\\{\begin{array}[]{ll}\zeta_{0},&\text{ if }t>0,\\\
\sum_{i=0}^{l-1}\zeta_{-i},&\text{ if
}-l\tau<t\leq-(l-1)\tau,l\in\\{1,2,\cdots,k\\},\\\
\sum_{i=0}^{k}\zeta_{-i},&\text{ if }t\leq-k\tau.\end{array}\right.$

Assertion 2. There exists a constant $L_{0}$ such that for $t=-j\tau-t_{0}$
satisfying $t_{0}\in[0,\tau)$ and $j\in\\{0,1,\cdots,k-1\\}$, the following
holds

$d\left(\phi_{t+s^{*}_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)\leq
L_{0}\eta.$

###### Proof.

We fix $t=-j\tau-t_{0}$ for some $j\in\\{0,1,\cdots,k-1\\}$ and
$t_{0}\in[0,\tau)$. Since $x_{-j}\in
W^{u}_{\epsilon^{\prime}}(\mathcal{S}^{R})$, we have

$\displaystyle
d\left(\phi_{-t_{0}}\left(x_{-j}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)\leq
Ce^{-\lambda t_{0}}d\left(x_{-j},\mathcal{S}^{R}\right)\leq 2C^{2}\eta.$ (5.7)

Note that $\tau-\zeta_{-j-1}-t_{0}\geq-1$ and $x_{-j-1}\in
W^{s}_{\epsilon^{\prime}}(\phi_{-\tau+\zeta_{-j-1}}(x_{-j}))\cap
W^{u}_{\epsilon^{\prime}}(\mathcal{S}^{R})$, we have

$\displaystyle\begin{split}&d\left(\phi_{\tau-\zeta_{-j-1}-t_{0}}\left(x_{-j-1}\right),\phi_{-t_{0}}\left(x_{-j}\right)\right)\\\
=&d\left(\phi_{\tau-\zeta_{-j-1}-t_{0}}\left(x_{-j-1}\right),\phi_{\tau-\zeta_{-j-1}-t_{0}}\phi_{-\tau+\zeta_{-j-1}}\left(x_{-j}\right)\right)\\\
\leq&e^{\beta}d\left(x_{-j-1},\phi_{-\tau+\zeta_{-j-1}}\left(x_{-j}\right)\right)\\\
\leq&2C^{2}e^{\beta}\eta,\end{split}$ (5.8)

where we used (2) and (3) of the basic canonical setting for the case
$\tau-\zeta_{-j-1}-t_{0}>0$ and $\tau-\zeta_{-j-1}-t_{0}\leq 0$, respectively.

Note that $|\zeta_{-l}|\ll 1,\tau\gg 1$ and $t_{0}\in[0,\tau)$, i.e.,
$\tau-\zeta_{-j-1}-t_{0}\geq-1$ and $\tau-\zeta_{-i}>\tau-1>1$. Since
$x_{-(k-l)}\in
W^{s}_{\epsilon^{\prime}}(\phi_{-\tau+\zeta_{-(k-l)}}(x_{-(k-j-1)}))\cap
W^{u}_{\epsilon^{\prime}}(\mathcal{S}^{R})$, we have

$\displaystyle\begin{split}&\sum_{l=0}^{k-j-2}d\left(\phi_{(k-j-l-1)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_{0}}\left(x_{-(k-l)}\right),\phi_{(k-j-l-2)\tau-\sum_{i=j+1}^{k-l-1}\zeta_{-i}-t_{0}}\left(x_{-(k-l-1)}\right)\right)\\\
&\leq\sum_{l=0}^{k-j-2}Ce^{-\lambda\left(\left(k-j-l\right)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_{0}\right)}d\left(x_{-(k-l)},\phi_{-\tau+\zeta_{-(k-l)}}\left(x_{-(k-j-1)}\right)\right)\\\
&\leq\sum_{l=0}^{k-j-2}Ce^{-\lambda\left(\left(k-j-l\right)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_{0}\right)}4C\eta\\\
&\leq 4C^{2}\eta
e^{-\lambda(2\tau-\zeta_{-j-1}-\zeta_{-j-2}-t_{0})}\sum_{l=0}^{k-l-1}e^{-\lambda\cdot(\tau-1)\cdot
l}\\\ &\leq 4C^{2}\eta e^{-\lambda(\tau-2)}\frac{1}{1-e^{-\lambda}}\\\
&=4C^{2}\eta\frac{1}{e^{\lambda}-1},\end{split}$ (5.9)

where we used 1b) of the basic canonical setting. Combining (5.7), (5.8) and
(LABEL:eq-an-4), we have that for $t=-j\tau-t_{0}$

$\displaystyle
d\left(\phi_{t+s^{*}_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)$
$\displaystyle=$ $\displaystyle
d\left(\phi_{-j\tau+\sum_{i=0}^{j}\zeta_{-i}-t_{0}}\left(x^{(-k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)$
$\displaystyle=$ $\displaystyle
d\left(\phi_{(k-j)\tau-\sum_{i=j+1}^{k}\zeta_{-i}-t_{0}}\left(x_{-k}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{l=0}^{k-j-2}d\left(\phi_{(k-j-l-1)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_{0}}\left(x_{-(k-l)}\right),\phi_{(k-j-l-2)\tau-\sum_{i=j+1}^{k-l-1}\zeta_{-i}-t_{0}}\left(x_{-(k-l-1)}\right)\right)$
$\displaystyle+d(\phi_{\tau-\zeta_{-j-1}-t_{0}}(x_{-j-1}),\phi_{-t_{0}}(x_{-j}))+d\left(\phi_{-t_{0}}(x_{-j}),\phi_{-t_{0}}(\mathcal{S}^{R})\right)$
$\displaystyle\leq$ $\displaystyle L_{0}\eta,$

where $L_{0}=2C^{2}(\frac{2}{e^{\lambda}-1}+e^{\beta}+1)$. This ends the proof
of Assertion 2. ∎

Now for $k\in\mathbb{N}$, we define $\bar{s}_{-k}:\mathbb{R}\to\mathbb{R}$ by

$\bar{s}_{-k}(t)=\left\\{\begin{array}[]{ll}\zeta_{0},&\text{ if }t>0,\\\
\sum_{i=0}^{l-1}\zeta_{-i}-\frac{t+(l-1)\tau}{\tau}\zeta_{-l},&\text{ if
}-l\tau<t\leq-(l-1)\tau,l\in\\{1,2,\cdots,k\\},\\\
\sum_{i=0}^{k}\zeta_{-i},&\text{ if }t\leq-k\tau.\end{array}\right.$

It is clear that $\bar{s}_{-k}$ is Lipschitz continuous with

$Lip(\bar{s}_{-k})\leq\frac{\max_{i\in\\{0,1,2,\cdots,k\\}}|\zeta_{i}|}{\tau}\leq\frac{2C\eta}{\tau},$

and

$|\bar{s}_{-k}(t)-s^{*}_{-k}(t)|\leq\max_{i\in\\{0,1,2,\cdots,k\\}}|\zeta_{i}|\leq
2C\eta.$

Therefore, by Assertion 2, when $t=-j\tau-t_{0}$ for some
$j\in\\{0,1,2,\cdots,k-1\\}$ and $t_{0}\in[0,\tau)$, one has that

$\displaystyle\begin{split}&d\left(\phi_{t+\bar{s}_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)\\\
\leq&d\left(\phi_{t+\bar{s}_{-k}(t)}\left(x^{(-k)}\right),\phi_{t+s^{*}_{-k}(t)}\left(x^{(-k)}\right)\right)+d\left(\phi_{t+s^{*}_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)\\\
\leq&L_{1}\eta,\end{split}$ (5.10)

where $L_{1}=L_{0}+2C^{2}.$ Now for $k\in\mathbb{N}$, we define
${s}_{-k}:\mathbb{R}\to\mathbb{R}$ by

${s}_{-k}(t)=\bar{s}_{-2k}(t-k\tau)-\sum_{i=0}^{k}\zeta_{-i}.$

It is clear that ${s}_{-k}(0)=0$. On the other hand, we note
$y_{k}=\phi_{-\tau k+\sum_{i=0}^{k}\zeta_{-i}}(x^{(-2k)})$. Thus, when
$t=-j\tau-t_{0}$ for some $j\in\\{-k,-k+1,\cdots,k-1\\}$ and
$t_{0}\in[0,\tau)$, (LABEL:E:BarSEst) implies that

$\displaystyle
d\left(\phi_{t+{s}_{-k}(t)}(y_{k}),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)=d\left(\phi_{t-\tau
k+\bar{s}_{-2k}(t-\tau
k)}\left(x^{(-2k)}\right),\phi_{-t_{0}}\left(\mathcal{S}^{R}\right)\right)\leq
L_{1}\eta.$

Notice that $s_{-k}$ are Lipschitz with
$Lip(s_{-k})\leq\frac{2C\eta}{\tau}\ll\eta$ for all $k\in\mathbb{N}$. Applying
the Ascoli-Azelá theorem, there exists a subsequence
$(s_{-k_{i}})_{i=1}^{+\infty}$ that converges to a Lipschitz continuous
function $\hat{s}:\mathbb{R}\to\mathbb{R}$ with
$Lip(\hat{s})\leq\frac{2C\eta}{\tau}\ll\eta$ and $\hat{s}(0)=0$. Without
losing any generality, we assume that $y_{k_{i}}\to y$ as $i\to+\infty$. By
the continuity, if $t=-j\tau-t_{0}$ for some $j\in\mathbb{Z}$ and
$t_{0}\in[0,\tau)$, then

$\displaystyle d(\phi_{t+\hat{s}(t)}(y),\phi_{-t_{0}}(\mathcal{S}^{R}))\leq
L_{1}\eta.$

That is to say, if $t=-(j+1)\tau+(\tau-t_{0})$ for some $j\in\mathbb{Z}$ and
$t_{0}\in[0,\tau)$, then

$\displaystyle d\left(\phi_{t+\hat{s}(t)}(y),\phi_{\tau-
t_{0}}(\mathcal{S}^{L})\right)\leq L_{1}\eta.$

Note that $y_{k_{i}}\in\Lambda$ for each $i\in\mathbb{N}$, thus $y\in\Lambda$.
Let $t_{1}=\tau-t_{0}$, then the proof of Claim A is completed. ∎

###  5.3. Proof of Lemma 3.7

In this section, we mainly prove a version of the so called Mañè-Conze-
Guivarc’h-Bousch’s Lemma. The proof partially follows Bousch’s arguments in
[Bousch_Mane].

####  5.3.1. Integration along segment

Recall that, for a continuous function $u$ and a segment $\mathcal{S}$ of
$\Phi$, the integration of $u$ along $\mathcal{S}$ is defined by

$\left\langle\mathcal{S},u\right\rangle:=\int_{a}^{b}u(\phi_{t}(x))dt.$

######  Lemma 5.1.

Let $u:M\to\mathbb{R}$ be an $\alpha$-Hölder function with
$\beta(u;1,\Lambda,\Phi)\geq 0$. Then for a segment $\mathcal{S}$ of
$\Phi|_{\Lambda}$ satisfying $|\mathcal{S}|\geq K$ and
$d(\mathcal{S}^{L},\mathcal{S}^{R})\leq\delta^{\prime}$, the following holds

$\left\langle\mathcal{S},u\right\rangle\geq-
K_{1}d^{\alpha}(\mathcal{S}^{L},\mathcal{S}^{R}),$

where $K_{1}=\frac{(CL)^{\alpha}}{\lambda\alpha}\|u\|_{\alpha}+L\|u\|_{0}$.

###### Proof.

Since $d(\mathcal{S}^{L},\mathcal{S}^{R})\leq\delta^{\prime}$ and
$|\mathcal{S}|\geq K$, by Anosov Closing Lemma, there exists a periodic
segment $\mathcal{O}$ of $\Phi|_{\Lambda}$ such that

$||\mathcal{S}|-|\mathcal{O}||\leq Ld(\mathcal{S}^{L},\mathcal{S}^{R})$

and

$d(\phi_{t}(\mathcal{O}^{L}),\phi_{t}(\mathcal{S}^{L}))\leq
Ld(\mathcal{S}^{L},\mathcal{S}^{R})\text{ for all }0\leq
t\leq\max(|\mathcal{S}|,|\mathcal{O}|).$

Therefore, Letting $v=v(\mathcal{O}^{L},\mathcal{S}^{L})$ as in Lemma 3.4, one
has that

$\displaystyle\left\langle\mathcal{S},u\right\rangle-\left\langle\mathcal{O},u\right\rangle$
$\displaystyle=\int_{0}^{|\mathcal{O}|}u(\phi_{t}(\mathcal{S}^{L}))-u(\phi_{t}\phi_{v}(\mathcal{O}^{L}))dt+\int_{|\mathcal{O}|}^{|\mathcal{S}|}u(\phi_{t}(\mathcal{S}^{L}))dt$
$\displaystyle\geq-\|u\|_{\alpha}\int_{0}^{|\mathcal{O}|}d^{\alpha}(u(\phi_{t}(\mathcal{S}^{L})),u(\phi_{t}\phi_{v}(\mathcal{O}^{L})))dt-\|u\|_{0}||\mathcal{S}|-|\mathcal{O}||$
$\displaystyle\geq-\|u\|_{\alpha}\int_{0}^{|\mathcal{O}|}\big{(}Ce^{-\lambda\min(t,T-t)}Ld(\mathcal{S}^{L},\mathcal{S}^{R})\big{)}^{\alpha}dt-\|u\|_{0}Ld(\mathcal{S}^{L},\mathcal{S}^{R})$
$\displaystyle\geq-\left(\frac{(CL)^{\alpha}}{\lambda\alpha}\|u\|_{\alpha}+L\|u\|_{0}\right)d^{\alpha}(\mathcal{S}^{L},\mathcal{S}^{R}),$

where we used the assumption $0<\alpha\leq 1$ and
$0<d(\mathcal{S}^{L},\mathcal{S}^{R})<\delta^{\prime}\ll 1$. Then the Lemma is
immediately from the fact $\left\langle\mathcal{O},u\right\rangle\geq 0$ since
$\beta(u;1,\Lambda,\Phi)\geq 0$. This ends the proof. ∎

######  Lemma 5.2.

Let $\mathcal{P}$ be a finite partition of $M$ with diameter smaller than
$\delta^{\prime}$ and $u:M\to\mathbb{R}$ be an $\alpha$-Hölder function with
$\beta(u;1,\Lambda,\Phi)\geq 0$. Then for a given segment $\mathcal{S}$ of
$\Phi|_{\Lambda}$, the following holds

$\left\langle\mathcal{S},u\right\rangle\geq-K_{2}\delta^{\prime\alpha},$

where
$K_{2}=\sharp\mathcal{P}\cdot\left(\frac{K\|u\|_{0}}{\delta^{\prime\alpha}}+K_{1}\right)$
and $K$, $K_{1}$ are as in Lemma 3.5 and 5.1 respectively.

###### Proof.

For $x\in M$, denote $\mathcal{P}(x)$ the element in $\mathcal{P}$ which
contains $x$. Assume $|\mathcal{S}|=(n-1)K+r$ for some $n\geq 1$ and $0\leq
r<K$. Let $t_{i}=iK$ for $0\leq i\leq n-1$ and $t_{n}=|\mathcal{S}|$. We
define the function $w:\mathbb{N}\to[0,n]\cap\mathbb{N}$ inductively by
letting

$w(0)=0$ $w(k)=\min\\{\eta(w(k-1))+1,n\\}.$

where $\eta:[0,n-1]\cap\mathbb{N}\to[0,n-1]\cap\mathbb{N}$ is the function
that maps each $i$ to the largest $j\in[0,n-1]\cap\mathbb{N}$ such that
$\mathcal{P}(\phi_{t_{i}}(\mathcal{S}^{L}))=\mathcal{P}(\phi_{t_{j}}(\mathcal{S}^{L}))$.
Let $s\geq 0$ be the smallest integer for which $\eta(w(s))=n-1$. Then
$\mathcal{P}(\phi_{t_{w(i)}}(\mathcal{S}^{L}))\neq\mathcal{P}(\phi_{t_{w(j)}}(\mathcal{S}^{L}))$
for $0\leq i<j\leq s$ which implies $s\leq\sharp\mathcal{P}$. For $0\leq j\leq
s$, we have two cases: If $\eta(w(j))=w(j)$

$\displaystyle\int_{t_{w(j)}}^{t_{\eta(w(j))}}u(\phi_{t}(x))dt=0\text{ and
}\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_{t}(x))dt\geq-K\|u\|_{0}.$
(5.11)

If $\eta(w(j))>w(j)$, by using Lemma 5.1,

$\displaystyle\int_{t_{w(j)}}^{t_{\eta(w(j))}}u(\phi_{t}(x))dt\geq-\left(\frac{(CL)^{\alpha}}{\lambda\alpha}\|u\|_{\alpha}+L\|u\|_{0}\right)\delta^{\prime\alpha},$
(5.12)

where we use the fact
$d(\phi_{t_{i}}(\mathcal{S}^{L}),\phi_{t_{j}}(\mathcal{S}^{L}))<\delta^{\prime}$
since
$\mathcal{P}(\phi_{t_{i}}(\mathcal{S}^{L}))=\mathcal{P}(\phi_{t_{j}}(\mathcal{S}^{L}))$.
On the other hand, as in (5.11),

$\displaystyle\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_{t}(\mathcal{S}^{L}))dt\geq-K\|u\|_{0}.$
(5.13)

Combining (5.11), (5.12) and (5.13), one has

$\displaystyle\left\langle\mathcal{S},u\right\rangle=\sum_{j=0}^{s-1}\int_{t_{w(j)}}^{t_{\eta(w(j))}}+\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_{t}(\mathcal{S}^{L}))dt$
$\displaystyle\geq$
$\displaystyle-s\left(K\|u\|_{0}+\left(\frac{(CL)^{\alpha}}{\lambda\alpha}\|u\|_{\alpha}+L\|u\|_{0}\right)\delta^{\prime\alpha}\right)$
$\displaystyle\geq$
$\displaystyle-\sharp\mathcal{P}\cdot\left(K\|u\|_{0}+\left(\frac{(CL)^{\alpha}}{\lambda\alpha}\|u\|_{\alpha}+L\|u\|_{0}\right)\delta^{\prime\alpha}\right),$

which completes the proof. ∎

In the following, we deal with the so called shadowing property for two finite
time segments, which will allow one to use one segment to shadow two segments
of which the ending point of one segment is close to the beginning point of
the other. Let $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ be two segments of
$\Phi|_{\Lambda}$, suppose that

$d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})<\delta^{\prime}.$

Then there exist $v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$ and
$w(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})=W^{s}_{\epsilon^{\prime}}(\phi_{v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})}(\mathcal{S}_{2}^{L}))\cap
W^{u}_{\epsilon^{\prime}}(\mathcal{S}_{1}^{R})$. Define a new segment
$\mathcal{S}_{1}*\mathcal{S}_{2}:\left[-|\mathcal{S}_{1}|,|\mathcal{S}_{2}|-v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})\right]$
by letting

$\mathcal{S}_{1}*\mathcal{S}_{2}(t)=\phi_{t}\left(w(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})\right)\
\forall
t\in\left[-|\mathcal{S}_{1}|,|\mathcal{S}_{2}|-v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})\right].$
(5.14)

We remark here that the definition of $\mathcal{S}_{1}*\mathcal{S}_{2}$ above
is not the unique way for describing the shadowing property. Nevertheless, it
is the most convenient way for the rest of the proof.

######  Lemma 5.3.

Given $0<\alpha\leq 1$ and a large constant $\gamma=\gamma(\alpha)\gg 1$
satisfying that $2C^{2\alpha}e^{-\frac{\gamma\alpha\lambda}{2}}\ll 1$, when
two segments $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ of $\Phi|_{\Lambda}$
satisfy the following

$d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\leq\delta^{\prime}\text{ and
}\min\\{|\mathcal{S}_{1}|,|\mathcal{S}_{2}|\\}\geq\gamma,$

then for all $u\in\mathcal{C}^{0,\alpha}(M)$,

$\frac{\left|\left\langle\mathcal{S}_{1}*\mathcal{S}_{2},u\right\rangle-\left\langle\mathcal{S}_{1},u\right\rangle-\left\langle\mathcal{S}_{2},u\right\rangle\right|}{d^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})-d^{\alpha}((\mathcal{S}_{1}*\mathcal{S}_{2})^{R},\mathcal{S}_{2}^{R})-d^{\alpha}((\mathcal{S}_{1}*\mathcal{S}_{2})^{L},\mathcal{S}_{1}^{L})}\leq
K_{3},$

where
$K_{3}=\frac{C\|u\|_{0}+\frac{2C^{2\alpha}\|u\|_{\alpha}}{\lambda\alpha}}{1-2C^{2\alpha}e^{-{(\gamma-1)\alpha\lambda}}}$
and the denominator of the left side of the above inequality is always
positive by the choice of $\gamma$.

###### Proof.

Fix $\alpha,\gamma,u,\mathcal{S}_{1},\mathcal{S}_{2}$ as in this Lemma. Note
$v=v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$,
$w=w\left(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R}\right)$ and

$\widetilde{\mathcal{S}}_{2}:[0,|\mathcal{S}_{2}|-v]:t\to\phi_{t+v}(\mathcal{S}_{2}^{L}).$

Thus, we have

$\displaystyle\left|\left\langle\mathcal{S}_{1}*\mathcal{S}_{2},u\right\rangle-\left\langle\widetilde{\mathcal{S}}_{2},u\right\rangle-\left\langle\mathcal{S}_{1},u\right\rangle\right|$
$\displaystyle=$
$\displaystyle\left|\int_{0}^{|\mathcal{S}_{2}|-v}u(\phi_{t}(w))-u(\phi_{t}\phi_{v}(\mathcal{S}_{2}^{L}))dt+\int_{0}^{|\mathcal{S}_{1}|}u(\phi_{-t}(w))-u(\phi_{-t}(\mathcal{S}_{1}^{R}))dt\right|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{|\mathcal{S}_{2}|-v}\|u\|_{\alpha}d^{\alpha}(\phi_{t}(w),\phi_{t}\phi_{v}(\mathcal{S}_{2}^{L}))dt+\int_{0}^{|\mathcal{S}_{1}|}\|u\|_{\alpha}d^{\alpha}(\phi_{-t}(w),\phi_{-t}(\mathcal{S}_{1}^{R}))dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{|\mathcal{S}_{2}|-v}\|u\|_{\alpha}(Ce^{-\lambda
t})^{\alpha}d^{\alpha}(w,\mathcal{S}_{2}^{L})dt+\int_{0}^{|\mathcal{S}_{1}|}\|u\|_{\alpha}(Ce^{-\lambda
t})^{\alpha}d^{\alpha}(w,\mathcal{S}_{1}^{R})dt$ $\displaystyle\leq$
$\displaystyle
2\|u\|_{\alpha}\frac{C^{2\alpha}}{\lambda\alpha}d^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

and

$\displaystyle\left|\left\langle\widetilde{\mathcal{S}}_{2},u\right\rangle-\left\langle\mathcal{S}_{2},u\right\rangle\right|\leq\|u\|_{0}|v|\leq\|u\|_{0}Cd(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\leq\|u\|_{0}Cd^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}).$

Therefore

$\left|\left\langle\mathcal{S}_{1}*\mathcal{S}_{2},u\right\rangle-\left\langle\mathcal{S}_{1},u\right\rangle-\left\langle\mathcal{S}_{2},u\right\rangle\right|\leq\left(C\|u\|_{0}+\frac{2C^{2\alpha}\|u\|_{\alpha}}{\lambda\alpha}\right)d^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})$
(5.15)

On the other hand, one has that

$d^{\alpha}((\mathcal{S}_{1}*\mathcal{S}_{2})^{L},\mathcal{S}_{2}^{L})\leq
C^{\alpha}e^{-\alpha\lambda(\gamma-v)}d^{\alpha}(w,\mathcal{S}_{2}^{L})\leq
C^{2\alpha}e^{-\alpha\lambda(\gamma-1)}d^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

and

$d^{\alpha}((\mathcal{S}_{1}*\mathcal{S}_{2})^{R},\mathcal{S}_{2}^{R})\leq
C^{\alpha}e^{-\alpha\lambda\gamma}d^{\alpha}(w,\mathcal{S}_{1}^{R})\leq
C^{2\alpha}e^{-\alpha\lambda\gamma}d^{\alpha}(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

which combining with (5.15) and the choice of $\gamma$ implies what needed,
thus accomplish the proof. ∎

####  5.3.2. Proof of Lemma 3.7

Before the main proof, we first state a technical Lemma which can be deduced
from the Lemma 1.1 of [Bousch_Mane].

######  Lemma 5.4.

Given $0<\alpha\leq 1,A>0,\gamma\in\mathbb{R}$ and a continuous function
$u:M\to\mathbb{R}$, the following are equivalent

  * (1).

For all $n\geq 1$ and $x_{i}\in M,{i\in\mathbb{Z}/n\mathbb{Z}}$,

$\displaystyle\sum_{i\in\mathbb{Z}/n\mathbb{Z}}u(x_{i})+A\sum_{i\in\mathbb{Z}/n\mathbb{Z}}d^{\alpha}(\phi_{\gamma}x_{i},x_{i+1})\geq
0.$ (5.16)

  * (2).

There exists an $\alpha$-Hölder function $v:M\to\mathbb{R}$ with
$\|v\|_{\alpha}\leq A$ such that $u\geq v\circ\phi_{\gamma}-v$.

Now we prove Lemma 3.7.

###### Proof.

Let $K_{1},K_{2},K_{3}$ be the constants as in Lemmas 5.1, 5.2 and 5.3. We fix
a $\gamma>N_{0}$ satisfying the condition in Lemma 5.3 and a large number $Q$
such that

$Q>\max\\{K_{1},K_{2},K_{3}\\}.$

For $n\geq 1$, we note $i^{(n)}=i+n\mathbb{Z}\in\mathbb{Z}/n\mathbb{Z}$ for
$i\in[0,n-1]\cap\mathbb{Z}$. Now we fix an integer $n\geq 1$ and points
$x_{i^{(n)}}\in\Lambda,i^{(n)}\in\mathbb{Z}/n\mathbb{Z}$. Note

$\mathcal{S}_{i^{(n)}}:[0,\gamma]\to\Lambda:t\to\phi_{t}(x_{i^{(n)}})\text{
for }i^{(n)}\in\mathbb{Z}/n\mathbb{Z},$
$\mathcal{L}^{(n)}=\\{\mathcal{S}_{i^{(n)}},i^{(n)}\in\mathbb{Z}/n\mathbb{Z}\\}$

and

$\displaystyle\Sigma^{(n)}=\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}\left\langle\mathcal{S}_{i^{(n)}},u\right\rangle+Q\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}d^{\alpha}(\mathcal{S}_{i^{(n)}}^{R},\mathcal{S}_{i^{(n)}+1^{(n)}}^{L}).$

If there is some $j^{(n)}\in\mathbb{Z}/n\mathbb{Z}$ such that
$d(\mathcal{S}_{j^{(n)}}^{R},\mathcal{S}_{j^{(n)}+1^{(n)}}^{L})<\delta^{\prime}$,
just take

$\mathcal{S}_{1^{(n-1)}}=\mathcal{S}_{j^{(n)}}*\mathcal{S}_{j^{(n)}+1^{(n)}}\text{
and }\mathcal{S}_{i^{(n-1)}}=\mathcal{S}_{j^{(n)}+i^{(n)}}\text{ for
}i=2,3,\cdots n-1$
$\mathcal{L}^{(n-1)}=\\{\mathcal{S}_{i^{(n-1)}},i^{(n-1)}\in\mathbb{Z}/(n-1)\mathbb{Z}\\}$

and

$\displaystyle\Sigma^{(n-1)}=\sum_{i^{(n-1)}\in\mathbb{Z}/(n-1)\mathbb{Z}}\left\langle\mathcal{S}_{i^{(n-1)}},u\right\rangle+Q\sum_{i^{(n-1)}\in\mathbb{Z}/n\mathbb{Z}}d^{\alpha}(\mathcal{S}_{i^{(n-1)}}^{R},\mathcal{S}_{i^{(n-1)}+1^{(n-1)}}^{L}).$

Note that by Lemma 5.3

$\displaystyle\Sigma^{(n)}-\Sigma^{(n-1)}$ $\displaystyle\geq$
$\displaystyle-\left|\left\langle\mathcal{S}_{j^{(n)}}*\mathcal{S}_{j^{(n)}+1^{(n)}},u\right\rangle-\left\langle\mathcal{S}_{j^{(n)}},u\right\rangle-\left\langle\mathcal{S}_{j^{(n)}+1^{(n)}},u\right\rangle\right|$
$\displaystyle+Qd^{\alpha}\left(\mathcal{S}_{j^{(n)}}^{R},\mathcal{S}_{(j+1)^{(n)}}^{L}\right)$
$\displaystyle-Q\left(d^{\alpha}\left(\mathcal{S}_{j^{(n)}}^{L},(\mathcal{S}_{j^{(n)}}*\mathcal{S}_{(j+1)^{(n)}})^{L}\right)-d^{\alpha}\left(\mathcal{S}_{(j+1)^{(n)}}^{R},(\mathcal{S}_{j^{(n)}}*\mathcal{S}_{(j+1)^{(n)}})^{R}\right)\right)$
$\displaystyle\geq 0.$

That is

$\displaystyle\Sigma^{(n)}\geq\Sigma^{(n-1)}.$ (5.17)

Repeat the above process until $\mathcal{L}^{(1)}$ with
$d(\mathcal{S}_{1^{(1)}}^{R},\mathcal{S}_{1^{(1)}}^{L})<\delta^{\prime}$ OR
some $m\in[1,n]\cap\mathbb{N}$ with

$d\left(\mathcal{S}_{j^{(m)}}^{R},\mathcal{S}_{(j+1)^{(m)}}^{L}\right)\geq\delta^{\prime}\text{
for all }j\in\mathbb{Z}/m\mathbb{Z}.$

In the case that the process ends at $\mathcal{L}^{(1)}$ with
$d(\mathcal{S}_{1^{(1)}}^{R},\mathcal{S}_{1^{(1)}}^{L})<\delta^{\prime}$. We
have by Lemma 5.1 that

$\displaystyle\begin{split}\Sigma^{(1)}&=\left\langle\mathcal{S}_{1^{(1)}},u\right\rangle+Qd^{\alpha}(\mathcal{S}_{1^{(1)}}^{R},\mathcal{S}_{1^{(1)}}^{L})\\\
&\geq-
K_{1}d^{\alpha}(\mathcal{S}_{1^{(1)}}^{R},\mathcal{S}_{1^{(1)}}^{L})+Qd^{\alpha}(\mathcal{S}_{1^{(1)}}^{R},\mathcal{S}_{1^{(1)}}^{L})\\\
&\geq 0.\end{split}$ (5.18)

In the case that the process ends at some $m\in[1,n]\cap\mathbb{N}$ with

$d\left(\mathcal{S}_{j^{(m)}}^{R},\mathcal{S}_{(j+1)^{(m)}}^{L}\right)\geq\delta^{\prime}\text{
for all }j^{(m)}\in\mathbb{Z}/m\mathbb{Z}.$

We have by Lemma 5.2 that

$\displaystyle\begin{split}\Sigma^{(m)}&=\sum_{i^{(m)}\in\mathbb{Z}/m\mathbb{Z}}\left\langle\mathcal{S}_{i^{(m)}},u\right\rangle+Q\sum_{i^{(m)}\in\mathbb{Z}/m\mathbb{Z}}d^{\alpha}(\mathcal{S}_{i^{(m)}}^{R},\mathcal{S}_{i^{(m)}+1^{(m)}}^{L})\\\
&\geq-mK_{2}\delta^{\prime\alpha}+mQ\delta^{\prime\alpha}\\\ &\geq
0.\end{split}$ (5.19)

Combining the inequality (5.18), (5.19) and the fact
$\Sigma^{(n)}\geq\Sigma^{(n-1)}\geq\Sigma^{(n-2)}\geq\cdots$ by (5.17), one
has

$\Sigma^{(n)}\geq 0.$

Then

$\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}u_{\gamma}(x_{i^{(n)}})+\frac{Q}{\gamma}\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}d^{\alpha}(\mathcal{S}^{R}_{i^{(n)}},\mathcal{S}^{L}_{i^{(n)}+1^{(n)}})=\frac{\Sigma^{(n)}}{\gamma}\geq
0.$

By Lemma 5.4, there is an $\alpha$-Hölder function $v$ on $\Lambda$ with
$\|v\|_{\alpha}\leq\frac{Q}{\gamma}$ such that

$u_{\gamma}|_{\Lambda}\geq v\circ\phi_{\gamma}|_{\Lambda}-v.$

This ends the proof. ∎

Finally , we give the proof of Lemma 3.9.

###### Proof of Lemma 3.9.

(1). By Lemma 3.7, we only need to show that

$\int u-\beta(u;\psi,\Lambda,\Phi)\psi d\mu\geq 0\text{ for all
}\mu\in\mathcal{M}(\Phi|_{\Lambda}).$

It is immediately from the fact

$\frac{\int ud\mu}{\int\psi d\mu}\geq\beta(u;\psi,\Lambda,\Phi)\text{ for all
}\mu\in\mathcal{M}(\Phi|_{\Lambda})$

since $\psi$ is strictly positive.

(2). Given a probability measure
$\mu\in\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)$, one has

$\int
u_{\gamma}+v\circ\phi_{\gamma}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}d\mu=\int
u-\beta(u;\psi,\Lambda,\Phi)\psi d\mu=0.$

Combining (1) and the fact
$u_{\gamma}|_{\Lambda}+v\circ\phi_{\gamma}|_{\Lambda}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}|_{\Lambda}$
is continuous on $\Lambda$, one has

$supp(\mu)\subset\\{x\in\Lambda:(u_{\gamma}+v\circ\phi_{\gamma}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma})|_{\Lambda}(x)=0\\}.$

Therefore,

$Z_{u,\psi}\subset\left\\{x\in\Lambda:(u_{\gamma}|_{\Lambda}+v\circ\phi_{\gamma}|_{\Lambda}-v-\beta(u;\psi,\Lambda,\Phi)\psi_{\gamma}|_{\Lambda})(x)=0\right\\}.$

This ends the proof. ∎

###  5.4. Proof of Lemma 3.11

In this section, we mainly prove the periodic approximation. The proof
partially follows the arguments in [BQ].

####  5.4.1. Joining of segments

Recall that the definition of the joining of two segments
$\mathcal{S}_{1}\star\mathcal{S}_{2}$ are given by (5.14), we give some
properties of jointed segments.

######  Lemma 5.5.

If two segments $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ satisfy
$|\mathcal{S}_{1}|\geq 1$ and
$d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\leq\delta^{\prime}$, then

  * (1).

$\max_{x\in\mathcal{S}_{1}*\mathcal{S}_{2}}d(x,\mathcal{S}_{1}\cup\mathcal{S}_{2})\leq
C^{3}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L});$

  * (2).

$|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-1\leq|\mathcal{S}_{1}*\mathcal{S}_{2}|\leq|\mathcal{S}_{1}|+|\mathcal{S}_{2}|+1.$

###### Proof.

(1). Note $v=v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$,
$w=w(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$ and

$\widetilde{\mathcal{S}}_{2}:[v,|\mathcal{S}_{2}|]:t\to\phi_{t}(\mathcal{S}_{2}^{L}).$

Then for $t\in[-|\mathcal{S}_{1}|,0]$,

$d(\phi_{t}(w),\mathcal{S}_{1})\leq
d(\phi_{t}(w),\phi_{t}(\mathcal{S}_{1}^{R}))\leq Ce^{\lambda
t}d(w,\mathcal{S}_{1}^{R})\leq C^{2}e^{\lambda
t}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\leq
C^{2}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

where we used $w\in W_{\epsilon}^{u}(\mathcal{S}_{1}^{R}).$ For
$t\in[0,|\mathcal{S}_{2}|-v]$,

$d(\phi_{t}(w),\widetilde{\mathcal{S}}_{2})\leq
d(\phi_{t}(w),\phi_{t}(w)\phi_{v}(\mathcal{S}_{2}^{L}))\leq Ce^{-\lambda
t}d(w,\phi_{v}(\mathcal{S}_{2}^{L}))\leq C^{2}e^{-\lambda
t}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}).$

where we used $w\in W_{\epsilon}^{s}(\phi_{v}(\mathcal{S}_{2}^{L})).$ Thus,
for $t\in[0,|\mathcal{S}_{2}|-v]$,

$\displaystyle d(\phi_{t}(w),\mathcal{S}_{2})$ $\displaystyle\leq
d(\phi_{t}(w),\widetilde{\mathcal{S}}_{2})+\max_{x\in\widetilde{\mathcal{S}}_{2}}d(\mathcal{S}_{2},\widetilde{\mathcal{S}}_{2})$
$\displaystyle\leq C^{2}e^{-\lambda
t}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})+d(\mathcal{S}_{2}^{R},\phi_{v}(\mathcal{S}_{2}^{R}))$
$\displaystyle\leq C^{3}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

where we used $C\gg 1$ and 1b) of the basic cononical setting. Thus, by
summing up,

$\max_{x\in\mathcal{S}_{1}*\mathcal{S}_{2}}d(x,\mathcal{S}_{1}\cup\mathcal{S}_{2})=\max_{t\in[-|\mathcal{S}_{1}|,|\mathcal{S}_{2}|-v]}d(\phi_{t}(w),\mathcal{S}_{1}\cup\mathcal{S}_{2})\leq
C^{3}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}).$

(2). One has

$|\mathcal{S}_{1}*\mathcal{S}_{2}|=|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-v\geq|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-Cd(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\geq|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-C\delta^{\prime}>|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-1,$

where we used the assumption $\delta^{\prime}\ll\frac{1}{C}.$ On the other
hand, one also has that

$|\mathcal{S}_{1}*\mathcal{S}_{2}|=|\mathcal{S}_{1}|+|\mathcal{S}_{2}|-v\leq|\mathcal{S}_{1}|+|\mathcal{S}_{2}|+C\delta^{\prime}\leq|\mathcal{S}_{1}|+|\mathcal{S}_{2}|+1.$

This ends the proof. ∎

######  Lemma 5.6.

There exists a large constant $P_{0}>1$ such that if two segments
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ satisfy $|\mathcal{S}_{1}|\geq P_{0}$,
$|\mathcal{S}_{2}|\geq P_{0}$ and
$d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\leq\delta^{\prime}$, then

$d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L})\geq
2d(\mathcal{S}_{1}^{L},(\mathcal{S}_{1}*\mathcal{S}_{2})^{L})+2d((\mathcal{S}_{1}*\mathcal{S}_{2})^{R},\mathcal{S}_{2}^{R}).$

###### Proof.

First we fix a large constant $P_{0}\gg 1$ such that
$C^{2}e^{-\lambda(P_{0}-1)}+C^{2}e^{-\lambda P_{0}}<\frac{1}{2}$. Fix two
segments $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ as in Lemma. Note
$v=v(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$ and
$w=w(\mathcal{S}_{2}^{L},\mathcal{S}_{1}^{R})$. Then

$\displaystyle d(\mathcal{S}_{1}^{L},(\mathcal{S}_{1}*\mathcal{S}_{2})^{L})$
$\displaystyle=d(\phi_{-|\mathcal{S}_{1}|}(\mathcal{S}_{1}^{R}),\phi_{-|\mathcal{S}_{1}|}(w))$
$\displaystyle\leq Ce^{-\lambda|\mathcal{S}_{1}|}d(\mathcal{S}_{1}^{R},w)$
$\displaystyle\leq C^{2}e^{-\lambda
P_{0}}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

where we used $w\in W_{\epsilon}^{u}(\mathcal{S}_{1}^{R}).$ On the other hand,

$\displaystyle d(\mathcal{S}_{2}^{R},(\mathcal{S}_{1}*\mathcal{S}_{2})^{R})$
$\displaystyle=d(\phi_{|\mathcal{S}_{2}|-v}\phi_{v}(\mathcal{S}_{2}^{L}),\phi_{|\mathcal{S}_{2}|-v}(w))$
$\displaystyle\leq
Ce^{-\lambda(|\mathcal{S}_{2}|-v)}d(\phi_{v}(\mathcal{S}_{2}^{L}),w)$
$\displaystyle\leq
C^{2}e^{-\lambda(P_{0}-1)}d(\mathcal{S}_{1}^{R},\mathcal{S}_{2}^{L}),$

where we used $w\in W_{\epsilon}^{u}(\phi_{v}(\mathcal{S}_{2}^{L})).$ By
assumption, we have

$\displaystyle
d(\mathcal{S}_{1}^{L},(\mathcal{S}_{1}*\mathcal{S}_{2})^{L})+d(\mathcal{S}_{2}^{R},(\mathcal{S}_{1}*\mathcal{S}_{2})^{R})$
$\displaystyle\leq
C^{2}e^{-\lambda(P_{0}-1)}d(\mathcal{S}_{1}^{R},\mathcal{S}_{1}^{L})+C^{2}e^{-\lambda
P_{0}}d(\mathcal{S}_{1}^{R},\mathcal{S}_{1}^{L})$
$\displaystyle\leq\frac{1}{2}d(\mathcal{S}_{1}^{R},\mathcal{S}_{1}^{L}).$

This ends the proof. ∎

####  5.4.2. Periodic approximation

For integer $n\geq 1$, let $\Sigma_{n}=\\{0,1,2,\cdot,n-1\\}^{\mathbb{N}}$ and
$\sigma$ be a shift on $\Sigma_{n}$. Assume $F$ is a subset of $\bigcup_{i\geq
1}\\{0,1,2,\cdot,n-1\\}^{i}$, then the subshift with forbidden $F$ is denoted
by $(Y_{F},\sigma)$ where

$Y_{F}=\left\\{x\in\\{0,1,2,\cdot,n-1\\}^{\mathbb{N}},w\text{ does not appear
in }x\text{ for all }w\in F\right\\}.$

The following lemma is Lemma 5 of [BQ], which will be used later.

######  Lemma 5.7 ([BQ]).

Suppose that $(Y,\sigma)$ is a shift of finite type $($with forbidden words of
length 2$)$ with $M$ symbols and entropy $h$. Then $(Y,\sigma)$ contains a
periodic point of period at most $1+Me^{(1-h)}$.

Now we are ready to prove Lemma 3.11, which partially follow the argument in
[BQ].

######  Proof of Lemma 3.11.

Fix a positive constant
$\delta^{\prime\prime}\ll\frac{\delta^{\prime}}{C^{10}e^{\beta}}$. Let
$\mathcal{P}=\\{B_{1},B_{2},\cdots,B_{m}\\}$ be a finite partition of
$\Lambda$ with diameter smaller than $\delta^{\prime\prime}$. For
$x\in\Lambda$, $\widehat{x}\in\\{1,2,3,\cdots,m\\}^{\mathbb{N}}$ is defined by

$\widehat{x}(n)=j\text{ if }\phi_{n}(x)\in B_{j},n=0,1,2,\cdots.$

Denote $\widehat{Z}=\\{\widehat{x}:x\in Z\\}$ and $W_{n}$ the collection of
length $n$ string that appears in $\widehat{Z}$. One has

$\sharp W_{n}=K_{n}e^{nh}$

where $h=h_{top}(\overline{\widehat{Z}},\sigma)$ and $K_{n}$ grows at a
subexponential rate. Let

$Y_{n}=\\{y_{0}y_{1}y_{2}\cdots\in W_{n}^{\mathbb{N}}:y_{i}y_{i+1}\in
W_{2n}\text{ for all }i\in\mathbb{N}\\}$

and $(Y_{n},\sigma_{n})$ is the 1-step shift of finite type on $W_{n}$. From
Lemma 5.7, the shortest periodic orbit in $Y_{n}$ is at most
$1+K_{n}e^{nh}e^{1-nh}=1+eK_{n}$. Denote one of the shortest periodic orbit in
$Y_{n}$ by $z_{0}z_{1}z_{2}\cdots z_{p_{n}-1}z_{0}z_{1}z_{2}\cdots$ for some
$p_{n}\leq 1+eK_{n}$ and $z_{i}\in W_{n},i=0,1,2,\cdots,p_{n}-1$. For
$i=0,1,2,\cdots,p_{n}-1$, there is $x_{i}\in Z$ such that the leading $2n$
string of $\widehat{x_{i}}$ is $z_{i}z_{i+1}$ (we note $z_{p_{n}}=z_{0}$,
$x_{p_{n}}=x_{0}$, $\mathcal{S}_{p_{n}}=\mathcal{S}_{0},\cdots$). Choose
segments $\mathcal{S}_{i}$ by

$\mathcal{S}_{i}:[\frac{n}{2},\frac{3n}{2}+v_{i}]\to
M:t\to\phi_{t}(x_{i})\text{ for }i=0,1,2,\cdots,p_{n}-1,$

where $v_{i}=v(\phi_{n}(x_{i}),x_{i+1})$. We have the following Claim.

Claim Q1.  $d(\mathcal{S}_{i}^{R},\mathcal{S}_{i+1}^{L})\leq
2C^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}$ for
$i=0,1,2,\cdots,p_{n}-1$.

###### Proof of Claim Q1..

Note that the leading $2n$ string of $\widehat{x_{i}}$ is $z_{i}z_{i+1}$ and
leading $n$ string of $\widehat{x_{i+1}}$ is $z_{i+1}$, which means

$\mathcal{P}(\phi_{n+j}(x_{i}))=\mathcal{P}(\phi_{j}(x_{i+1}))\text{ for
}j=0,1,\cdots,n-1.$

Therefore,

$d(\phi_{n+j}(x_{i}),\phi_{j}(x_{i+1}))<\delta^{\prime\prime}\text{ for
}j=0,1,\cdots,n-1.$

Thus

$d(\phi_{n+t}(x_{i}),\phi_{t}(x_{i+1}))<Ce^{\beta}\delta^{\prime\prime}<\delta^{\prime}\text{
for }t\in[0,n].$

Then by Lemma 3.4, we have

$\displaystyle
d\left(\phi_{\frac{3n}{2}+v_{i}}(x_{i}),\phi_{\frac{n}{2}}(x_{i+1})\right)$
$\displaystyle\leq
C^{2}e^{-\frac{n\lambda}{2}}(d(\phi_{n}(x_{i}),x_{i+1})+d(\phi_{2n}(x_{i}),\phi_{n}(x_{i+1})))$
$\displaystyle\leq 2C^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

This ends the proof of Claim Q1. ∎

Now we define segments $\overline{\mathcal{S}}_{i}$ recursively for $0\leq
i\leq p_{n}-1$ by $\overline{\mathcal{S}}_{0}:=\mathcal{S}_{0}$ and

$\overline{\mathcal{S}}_{i}:=\overline{\mathcal{S}}_{i-1}*\mathcal{S}_{i}\text{
for }1\leq i\leq p_{n}-1.$

Based on Claim Q1, we have the following claim.

Claim Q2. There is a positive integer $N$ such that for any $n\geq N$, one has

  * (1).

$\overline{\mathcal{S}}_{i}$ is well defined for $0\leq i\leq p_{n}-1;$

  * (2).

$d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})\leq
4C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}<\delta^{\prime}$ for
$0\leq i\leq p_{n}-2;$

  * (3).

$d(\overline{\mathcal{S}}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{p_{n}-1}^{L})\leq
2C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}<\min\left\\{\delta^{\prime},\frac{1}{L}\right\\}$,
where $L$ is as in Lemma 3.5;

  * (4).

$(n-1)p_{n}\leq|\overline{\mathcal{S}}_{p_{n}-1}|\leq(n+1)p_{n};$

  * (5).

$\max_{x\in\overline{\mathcal{S}}_{p_{n}-1}}d(x,Z)\leq
C^{4}p_{n}^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

###### Proof of Claim Q2..

Since $p_{n}$ grows at a subexponential rate, we can take $N$ large enough
such that

$\displaystyle N>P_{0}\text{ and
}4p_{n}C^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}<\min\left\\{\delta^{\prime},\frac{1}{L}\right\\}\text{
for all }n\geq N,$ (5.20)

where $P_{0}$ is the constant as in Lemma 5.6. For $0\leq i\leq p_{n}-2$, we
define

$\chi(i)=d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})+d(\mathcal{S}_{i+1}^{R},\mathcal{S}_{i+2}^{L})+\cdots+d(\mathcal{S}_{p_{n}-2}^{R},\mathcal{S}_{p_{n}-1}^{L})+d(\mathcal{S}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{i}^{L}).$

By Claim Q1,

$\chi(0)\leq 2C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

Now we are to show that $\chi(i)$ and $\overline{\mathcal{S}}_{i}$ are well
defined, which satisfy that

$\chi(i)\leq\delta^{\prime}\text{ and
}|\overline{\mathcal{S}}_{i}|>P_{0}\text{ for }i=0,1,2,\cdots,p_{n}-2.$

These are clearly true for $i=0$. Now we assume that these are true for some
$i\in\\{0,1,2\cdots,p_{n}-2\\}$. Then for $i+1$, since
$\chi(i)\leq\delta^{\prime}$, one has
$d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})\leq\delta^{\prime}$.
Thus, we can join $\overline{\mathcal{S}}_{i}$ and $\mathcal{S}_{i+1}$ by
letting

$\overline{\mathcal{S}}_{i+1}=\overline{\mathcal{S}}_{i}*\mathcal{S}_{i+1}.$

It is clear that $|\overline{\mathcal{S}}_{i+1}|>P_{0}$ by Lemma 5.5 (2).

On the other hand, by triangle inequality and Lemma 5.6, one has that

$\displaystyle\begin{split}&\chi(i)-\chi(i+1)\\\
=&d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})+d(\mathcal{S}_{i+1}^{R},\mathcal{S}_{i+2}^{L})-d(\overline{\mathcal{S}}_{i+1}^{R},\mathcal{S}_{i+2}^{L})+d(\mathcal{S}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{i}^{L})-d(\mathcal{S}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{i+1}^{L})\\\
\geq&d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})-d(\overline{\mathcal{S}}_{i}^{L},\overline{\mathcal{S}}_{i+1}^{L})-d(\overline{\mathcal{S}}_{i+1}^{R},\mathcal{S}_{i+1}^{R})\\\
\geq&\frac{1}{2}d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L}).\end{split}$
(5.21)

Therefore, $\chi(i+1)\leq\chi(i)\leq\delta^{\prime}$. By induction, we can
finish the proof of (1).

By (LABEL:Q-315), one has

$d(\overline{\mathcal{S}}_{i}^{R},\mathcal{S}_{i+1}^{L})\leq
2\chi(i)\leq\cdots\leq 2\chi(0)\leq
4C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}\text{ for
}i=0,1,2\cdots,p_{n}-2,$

and we can deduce the following by triangle inequality:

$d(\overline{\mathcal{S}}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{p_{n}-1}^{L})\leq\chi(p_{n}-2)\leq\cdots\chi(0)\leq
2C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

This ends the proof of (2) as well as (1), (3), and (4) follows from Lemma 5.5
(2).

Now we denote
$D_{i}:=\overline{\mathcal{S}}_{i}\bigcup\cup_{j=i+1}^{p_{n}-1}\mathcal{S}_{j}$
for $i=0,1,2,\cdots,p_{n}-1.$ Then by Lemma 5.5 (1),

$\max_{x\in
D_{i+1}}d(x,D_{i})\leq\max_{x\in\overline{\mathcal{S}}_{i}*\mathcal{S}_{i+1}}d(x,\overline{\mathcal{S}}_{i}\cup\mathcal{S}_{i+1})\leq
C^{3}d(\mathcal{S}_{i}^{R},\mathcal{S}_{i+1}^{L})\leq
2C^{5}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

Therefore, by triangle inequality and the fact that $D_{0}\subset Z$,

$\max_{x\in\overline{\mathcal{S}}_{p_{n}-1}}d(x,Z)\leq\max_{x\in
D_{0}}d(x,Z)+\sum_{i=0}^{p_{n}-2}\max_{x\in D_{i+1}}d(x,D_{i})\leq
2C^{5}p_{n}^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

This ends the proof of Claim Q2. ∎

Recall that $P_{0}$ is the constant as in Lemma 5.6, $K,L$ are the constants
as in Lemma 3.5 and $N$ is the constant as in Claim Q2. We fix an integer
$n>\max(P_{0},K,N)+1$ and let $\overline{\mathcal{S}}_{p_{n}-1}$ be the
segment as in Claim Q2. Then by Claim Q2 (3),

$|\overline{\mathcal{S}}_{p_{n}-1}|>K\text{ and
}d(\overline{\mathcal{S}}_{p_{n}-1}^{R},\overline{\mathcal{S}}_{p_{n}-1}^{L})\leq\delta^{\prime}.$

Applying the Anosov Closing Lemma, we have a periodic segment
$\mathcal{O}_{n}$ such that

$\displaystyle\left||\overline{\mathcal{S}}_{p_{n}-1}|-|\mathcal{O}_{n}|\right|\leq
Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right)$
(5.22)

and

$\displaystyle
d\left(\phi_{t}(\mathcal{O}_{n}^{L}),\phi_{t}(\overline{\mathcal{S}}_{p_{n}-1}^{L})\right)\leq
Ld(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R})\
\forall
t\in\left[0,\max\left(|\overline{\mathcal{S}}_{p_{n}-1}|,|\mathcal{O}_{n}|\right)\right].$
(5.23)

We claim the following:

Claim Q3.
$\max_{x\in\mathcal{O}_{n}}d(x,Z)\leq(2C^{3}Lp_{n}+C^{4}p_{n}^{2})e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

###### Proof of Claim Q3.

If $|\mathcal{O}|\leq|\overline{\mathcal{S}}_{p_{n}-1}|$, by (5.23),

$\max_{x\in\mathcal{O}}d(x,\overline{\mathcal{S}}_{p_{n}-1})\leq
Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right).$

If $|\mathcal{O}|>|\overline{\mathcal{S}}_{p_{n}-1}|$, note
$t_{*}=\min(t,|\overline{\mathcal{S}}_{p_{n}-1}|)$ for
$t\in[0,|\mathcal{O}|]$. Then, by (5.22) and (5.23), one has that for
$t\in[0,|\mathcal{O}|]$

$\displaystyle
d\left(\phi_{t}(\mathcal{O}^{L}),\overline{\mathcal{S}}_{p_{n}-1}\right)$
$\displaystyle\leq
d\left(\phi_{t}(\mathcal{O}^{L}),\phi_{t_{*}}(\mathcal{O}^{L})\right)+d\left(\phi_{t_{*}}(\mathcal{O}^{L}),\phi_{t_{*}}(\overline{\mathcal{S}}_{p_{n}-1}^{L})\right)$
$\displaystyle\leq
CLd\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right)+Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right)$
$\displaystyle\leq
C^{2}Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right),$

where $C\gg 1$ as in Remark 3.2. Therefore,

$\max_{x\in\mathcal{O}}d\left(x,\overline{\mathcal{S}}_{p_{n}-1}\right)=\max_{t\in[0,|\mathcal{O}|]}d\left(\phi_{t}(\mathcal{O}^{L}),\overline{\mathcal{S}}_{p_{n}-1}\right)\leq
C^{2}Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right).$

Combining with (3), (5) of Claim Q2, we have

$\displaystyle\max_{x\in\mathcal{O}}d(x,Z)$
$\displaystyle\leq\max_{x\in\mathcal{O}}d\left(x,\overline{\mathcal{S}}_{p_{n}-1}\right)+\max_{x\in\overline{\mathcal{S}}_{p_{n}-1}}d(x,Z)$
$\displaystyle\leq C^{2}L\cdot
2C^{2}p_{n}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}+C^{5}p_{n}^{2}e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}$
$\displaystyle=(2C^{4}Lp_{n}+C^{5}p_{n}^{2})e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}.$

This ends the proof of Claim Q3. ∎

By Claim Q3 and (5.22), we have

$\displaystyle d_{\alpha,Z}(\mathcal{O}_{n})$
$\displaystyle\leq|\mathcal{O}_{n}|\left(\max_{x\in\mathcal{O}}d(x,Z)\right)^{\alpha}$
$\displaystyle\leq\left(|\overline{\mathcal{S}}_{p_{n}-1}|+Ld\left(\overline{\mathcal{S}}_{p_{n}-1}^{L},\overline{\mathcal{S}}_{p_{n}-1}^{R}\right)\right)\cdot\left((2C^{4}Lp_{n}+C^{5}p_{n}^{2})e^{-\frac{n\lambda}{2}}\delta^{\prime\prime}\right)^{\alpha}$
$\displaystyle\leq H_{n}e^{-\frac{n\alpha\lambda}{2}},$

where
$H_{n}=\big{(}(2C^{4}Lp_{n}+C^{5}p_{n}^{2})\delta^{\prime\prime}\big{)}^{\alpha}\cdot((n+1)p_{n}+1)$.
Note that $H_{n}$ grows at a subexponential rate as $n$ increases as $p_{n}$
does. Hence

$\limsup_{P\to+\infty}P^{k}\min_{\mathcal{O}\in\mathcal{O}^{P}_{\Lambda}}d_{\alpha,Z}(\mathcal{O})\leq\limsup_{n\to+\infty}((n+1)p_{n}+1)^{k}\cdot
H_{n}e^{-\frac{n\alpha\lambda}{2}}=0,$

where we used the fact that $p_{n},H_{n}$ grow at a subexponential rate as $n$
increases and $|\mathcal{O}_{n}|\leq np_{n}+1$. The proof of Lemma 3.11 is
completed. ∎

##  6\. Further discussions on the case of $C^{s,\alpha}$-observables

For $s\in\mathbb{N}$, $0\leq\alpha\leq 1$ and a strictly positive function
$\psi$ on $M$, $Per^{*}_{s,\alpha}(M,\psi)$ is defined as the collection of
$C^{s,\alpha}$-continuous functions on $M$, such that for each $u\in
Per^{*}_{s,\alpha}(M,\psi)$, $\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)$ contains
at least one periodic measure. And $Loc_{s,\alpha}(M,\psi)$ is defined by

$\displaystyle Loc_{s,\alpha}(M,\psi):=\\{$ $\displaystyle u\in
Per^{*}_{s,\alpha}(M,\psi):\text{ there is }\varepsilon>0\text{ such that }$
$\displaystyle\mathcal{M}_{min}(u+h;\psi,\Lambda,\Phi)=\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)\text{
for all }\|h\|_{r,\alpha}<\varepsilon\\}.$

In the case $s\geq 1$ and $\alpha>0$ or $s\geq 2$, we do not have analogous
result like Proposition 4.7. However, we have the following weak version.

######  Proposition 6.1.

Let $\mathcal{O}$ be a periodic segment of $\Phi|_{\Lambda}$ with
$D(\mathcal{O})>0$ and $u\in\mathcal{C}(M)$ with $u|_{\mathcal{O}}=0$ and
$u|_{M/\mathcal{O}}>0$. Then there exists a constant $\varrho>0$ such that the
probability measure

$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\Lambda,\Phi),$

where $h$ is any $\mathcal{C}^{0,1}(M)$ function with $\|h\|_{1}<\varrho$.

######  Remark 6.2.

As in Remark 4.12, for $s\in\mathbb{N}$ and $0\leq\alpha\leq 1$, we let
$\widetilde{w}\in\mathcal{C}^{s,\alpha}(M)$ such that
$\|\widetilde{w}\|_{s,\alpha}<\varepsilon$, $\widetilde{w}|_{\mathcal{O}}=0$
and $\widetilde{w}|_{M\setminus\mathcal{O}}>0$. Then $\mu_{\mathcal{O}}$ is
the unique measure in $\mathcal{M}_{min}(u+\widetilde{w}+h;\psi,\Lambda,\Phi)$
whenever $\|h\|_{s,\alpha}<\varrho$. The Proposition shows that there is an
open subset of $\mathcal{C}^{s,\alpha}({M})$ near $u$ such that functions in
the open set have the same unique minimizing measure with respect to $\psi$
and the probability measure supports on a periodic orbit.

By using Remark 6.2, we have the following result.

######  Theorem 6.3.

$Loc_{s,\alpha}(M,\psi)$ is an open dense subset of
$Per^{*}_{s,\alpha}(M,\psi)$ w.r.t. $\|\cdot\|_{s,\alpha}$ for integer $s\geq
1$ and real number $0\leq\alpha\leq 1$.

###### Proof.

Given $s\geq 1$ and $0\leq\alpha\leq 1$. The openness is clearly true. We
prove $Loc_{s,\alpha}(M,\psi)$ is dense in $Per^{*}_{s,\alpha}(M,\psi)$ w.r.t.
$\|\cdot\|_{s,\alpha}$. Since

$\int ud\mu=\int\bar{u}d\mu\text{ for all
}\mu\in\mathcal{M}(\Phi|_{\Lambda}),$

we have
$\mathcal{M}_{min}(u;\psi,\Lambda,\Phi)=\mathcal{M}_{min}(\bar{u};\psi,\Lambda,\Phi)$.
Then the theorem follows from Remark 6.2 immediately. ∎

###  6.1. Proof of Proposition 6.1

Now we finish the proof of Proposition 6.1.

###### Proof of Proposition 6.1.

Let $\mathcal{O}$ be a periodic segment of $\Phi|_{\Lambda}$ and
$u\in\mathcal{C}(M)$ with $u|_{\mathcal{O}}=0$ and $u|_{M/\mathcal{O}}>0$. For
$0\leq\rho\leq D(\mathcal{O})$, we note
$\theta(\rho)=\min\\{u(x):d(x,\mathcal{O})\geq\rho,x\in M\\}.$ It is clear
that $\theta(0)=0$, $\theta(\rho)>0$ for $\rho\neq 0$ and $\theta$ is non-
decreasing. Since $D(\mathcal{O})>0$ by assumption, there are two constants
$\rho_{1},\rho_{2}$ satisfy

$\displaystyle 0<\rho_{1}<\rho_{2}<\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}.$
(6.1)

Next we will show that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\Lambda,\Phi)$ for all $h\in
C^{0,1}(M)$ with $\|h\|_{1}<\varrho$, where the constant $\varrho$ is positive
and

$\displaystyle\varrho<\frac{1}{2}\min\left\\{\frac{\psi_{min}\theta(\rho_{1})}{1+\psi_{min}},\frac{\theta(\frac{D(\mathcal{O})}{4C^{2}e^{2\beta}})}{(1+\frac{\|\psi\|_{1}}{\psi_{min}})\cdot\frac{4C^{3}\rho_{2}}{\lambda}+|\mathcal{O}|\cdot\frac{1+\psi_{min}}{\psi_{min}}+\frac{1+\psi_{min}}{\psi_{min}}}\right\\}.$
(6.2)

Now we fix a function $h$ as above. Note $G=u+h-a_{\mathcal{O}}\psi$ where

$\displaystyle\begin{split}a_{\mathcal{O}}&=\frac{\left\langle\mathcal{O},u+h\right\rangle}{\left\langle\mathcal{O},\psi\right\rangle}\leq\frac{\|h\|_{0}}{\psi_{min}}.\end{split}$
(6.3)

Then $\frac{\int Gd\mu}{\int\psi d\mu}=\frac{\int u+hd\mu}{\int\psi
d\mu}-a_{\mathcal{O}}.$ Therefore, to show that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\Lambda,\Phi)$, it is enough
to show that

$\int Gd\mu\geq 0\text{ for all }\mu\in\mathcal{M}^{e}(\Phi|_{\Lambda}),$

where we used the assumption $\psi$ is strictly positive and the fact $\int
Gd\mu_{\mathcal{O}}=0$. Now we let $Area_{1}:=\\{y\in
M:d(y,\mathcal{O})\leq\rho_{1}\\}.$ We have the following claim.

Claim F1. $Area_{1}$ contains all $x\in M$ with $G(x)\leq 0$.

###### Proof of Claim F1.

For $x\notin Area_{1}$, we have

$\displaystyle G(x)$
$\displaystyle=u(x)+h(x)-a_{\mathcal{O}}\psi\geq\theta(\rho_{1})-\|h\|_{0}-a_{\mathcal{O}}\|\psi\|_{0}\geq\theta(\rho_{1})-\frac{1+\psi_{min}}{\psi_{min}}\|h\|_{0}>0.$

where we used (6.2) and (6.3). This ends the proof of Claim F1.∎

Note $Area_{2}=\\{y\in M:d(y,\mathcal{O})\leq\rho_{2}\\}.$ It is clear that
$Area_{1}$ is in the interior of $Area_{2}$. Thus, $d(Area_{1},M\setminus
Area_{2})>0$. Therefore, by Claim F1, we can fix a constant $0<\tau<1$ such
that $G(\phi_{t}(x))>0$ for all $x\in M\setminus Area_{2}$ and $|t|\leq\tau$.

Claim F2. If $z\in\Lambda$ is not a generic point of $\mu_{\mathcal{O}}$, then
there is $m\geq\tau$ such that $\int_{0}^{m}G(\phi_{t}(z))dt>0$.

Next we prove Proposition 6.1 by assuming the validity of Claim F2, proof of
which is left to the next subsection. Same as the argument at the beginning of
the proof, it is enough to show that for all
$\mu\in\mathcal{M}^{e}(\Phi|_{\Lambda})$

$\int Gd\mu\geq 0.$

Given $\mu\in\mathcal{M}^{e}(\Phi|_{\Lambda})$, in the case
$\mu=\mu_{\mathcal{O}}$, it is obviously true. In the case
$\mu\neq\mu_{\mathcal{O}}$, just let $z$ be a generic point of $\mu$. Note
that $z$ is not a generic point of $\mu_{\mathcal{O}}$. By Claim F2, we have
$t_{1}\geq\tau$ such that

$\int_{0}^{t_{1}}G(\phi_{t}(z))dt>0.$

Note that $\phi_{t_{1}}(z)$ is still not a generic point of
$\mu_{\mathcal{O}}$. Apply Claim F2 again, we have $t_{2}\geq t_{1}+\tau$ such
that

$\int_{t_{1}}^{t_{2}}G(\phi_{t}(z))dt>0.$

By repeating the above process, we have $0\leq t_{1}<t_{2}<t_{3}<\cdots$ with
the gap not less than $\tau$ such that

$\int_{t_{i}}^{t_{i+1}}G(\phi_{t}(z))dt>0\text{ for }i=0,1,2,3,\cdots,$

where $t_{0}=0$. Therefore,

$\displaystyle\int Gd\mu$
$\displaystyle=\lim_{m\to+\infty}\frac{1}{m}\int_{0}^{m}G(\phi_{t}(z))dt$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{t_{i}}\left(\int_{t_{0}}^{t_{1}}G(\phi_{t}(z))dt+\int_{t_{1}}^{t_{2}}G(\phi_{t}(z))dt+\cdots+\int_{t_{i-1}}^{t_{i}}G(\phi_{t}(z))dt\right)$
$\displaystyle\geq 0.$

That is, $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\Lambda,\Phi)$. This
ends the proof of the Proposition. ∎

###  6.2. Proof of Claim F2

Assume that $z$ is not a generic point of $\mu_{\mathcal{O}}$, if $z\notin
Area_{2}$, let $m=\tau$, we have nothing to prove since $G(\phi_{t}(z))>0$ for
all $|t|\leq\tau$. Now we assume that $z\in Area_{2}$. In the case that
$d(\phi_{t}(z),\mathcal{O})<\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}$ for all
$t\geq 0$, by Lemma 4.2, $z$ is a generic point of $\mu_{\mathcal{O}}$, which
contradicts to our assumption. Hence, there must be some $m_{1}>0$ such that

$d(\phi_{m_{1}}(z),\mathcal{O})\geq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}.$

We can assume $m_{2}>0$ be the smallest time such that

$\displaystyle
d(\phi_{m_{2}}(z),\mathcal{O})\geq\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}.$
(6.4)

The existence of $m_{2}$ is ensured by (6.1). Then for $0\leq t\leq 1$,

$d(\phi_{m_{2}-t}(z),\mathcal{O})\geq\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}.$

Then

$\displaystyle\begin{split}\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt&=\int_{m_{2}-1}^{m_{2}}{u}(\phi_{t}(z))+h(\phi_{t}(z))-a_{\mathcal{O}}\psi(\phi_{t}(z))dt\\\
&\geq\int_{m_{2}-1}^{m_{2}}\theta\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)-\|h\|_{0}-a_{\mathcal{O}}\|\psi\|_{0}dt\\\
&\geq\theta\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)-\frac{1+\psi_{min}}{\psi_{min}}\|h\|_{0}.\\\
\end{split}$ (6.5)

where we used (6.3) and the definition of $\theta(\cdot)$. On the other hand,
one has that $\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}>\rho_{2}$, which implies
that

$\displaystyle\phi_{m_{2}-t}(z)\notin Area_{2}\text{ for all }0\leq t\leq 1.$
(6.6)

Since $Area_{2}$ is compact, we can take $m_{3}$ the largest time with $0\leq
m_{3}\leq m_{2}$ such that

$\phi_{m_{3}}(z)\in Area_{2},$

where we use the assumption $z\in Area_{2}$. By (6.6), it is clear that
$m_{3}<m_{2}-1$. Then by Claim F1 and the fact that $Area_{1}\subset
Area_{2}$,

$\displaystyle G(\phi_{t}(z))>0\text{ for all }m_{3}<t<m_{2}-1.$ (6.7)

Since $m_{3}<m_{2}$, one has by (6.4) that

$d(\phi_{t}(z),\mathcal{O})<\frac{D(\mathcal{O})}{4C^{2}e^{\beta}}<\delta^{\prime}\text{
for all }0\leq t\leq m_{3}.$

Therefore, by Lemma 4.2, there is $y_{0}\in\mathcal{O}$ such that

$d(\phi_{t}(z),\phi_{t}(y_{0}))\leq
Cd(\phi_{t}(z),\mathcal{O})\leq\frac{D(\mathcal{O})}{4Ce^{\beta}}\text{ for
all }t\in[0,m_{3}].$

Also notice that

$d(z,y_{0})\leq C\rho_{2}\text{ and
}d(\phi_{m_{3}}(z),\phi_{m_{3}}(y_{0}))\leq C\rho_{2},$

where we used $z,\phi_{m_{3}}(z)\in Area_{2}$. By using Lemma 3.4, we have for
all $0\leq t\leq m_{3}$,

$\displaystyle d(\phi_{t}(z),\phi_{t}\phi_{v}(y_{0}))$ $\displaystyle\leq
C^{2}e^{-\lambda\min(t,m_{3}-t)}(d(z,y_{0})+d(\phi_{m_{3}}(z),\phi_{m_{3}}(y_{0})))$
$\displaystyle\leq 2C^{3}e^{-\lambda\min(t,m_{3}-t)}\rho_{2},$

where $v=v(y_{0},z)$. Hence,

$\displaystyle\begin{split}\int_{0}^{m_{3}}d(\phi_{t}(z),\phi_{t}\phi_{v}(y_{0}))dt&\leq\int_{0}^{m_{3}}2C^{3}\rho_{2}(e^{-\lambda
t}+e^{-\lambda(m_{3}-t)})dt\leq\frac{4C^{3}\rho_{2}}{\lambda}.\end{split}$

Since $u(\phi_{t}(y_{0}))=0$ for all $t\in\mathbb{R}$ and $u\geq 0$, one has

$\displaystyle\begin{split}&\int_{0}^{m_{3}}G(\phi_{t}(z))-G(\phi_{t+v}(y_{0}))dt\\\
=&\int_{0}^{m_{3}}{u}(\phi_{t}(z))+h(\phi_{t}(z))-{u}(\phi_{t+v}(y_{0}))-h(\phi_{t+v}(y_{0}))-a_{\mathcal{O}}(\psi(\phi_{t}(z))-\psi(\phi_{t+v}(y_{0})))dt\\\
\geq&\int_{0}^{m_{3}}h(\phi_{t}(z))-h(\phi_{t+v}(y_{0}))-a_{\mathcal{O}}(\psi(\phi_{t}(z))-\psi(\phi_{t+v}(y_{0})))dt\\\
\geq&-(\|h\|_{1}+|a_{\mathcal{O}}|\|\psi\|_{1})\int_{0}^{m_{3}}d(\phi_{t}(z),\phi_{t+v}(y_{0}))dt\\\
\geq&-(\|h\|_{1}+\frac{\|h\|_{0}\|\psi\|_{1}}{\psi_{min}})\cdot\frac{4C^{3}\rho_{2}}{\lambda}\\\
\geq&-\|h\|_{1}(1+\frac{\|\psi\|_{1}}{\psi_{min}})\cdot\frac{4C^{3}\rho_{2}}{\lambda}.\end{split}$
(6.8)

By assuming that $m_{3}=p|\mathcal{O}|+q$ for some nonnegative integer $p$ and
real number $0\leq q\leq|\mathcal{O}|$, one has by (6.4) that

$\displaystyle\begin{split}\int_{0}^{m_{3}}G(\phi_{t+v}(y_{0}))dt&=\int_{m_{3}-q}^{m_{3}}G(\phi_{t+v}(y_{0}))dt\geq-|\mathcal{O}|\cdot\frac{1+\psi_{min}}{\psi_{min}}\|h\|_{0},\end{split}$
(6.9)

where we used $\int Gd\mu_{\mathcal{O}}=0$. Combining (6.3), (6.5), (6.7),
(6.8) and (6.9), we have

$\displaystyle\ \ \ \int_{0}^{m_{2}}G(\phi_{t}(z))dt$
$\displaystyle\geq\int_{0}^{m_{3}}G(\phi_{t}(z))dt+\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt$
$\displaystyle=\int_{0}^{m_{3}}G(\phi_{t}(z))-G(\phi_{t+v}(y_{0}))dt+\int_{0}^{m_{3}}G(\phi_{t+v}(y_{0}))dt+\int_{m_{2}-1}^{m_{2}}G(\phi_{t}(z))dt$
$\displaystyle\geq-\|h\|_{1}(1+\frac{\|\psi\|_{1}}{\psi_{min}})\cdot\frac{4C^{3}\rho_{2}}{\lambda}-|\mathcal{O}|\cdot\frac{1+\psi_{min}}{\psi_{min}}\|h\|_{0}+\theta\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)-\frac{1+\psi_{min}}{\psi_{min}}\|h\|_{0}$
$\displaystyle=\theta\left(\frac{D(\mathcal{O})}{4C^{3}e^{2\beta}}\right)-\left((1+\frac{\|\psi\|_{1}}{\psi_{min}})\cdot\frac{4C^{3}\rho_{2}}{\lambda}+|\mathcal{O}|\cdot\frac{1+\psi_{min}}{\psi_{min}}+\frac{1+\psi_{min}}{\psi_{min}}\right)\|h\|_{1}$
$\displaystyle>0,$

where we used assumption (6.2). Therefore, $m=m_{2}$ is the time as required
since $m_{2}\geq 1>\tau$ by (6.7). This completes the proof of Claim F2.

## Acknowledgement

At the end, we would like to express our gratitude to Tianyuan Mathematical
Center in Southwest China, Sichuan University and Southwest Jiaotong
University for their support and hospitality.

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