# Ergodic optimization theory for a class of typical maps

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,
Huazhong University of Sciences and Technology,
Wuhan 430074, China<EMAIL_ADDRESS>
###### Abstract.

In this article, we consider the weighted ergodic optimization problem of a
class of dynamical systems $T:X\to X$ where $X$ is a compact metric space and
$T$ is Lipschitz continuous. We show that once $T:X\to X$ satisfies both the
Anosov shadowing property (ASP) and the Mañé-Conze-Guivarc’h-Bousch property
(MCGBP), the minimizing measures of generic Hölder observables are unique and
supported on a periodic orbit. Moreover, if $T:X\to X$ is a subsystem of a
dynamical system $f:M\to M$ (i.e. $X\subset M$ and $f|_{X}=T$) where $M$ is a
compact smooth manifold, the above conclusion holds for $C^{1}$ observables.

Note that a broad class of classical dynamical systems satisfies both ASP and
MCGBP, which includes Axiom A attractors, Anosov diffeomorphisms and uniformly
expanding maps. Therefore, the open problem proposed by Yuan and Hunt in [YH]
for $C^{1}$-observables is solved consequentially.

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 mainly studies the
problems relating to minimizing (or maximum) orbits, minimizing (or maximum)
invariant measures and minimizing (or maximum) ergodic averages.
This theory has strong connection with other fields, such as Anbry-Mather
theory [Co2, Mane] in Lagrangian Mechanics; ground state theory [BLL] in
thermodynamics formalism and multifractal analysis; and controlling chaos
[OGY90, SGOY] in control theory.

Let $(X,T)$ be a dynamical system and $u$ be a real-valued function on $X$.
For a given orbit of $T$, say $\mathcal{O}=\\{T^{i}x\\}_{i=0,1,\cdots}$, the
time average of $u$ along $\mathcal{O}$ is usually defined by
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)$ when it converges. An
orbit is called $u$-minimizing if the time average of $u$ along the orbit is
less than along any other orbits.
In a reasonably general case, $X$ is assumed compact and $T$, $u$ are assumed
continuous. Thereafter, $\mathcal{M}(X,T)$, the set of all $T$-invariant Borel
probability measures on $X$, becomes a non-empty convex and compact
topological space with respect to weak∗ topology. It is well known that, by
Birkhoff ergodic theorem, for a given $\mu\in\mathcal{M}(X,T)$, the time
averages of $u$ are well defined for orbits initiated on $\mu$-a.e. $x$.
Moreover, when $\mu$ is ergodic, time averages of $u$ become constant
($\mu$-a.e.) and equal to the space average $\int_{X}ud\mu$ which is also
called the ergodic average of $u$ with respect to $\mu$. Denote by
$\mathcal{M}^{e}(X,T)\subset\mathcal{M}(X,T)$ the collection of ergodic
measures, which is the set of the extremal points of $\mathcal{M}(X,T)$. By
the compactness and convexity of $\mathcal{M}(X,T)$, the minimizing (or
maximizing) ergodic average of $u$ always exists and can be achieved by some
ergodic measure which is called $u$-minimizing measure.

To understand minimizing measures is the main task of the classical ergodic
optimization theory. This kind of problem involves three main factors:
complexity of the systems, regularity of the observable functions, and
complexity of the minimizing measures. The well known Meta-Conjecture states
that when the systems is chaotic then the minimizing measures of generic
typical observables have low complexity. As a special case of the Meta-
Conjecture, a more precise conjecture, the Typical Periodic Optimization (TPO)
Conjecture, is proposed by Yuan and Hunt ([YH], 1999) first and later in a
general form, which is focusing on a special class of chaotic systems
including uniformly hyperbolic systems and uniformly expanding maps, and is
conjecturing that for a suitably continuous real-valued observable space $V$
(e.g. Lipschitz observable space), $V_{per}$ contains an open and dense
subset, where $V_{per}$ is the subspace of $V$ such that for each $u\in
V_{per}$ the set of $u$-minimizing measures contains at least a periodic
measure. TPO Conjecture is one of the fundamental questions raised in the
field of ergodic optimization theory, which has attracted sustained attention
and yielded considerable results for last two decades.

The first attempt towards the TPO Conjecture is due to Contreras, Lopes and
Thieullen [CLT]. They considered the case where $T$ is smooth, orientation-
preserving, uniformly expanding map of the circle, and $V$ is the
$\alpha$-Hölder functions space $C^{0,\alpha}$, and derived a weaker version
result.
Afterwards, there is a series of works for the case that $(X,T)$ is subshifts
of finite type: for example, Bousch [Bousch_Walters] proved that the TPO
Conjecture holds for Walter functions; Quas and Siefken [QS] asserted the
validity of the conjecture for the case where $T$ is a full shift and $V$ is
the space of ”super continuous” functions; Bochi and Zhang ([BZ]) solved the
case when $T$ is a one-side shift on two symbols and $V$ is a space of
functions with strong modulus of regularity; Morris [Morris_entropy] proved
that for a generic real-valued Hölder continuous function on a subshift of
finite type, the maximizing measure must have zero entropy.
For more comprehensive survey for the classical ergodic optimization theory,
we refer the readers to Jenkinson [Jenkinson_survey, Je2], to Bochi [B] and to
Baraviera, Leplaideur, Lopes [BLL] for a historical perspective of the
development in this area.
In the existing literature, the best result towards the TPO Conjecture is
obtained by Conteras [Contreras_EO], which built on work of [Bousch_Mane,
Morris_entropy, QS, YH]. The main result obtained in [Contreras_EO] states
that for a uniformly expanding map the minimizing measures of (topological)
generic Lipschitz observations are uniquely supported on periodic orbits.

The purpose of this paper is to investigate validity of the TPO conjecture for
a considerably broad class of typical dynamical systems such as Anosov
diffeomorphisms. Precisely, systems considered in this paper are assumed to be
Lipschitz and satisfy the so called Anosov Shadow property (abbr. ASP) and
Mañé-Conze-Guivarc’h-Bousch property (abbr. MCGBP), definitions of which are
given in Section 2. In the existing literature, Axiom A attractors, Anosov
diffeomorphisms and expanding maps all satisfy both ASP and MCGBP.

Summary of the main result: In this paper, we extend the validity of the TPO
Conjecture on two scopes: systems and observables. To avoid tediousness, we
summarize only part of the main result into the following theorem, the precise
statement of which will be given in Section 2.

Theorem A: For an Axiom A or uniformly expanding system and a (topologically)
generic observable function from Hölder function space, Lipschitz function
space or $C^{1,0}$ function space (if well defined) the minimizing measure is
unique and is supported on a periodic orbit.

This gives a positive answer to the conjecture proposed by Yuan and Hunt on
1999 (Conjecture 1.1 in [YH]) for Hölder, Lipschitz, and $C^{1}$ cases.

Remarks on the techniques of the proof: A major ingredient of the proof in
this paper is based on a closing lemma due to Bressaud and Quas [BQ], which
allow us to identify the suitable periodic orbits with ”good shape”. By the
”good shape”, roughly speaking, we mean that the periodic orbit should satisfy
three properties:1) period of the orbit should not be too large; 2) points of
the orbit should be distributed evenly, in another word, distance of
distinguished points from the orbit should not be too small; 3) orbit should
be close enough to a given invariant compact set. Based on these periodic
orbit with ”good shape”, one can construct a sequence of observables whose
minimizing measures are unique and periodic converges to a arbitrarily given
observable.

Comparing with the proofs of existing results such as Contreras’ proof in
[Contreras_EO], our proof follows a more direct way. For example, since
Contreras’ proof firstly went to an intermedia result of Morris
[Morris_entropy] which states that the minimizing measures of generic
Lipschitz observables have zero entropy, the methodology largely depends on
the uniformly expanding property of the system; In contrast, our proof is
devoted to searching the target periodic orbits directly, which automatically
avoids the entropy argument of the minimizing measures, thus the methodology
is applicable to a broader class of systems beyond expanding maps.

Organization of the paper: The structure of the paper is organized as follows.
In Section 2, we give the main setting and notions, and state the main result;
In Section 3, we give the proof of the main result (Theorem 2.1); In Section
4, we consider the case of observable functions with high regularity, for
which some partial results and remaining questions are presented; In Appendix
A, we briefly explain why Anosov diffeomorphisms being MCGBP for the sake of
completeness.

##  2\. settings and results

In this paper, we will study the typical optimization problem in weighted
ergodic optimization theory which has strong connection with zero temperature
limit. The current section is devoted to formulating the setting and stating
the result.

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

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

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

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

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

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

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

For each real number $\eta\geq 0$, we call a sequence
$\\{x_{i}\\}_{i=0}^{n-1}\subset X$ _a periodic $\eta$-pseudo-orbit_ of
$(X,T)$, if each $x_{i+1}$ belongs to an $\eta$-neighbourhood of $T(x_{i})$,
for all $i=0,\cdots,n-1\mod n$. With this convention, we say that $(X,T)$
satisfies Anosov Shadow property (abbr. ASP) and Mañé-Conze-Guivarc’h-Bousch
property (abbr. MCGBP), if

  1. A.

(ASP) There are positive constants $\lambda,\delta,C,L$ such that

    1. (1).

For $n\in\mathbb{N}$ and $x,y\in X$ with $d(T^{i}x,T^{i}y)\leq\delta$ for all
$0\leq i\leq n$, one has for all $0\leq k\leq n$, $d(T^{k}x,T^{k}y)\leq
Ce^{-\lambda\min(k,n-k)}(d(x,y)+d(T^{n}x,T^{n}y)).$

    2. (2).

For any $0\leq\eta\leq\delta$, $n\geq 1$ and a periodic $\eta$-pseudo-orbit
$\\{x_{i}\\}_{i=0}^{n-1}$, there is a periodic orbit
$\\{T^{i}x\\}_{i=0}^{m-1}$ with period $m$ such that $m|n$ and
$d(x_{i},T^{i}x)\leq L\eta,~{}~{}\forall 0\leq i\leq n-1$.

  2. B.

(MCGBP) For any $0<\alpha\leq 1$, there exists positive integer $K=K(\alpha)$
such that for all $u\in\mathcal{C}^{0,\alpha}(X)$, there is
$v\in\mathcal{C}^{0,\alpha}(X)$ such that

$\bar{u}:=u_{K}-v\circ T^{K}+v-\beta(u;{X},T)\geq 0$

where $u_{K}=\frac{1}{K}\sum_{i=0}^{K-1}u\circ T^{i}$ and
$\beta(u;{X},T)=\min_{\nu\in\mathcal{M}(X,T)}\int ud\nu$.

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

For the sake of completeness, we give a brief proof of Anosov diffeomorphisms
satisfying MCGBP in the Appendix Section A, while ASP is a standard property
of Anosov diffeomorphisms thus the proof of which is not repeated in this
paper.

In summary, let $\mathfrak{C}$ be the set of triple $(X,T,\psi)$ satisfying
the following properties:

  * H1)

$(X,d)$ is a compact metric space and $T:X\rightarrow X$ is Lipschitz
continuous;

  * H2)

$(X,T)$ satisfying ASP and MCGBP;

  * H3)

$\psi:X\to\mathbb{R}^{+}$ are continuous.

The main results obtained in this paper is summarized in the following:

######  Theorem 2.1.

Suppose $(X,T,\psi)\in\mathfrak{C}$, then the following hold:

  * I)

For $\alpha\in(0,1]$, if $\psi\in\mathcal{C}^{0,\alpha}(X)$, then there exists
an open and dense set $\mathfrak{P}\subset\mathcal{C}^{0,\alpha}(X)$ such that
for any $u\in\mathfrak{P}$, $(u,\psi)$-minimizing measure is uniquely
supported on a periodic orbit of $T$.

  * II)

If $(X,T)$ is a sub-system of a dynamical system $(M,f)$ (i.e. $X\subset M$
and $T=f|_{X}$) and $\psi\in\mathcal{C}^{0,1}(M)$, where $M$ is a compact
$\mathcal{C}^{\infty}$ Riemannian manifold , then there exists an open and
dense set $\mathfrak{P}\subset\mathcal{C}^{1,0}(M)$ such that for any
$u\in\mathfrak{P}$, the $(u|_{X},\psi|_{X})$-minimizing measure of $(X,T)$ is
uniquely supported on a periodic orbit of $T$, where $\mathcal{C}^{1,0}(M)$ is
the Banach space of continuous differentiable functions on $M$ endowed with
the standard $C^{1}$-norm.

######  Remark 2.2.

It is worth to point out that Theorem 2.1 only requires $(X,T)$ satisfy ASP
and MCGBP, which means that, in particular, neither topological transitivity
for Anosov diffeomorphisms (although it is conjectured that Anosov
diffeomorphisms are always topological transitive) nor non-wandering property
for Axiom A attractors are needed. If in addition $X\supset\text{supp}(\mu)$
for all $\mu\in\mathcal{M}(M,f)$ in Theorem 2.1 (II), then for any
$u\in\mathfrak{P}$, the $(u,\psi)$-minimizing measure of $(M,f)$ is also
uniquely supported on a periodic orbit of $f$.

On the other hand, the reason of adding the nonconstant 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, i.e., the
measure $\mu_{u,\psi}\in\mathcal{M}(X,T)$, which satisfies

$\mu_{u,\psi}:=\arg\max\left\\{h_{\nu}(T)+\frac{\int ud\nu}{\int\psi
d\nu}:~{}\forall\nu\in\mathcal{M}(X,T)\right\\},$ (2.2)

where $h_{\nu}$ is the Kolmogorov-Sinai entropy of $\nu$. Such weighted
equilibrium state arises naturally in the studies of non-conformal
multifractal analysis (e.g. high dimensional Lyapunov spectrum) for
asymptotically (sub)additive potentials, see works [BF, BCW, FH]. In fact,
when the ground state exists, (i.e., the limit
$\lim_{t\to+\infty}\mu_{tu,\psi}$ exists), then the limit formulates a special
candidate of $(-u,\psi)$-minimizing measure.

##  3\. Proof of Theorem 2.1

Before starting the proof, we introduce some notions at first for the sake of
convenience. For $(X,T,\psi)\in\mathcal{C}$ and a continuous function
$u:X\to\mathbb{R}$, define

$Z_{u,\psi}:=\overline{\cup_{\mu\in\mathcal{M}_{min}(u;\psi,{X},T)}supp(\mu)}.$
(3.1)

For $\alpha\in(0,1]$, a non-empty subset $Z$ of $X$ and a periodic orbit
$\mathcal{O}$ of $(X,T)$, define _the $\alpha$-deviation of $\mathcal{O}$ with
respect to $Z$_ by

$d_{\alpha,Z}(\mathcal{O})=\sum_{x\in\mathcal{O}}d^{\alpha}(x,Z),$

where we recall that $d$ is the metric on $X$. Let $\lambda,\delta,C,L$ be the
constants as in ASP and fix these notations. Define
$D:{X}\times{X}\to[0,+\infty)$ by

$D(x,y)=\left\\{\begin{array}[]{ll}\delta,&\text{ if }d(x,y)\geq\delta,\\\
d(x,y),&\text{ if }d(x,y)<\delta.\end{array}\right.$

By a periodic orbit $\mathcal{O}$ of $(X,T)$, the gap of $\mathcal{O}$ is
defined by

$D(\mathcal{O})=\left\\{\begin{array}[]{ll}\delta,&\text{ if
}\sharp\mathcal{O}=1,\\\ \min_{x,y\in\mathcal{O},x\not=y}D(x,y),&\text{ if
}\sharp\mathcal{O}>1.\end{array}\right.$ (3.2)

We will prove Part I) and Part II) of Theorem 2.1 separately.

###  3.1. Proof of Part I) of Theorem 2.1

The proof of Part I) of Theorem 2.1 mainly contains two steps:

  * Step 1.

We show how to construct periodic orbit $\mathcal{O}$ of $(X,T)$ to make the
ratio $\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z}(\mathcal{O})}$ as large as
needed. Such a periodic orbit will be a candidate to support the minimizing
measures of observables nearby $u$.

  * Step 2.

We show that for any given $u\in\mathcal{C}^{0,\alpha}(X)$ there exists an
open set $\mathcal{U}$ of observables being arbitrarily close to $u$, whose
minimizing measures are all supported on a single periodic orbit
$\mathcal{O}$.

The main aim of Step 1 can be summarized into the following Proposition:

######  Proposition 3.1.

Let $(X,T)$ satisfy H1) and ASP, $u:X\rightarrow\mathbb{R}$ and
$\psi:X\rightarrow\mathbb{R}^{+}$ are continuous. Then for any
$\alpha\in(0,1]$, a given $\tilde{L}>0$ and $T$-forward-invariant non-empty
subset $Z\subset X$ (i.e. $T(Z)\subset Z$), there exists an periodic orbit
$\mathcal{O}$ of $(X,T)$ such that

$\displaystyle\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z}(\mathcal{O})}>\hat{L}.$
(3.3)

Similarly, the main aim of Step 2 can be summarized into the following
Proposition:

######  Proposition 3.2.

Let $(X,T,\psi)\in\mathcal{C}$ (satisfying H1), H2) and H3)) and
$\psi,u\in\mathcal{C}^{0,\alpha}({X})$ for some $\alpha\in(0,1]$. Then for any
$\varepsilon>0$, there exist $\hat{L},\hat{\delta}>0$ which depend on
$\varepsilon,\alpha,u,\psi$ and system constants only such that the following
holds: If there is a periodic orbit $\mathcal{O}$ of $(X,T)$ satisfying

$\displaystyle\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z_{u,\psi}}(\mathcal{O})}>\hat{L},$
(3.4)

then the observable function $u_{\epsilon,h}:=u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h$ has a unique minimizing measure

$\mu_{\mathcal{O}}:=\frac{1}{\sharp\mathcal{O}}\sum_{x\in\mathcal{O}}\delta_{x}$

whenever $h\in\mathcal{C}^{0,\alpha}({X})$ with
$\|h\|_{\alpha}<10\epsilon\text{ and
}\|h\|_{0}<\frac{D^{\alpha}(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}.$

It is clear that the collection of $u_{\epsilon,h}$ in Proposition 3.2 forms
an non-empty open subset of $\mathcal{C}^{0,\alpha}(X)$, which is about
$\varepsilon$-apart from $u$. Since $\varepsilon$ can be taken arbitrarily
small and the existence of periodic orbit satisfying (3.4) are guaranteed by
Proposition 3.1, Part I) of Theorem 2.1 follows. Thus it remains to prove
Proposition 3.1 and 3.2.

####  3.1.1. Proof of Proposition 3.1

At first, we introduce a lemma giving a quantified estimate of the denseness
of periodic orbits, which can be viewed as a version of Quas and Bressaud’s
periodic approximation lemma.

######  Lemma 3.3.

Let $(X,T)$ be a dynamical system satisfying H1) and ASP, $Z$ be a nonempty
$T$-forward-invariant subset of $X$. Then for all $\alpha\in(0,1]$ and $k>0$,
one has

$\lim_{n\to\infty}n^{k}\min_{\mathcal{O}\in\mathcal{O}^{n}}d_{\alpha,Z}(\mathcal{O})=0,$

where $\mathcal{O}^{n}$ denote the collection of all periodic orbits of
$(X,T)$ with period not larger than $n$

###### Proof.

We follow the arguments in [BQ]. Before going to the proof, we need to state
two technical results first. The following lemma is Lemma 5 of [BQ].

Let $\Sigma_{n}=\\{0,1,2,\cdot,n-1\\}^{\mathbb{N}}$ and $\sigma$ is 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 noted by
$(Y_{F},\sigma)$ where

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

######  Lemma 3.4 ([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 $1+Me^{(1-h)}$.

######  Lemma 3.5.

Let $(X,T)$ be a dynamical system satisfying H1) and ASP, and $Z$ be a
nonempty subset of $X$. Then for any $0<\alpha\leq 1$, $0<\eta\leq\delta$,
$n\geq 0$, and periodic $\eta$-pseudo-orbit $\widetilde{\mathcal{O}}$ of
$(X,T)$ with period $n$, there exists a periodic orbit $\mathcal{O}$ of
$(X,T)$ with period $m$ such that $m|n$ and

$d_{\alpha,Z}(\mathcal{O})\leq
d_{\alpha,Z}(\widetilde{\mathcal{O}_{n}})+n(L\eta)^{\alpha},$

where $\delta,L$ are the constants as in ASP.

###### Proof.

For $n\geq 1$, assume $\widetilde{\mathcal{O}}=\\{x_{i}\\}_{i=0}^{n-1}$ is a
periodic $\eta$-pseudo-orbit with period $n$. By ASP, there is a periodic
orbit $\mathcal{O}=\\{x,Tx,\cdots,T^{m-1}x\\}$ such that $m|n$ and
$d(x_{i},T^{i}x)\leq L\eta$ for $0\leq i\leq n-1$. Therefore, for
$0<\alpha\leq 1$, one has

$d_{\alpha,Z}(\mathcal{O})=\sum_{i=0}^{m-1}d^{\alpha}(T^{i}x,Z)\leq\sum_{i=0}^{n-1}(d(x_{i},Z)+L\eta)^{\alpha}\leq
d_{\alpha,Z}(\widetilde{\mathcal{O}})+n(L\eta)^{\alpha}.$

This ends the proof. ∎

Now, we are ready to prove Lemma 3.3.

Fix $\alpha,k,Z$ as in the lemma and $\lambda,\delta,C,L$ are the constants as
in ASP. Let $\mathcal{P}=\\{P_{1},P_{2},\cdots,P_{m}\\}$ be a finite partition
of $X$ with diameter smaller than $\delta$. For $x\in X$,
$\widehat{x}\in\\{1,2,3,\cdots,m\\}^{\mathbb{N}}$ is defined by

$\widehat{x}(n)=j\text{ whenever }T^{n}x\in P_{j}\text{ and }n\in\mathbb{N}.$

Denote $\widehat{Z}=\\{\widehat{x}:x\in Z\\}$ and $W_{n}$ is the collection of
length $n$ strings that appear in $\widehat{Z}$. Then $K_{n}:=e^{-nh}\sharp
W_{n}$ grows at a subexponential rate, i.e.

$\lim_{n\to\infty}\frac{\log K_{n}}{n}=0,$

where $h=h_{top}(\overline{\widehat{Z}},\sigma)$. Denote

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

Let $(Y_{n},\sigma_{n})$ be the 1-step shift of finite type $W_{n}$. Then
$(\overline{\widehat{Z}},\sigma^{n})$can be considered as a subsystem of
$(Y_{n},\sigma_{n})$. Hence

$h_{top}(Y_{n},\sigma_{n})\geq h_{top}(\overline{\widehat{Z}},\sigma^{n})=nh.$

Thus from Lemma 3.4, the shortest periodic orbit in $Y_{n}$ is at most
$1+e^{1-nh}\sharp W_{n}=1+eK_{n}$. Denote one of the shortest periodic orbit
in $Y_{n}$ by $z_{1}z_{2}\cdots z_{p_{n}}z_{1}z_{2}\cdots$ for some $p_{n}\leq
1+eK_{n}$ and $z_{i}\in W_{n},i=1,2,\cdots,p_{n}$.

Now we construct a periodic pseudo-orbit in $Z$. For $i=1,2,\cdots p_{n}$,
there is $x_{i}\in Z$ such that the leading length $2n$ string of
$\widehat{x_{i}}$ is $z_{i}z_{i+1}$ (Note $z_{{p_{n}}+1}=z_{1}$). Hence,
$\widehat{T^{n}x_{i}}$ and $\widehat{x_{i+1}}$ have the same leading length
$n$ string which implies $d(T^{n+j}x_{i},T^{j}x_{i+1})<\delta$ $($Note
$x_{p_{n}+1}=x_{1}$$)$ for $j=0,1,2,\cdots,n-1$. By ASP,

$\displaystyle d(T^{n+[\frac{n}{2}]}x_{i},T^{[\frac{n}{2}]}x_{i+1})$
$\displaystyle<Ce^{-\lambda\min([\frac{n}{2}],n-1-[\frac{n}{2}])}\cdot(d(T^{n}x_{i},x_{i+1})+d(T^{2n-1}x_{i},T^{n-1}x_{i+1}))$
$\displaystyle\leq 2\delta Ce^{-\lambda([\frac{n}{2}]-1)}.$

Therefore, we select the periodic $2\delta
Ce^{-\lambda([\frac{n}{2}]-1)}$-pseudo-orbit $\widetilde{\mathcal{O}_{n}}$ in
$Z$ with periodic $n{p_{n}}$ by

$\\{T^{[\frac{n}{2}]}x_{1},T^{[\frac{n}{2}]+1}x_{1},\cdots,T^{n+[\frac{n}{2}]-1}x_{1},T^{[\frac{n}{2}]}x_{2},\cdots,T^{n+[\frac{n}{2}]-1}x_{2},T^{[\frac{n}{2}]}x_{3},\cdots,T^{n+[\frac{n}{2}]-1}x_{p_{n}}\\}.$

By ASP, while $n$ is sufficiently large, we have a periodic orbit
$\mathcal{O}_{n}$ with period $m_{n}$, such that $m_{n}|n{p_{n}}$

$d_{\alpha,Z}(\mathcal{O}_{n})\leq
d_{\alpha,Z}(\widetilde{\mathcal{O}_{n}})+n{p_{n}}(2\delta
CL)^{\alpha}e^{-\lambda\alpha([\frac{n}{2}]-1)}=n{p_{n}}(2\delta
CL)^{\alpha}e^{-\lambda\alpha([\frac{n}{2}]-1)},$

where we used lemma 3.5. Since $p_{n}\leq 1+eK_{n}$ and $K_{n}$ grows at a
subexponential rate, we obtain

$\displaystyle\limsup_{n\to\infty}n^{k}\min_{\mathcal{O}\in\mathcal{O}^{n}}d_{\alpha,Z}(\mathcal{O})$
$\displaystyle\leq\limsup_{n\to\infty}(\max\\{ip_{i}:{1\leq i\leq
n+1}\\})^{k}\cdot
n{p_{n}}\delta^{\alpha}e^{-\lambda\alpha([\frac{n}{2}]-1)}=0.$

This ends the proof. ∎

Let $(X,T)$ satisfy H1) and ASP, $u:X\rightarrow\mathbb{R}$ and
$\psi:X\rightarrow\mathbb{R}^{+}$ are continuous. Given $\alpha\in(0,1]$,
$\tilde{L}>0$ and a $T$-forward-invariant non-empty subset $Z\subset X$. Now,
we are ready to construct the required periodic orbit in Proposition 3.1
satisfying (3.3). Before the rigorous proof, we firstly introduce the idea of
the construction in a vague way: One can start with a periodic orbit
$\mathcal{O}_{0}$ with long enough period $n$ and a good approximation to $Z$
(say $d_{\alpha,Z}(\mathcal{O}_{0})<n^{-k}$ for some large $k$); Once the gap
of $\mathcal{O}_{0}$, $D(\mathcal{O}_{0})$, is too small to meet the
requirement, $\mathcal{O}_{0}$ can be decomposed into two pseudo periodic
orbits, one of which has at most half of the original period $n$; Such pseudo
orbits will provide a nearby periodic orbit with same period by ASP, say
$\mathcal{O}_{1}$; One can show that the ratio
$\frac{d_{\alpha,Z}(\mathcal{O}_{1})}{d_{\alpha,Z}(\mathcal{O}_{0})}$ is
bounded by a constant depending on system constants and $\tilde{L}$ only
rather than dependending on $d_{\alpha,Z}(\mathcal{O}_{0})$; Note that the
operation of decomposing periodic orbits into periodic orbits with period
halved can be done at most $\log_{2}n$ times; Therefore, by adjusting the
largeness of $n,k$, the above process will end at either a periodic orbit meet
the requirement of Proposition 3.1 or a fixed point, both of which will
clearly accomplish the proof.

Let $C,L,\lambda,\delta$ be as in ASP. Take $k\in\mathbb{N}$ large enough, on
which the condition will be proposed later. By Lemma 3.3, there exists a
periodic orbit $\mathcal{O}_{0}$ of $(X,T)$ with period $n$ large enough such
that

$d_{\alpha,Z}(\mathcal{O}_{0})<\tilde{L}_{0}n^{-k}\ll\delta,$ (3.5)

where $\tilde{L}_{0}=1$. If
$D^{\alpha}(\mathcal{O}_{0})>\tilde{L}d_{\alpha,Z}(\mathcal{O}_{0})$, the
proof is done. Otherwise, one has that

$D^{\alpha}(\mathcal{O}_{0})\leq\tilde{L}d_{\alpha,Z}(\mathcal{O}_{0})<\tilde{L}\tilde{L}_{0}n^{-k},$
(3.6)

which is required to be smaller than $\delta^{\alpha}$ by choosing $n,k$ large
enough. Therefore, there are $y\in\mathcal{O}_{0}$ and $1\leq n_{1}\leq n-1$
such that

$d(y,T^{n_{1}}y)<(\tilde{L}\tilde{L}_{0})^{\frac{1}{\alpha}}n^{-\frac{k}{\alpha}}<\delta.$

We split the periodic orbit $\mathcal{O}_{0}$ into two pieces of orbit by

$\displaystyle\mathcal{Q}^{0}_{0}=\\{y,Ty,\cdots,T^{n_{1}-1}y\\};$
$\displaystyle\mathcal{Q}^{1}_{0}=\\{T^{n_{1}}y,T^{n_{1}}y,\cdots,T^{n-1}y\\}.$

Note that each of the above segment of orbit induces a $\delta$-pseudo
periodic orbit, and moreover, period of one such $\delta$-pseudo periodic
orbit does not exceed $\frac{n}{2}$. Without losing any generality, we assume
that $n_{1}\leq\frac{n}{2}$.

By ASP, there exists a periodic orbit

$\mathcal{O}_{1}=\\{z_{1},Tz_{1},\cdots,T^{m_{1}-1}z_{1}\\}$

such that $T^{m_{1}}z=z$, $m_{1}|n_{1}$ and $d(T^{i}y,T^{i}z_{1})\leq
L(\tilde{L}\tilde{L}_{0})^{\frac{1}{\alpha}}n^{-\frac{k}{\alpha}}$ for all
$0\leq i\leq n_{1}-1$. Therefore, by ASP again, for all $0\leq i\leq n_{1}-1$,

$d(T^{i}y,T^{i}z_{1})\leq
Ce^{-\lambda\min\\{i,n_{1}-i\\}}2L(\tilde{L}\tilde{L}_{0})^{\frac{1}{\alpha}}n^{-\frac{k}{\alpha}},$

which $L(\tilde{L}\tilde{L}_{0})^{\frac{1}{\alpha}}n^{-\frac{k}{\alpha}}$ is
required to be smaller than $\delta$, that is,
$L^{\alpha}(\tilde{L}\tilde{L}_{0})n^{-k}<\delta^{\alpha}$ by choosing $n,k$
large enough.

Hence,

$\displaystyle\begin{split}d_{\alpha,Z}(\mathcal{O}_{1})\leq&d_{\alpha,Z}(\mathcal{Q}^{0}_{0})+\sum_{i=0}^{n_{1}-1}\left(Ce^{-\lambda\min\\{i,n_{1}-i\\}}2L\right)^{\alpha}(\tilde{L}\tilde{L}_{0})n^{-k}\\\
\leq&d_{\alpha,Z}(\mathcal{O}_{0})+\frac{2(2CL)^{\alpha}}{1-e^{-\lambda\alpha}}(\tilde{L}\tilde{L}_{0})n^{-k}\\\
<&\tilde{L}_{1}n^{-k},\end{split}$ (3.7)

where
$\tilde{L}_{1}=(1+\frac{2(2CL)^{\alpha}}{1-e^{-\lambda\alpha}}\tilde{L})\tilde{L}_{0}=1+\frac{2(2CL)^{\alpha}}{1-e^{-\lambda\alpha}}\tilde{L}$.

If $D^{\alpha}(\mathcal{O}_{1})>\tilde{L}d_{\alpha,Z}(\mathcal{O}_{1})$, the
proof is done. Otherwise, one repeats the above operation to get another
periodic orbit $\mathcal{O}_{2}$ with period $\leq\frac{n}{4}$. Note that, in
this case, in order to make the above process repeatable one only need

$\tilde{L}\tilde{L}_{1}n^{-k}<\delta^{\alpha}\text{ and
}L^{\alpha}\tilde{L}\tilde{L}_{1}n^{-k}<\delta^{\alpha},$

which is doable by choosing $n,k$ large enough. Suppose the above operation
can be executed $m$ times resulting at a periodic orbit $\mathcal{O}_{m}$.
Then, by applying the same argument inductively, one has that

$d_{\alpha,Z}(\mathcal{O}_{m})<\tilde{L}_{m}n^{-k},$

where
$\tilde{L}_{m}=(1+\frac{2(2CL)^{\alpha}}{1-e^{-\lambda\alpha}}\tilde{L})\tilde{L}_{m-1}=\cdots={\tilde{L}_{1}}^{m}$.
Since every operation will (at least) halve the period of the resulting
periodic orbit, such process has to end before $\left(\left[\frac{\log n}{\log
2}\right]+1\right)$-th operation. In order to make each operation doable, one
only need $n,k$ satisfying the following condition

$\tilde{L}\tilde{L}_{1}^{\frac{\log n}{\log
2}+1}n^{-k}=\tilde{L}\tilde{L}_{1}n^{-k+\frac{\log\tilde{L}_{1}}{\log
2}}<\delta^{\alpha}\text{ and }L^{\alpha}\tilde{L}\tilde{L}_{1}^{\frac{\log
n}{\log
2}+1}n^{-k}=L^{\alpha}\tilde{L}\tilde{L}_{1}n^{-k+\frac{\log\tilde{L}_{1}}{\log
2}}<\delta^{\alpha}.$

Note that, after the last operation being executed, there are two possible
cases: the resulting periodic orbit either meet the requirement of Proposition
3.1 or is a fixed point of $T$. In the second case, by the definition of
$D(\mathcal{O})$ (3.2), requirement of Proposition 3.1 is also met once one
additionally choose $n,k$ to satisfy that ${\tilde{L}_{1}}^{\frac{\log n}{\log
2}+1}n^{-k}<\delta^{\alpha}$, that is,
${\tilde{L}_{1}}n^{-k+\frac{\log\tilde{L}_{1}}{\log 2}+1}<\delta^{\alpha}$.

The proof of Proposition 3.1 is completed.

####  3.1.2.  Proof of Proposition 3.2

Before going to the proof of Proposition 3.2, we need to introduce a technical
lemma and some notions that play important roles in later proof.

######  Lemma 3.6.

Let $(X,T)$ be a dynamical system satisfying H1) and MCGBP. Then for all
$0<\alpha\leq 1$, strictly positive $\psi\in\mathcal{C}^{0,\alpha}({X})$ and
$u\in\mathcal{C}^{0,\alpha}({X})$, there is $v\in\mathcal{C}^{0,\alpha}({X})$
such that

  * (1)

$u_{K}-v\circ T^{K}+v-\beta(u;\psi,{X},T)\psi_{K}\geq 0;$

  * (2)

$Z_{u,\psi}\subset\\{x\in{X}:(u_{K}-v\circ
T^{K}+v-\beta(u;\psi,{X},T)\psi_{K})(x)=0\\},$

where $K=K(\alpha)$ is the natural number as in MCGBP and $Z_{u,\psi}$ is
given by (3.1).

###### Proof.

(1). By MCGBP, we only need to show that
$\beta(u-\beta(u;\psi,{X},T)\psi;X,T)=0$, that is,

$\min_{\mu\in\mathcal{M}({X},T)}\int u-\beta(u;\psi,{X},T)\psi d\mu=0.$

It is immediately from the fact

$\min_{\mu\in\mathcal{M}({X},T)}\frac{\int ud\mu}{\int\psi
d\mu}=\beta(u;\psi,{X},T),$

where we used the assumption $\psi$ is strictly positive.

(2). By a probability measure $\mu\in\mathcal{M}_{min}(u;\psi,{X},T)$, we have

$\int u_{K}-v\circ T^{K}+v-\beta(u;\psi,{X},T)\psi_{K}d\mu=\int
u-\beta(u;\psi,{X},T)\psi d\mu=0.$

Combining (1) and the fact $u_{K}-v\circ T^{K}+v-\beta(u;\psi,{X},T)\psi_{K}$
is continuous, we have

$supp(\mu)\subset\\{x\in{X}:(u_{K}-v\circ
T^{K}+v-\beta(u;\psi,{X},T)\psi_{K})(x)=0\\}.$

Therefore, by the continuity of $u_{K}-v\circ
T^{K}+v-\beta(u;\psi,{X},T)\psi_{K}$, one has that

$Z_{u,\psi}=\overline{\cup_{\mu\in\mathcal{M}_{min}(u;\psi,{X},T)}supp(\mu)}\subset\\{x\in{X}:(u_{K}-v\circ
T^{K}+v-\beta(u;\psi,{X},T)\psi_{K})(x)=0\\}.$

This ends the proof. ∎

######  Remark 3.7.

For convenience, in the following text, if we need to use lemma 3.6, we use
$\bar{u}$ to represent $u_{K}-v\circ T^{K}+v-\beta(u;\psi,{X},T)\psi_{K}$ for
short. Then, $\bar{u}\geq 0$ and
$Z_{u,\psi}\subset\\{x\in{X}:\bar{u}(x)=0\\}.$

Fix $\varepsilon,\alpha,\psi,u$ as in Proposition 3.2, $K,\bar{u}$ as in
remark (3.7), and $C$, $\delta$ as in ASP. By remark 3.7, one has that

$\bar{u}\geq 0\text{ and }Z_{u,\psi}\subset\\{x\in{X}:\bar{u}(x)=0\\}.$

In stead of investigating the minimizing measure of $\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h$, we consider a modified observable
$G:=\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+h-a_{\mathcal{O}}\psi_{K}$ which will provide
more conveniences, where

$a_{\mathcal{O}}:=\frac{\sum_{y\in\mathcal{O}}\left(\bar{u}(y)+\varepsilon
d^{\alpha}(y,\mathcal{O})+h(y)\right)}{\sum_{y\in\mathcal{O}}\psi_{K}(y)}.$

Clearly $\int Gd\mu_{\mathcal{O}}=0$. Note that, by the definition of
$u_{K}\left(:=\frac{1}{K}\sum_{i=0}^{K-1}u\circ T^{i}\right)$, for all
$\mu\in\mathcal{M}({X},T)$, one has that

$\displaystyle\frac{\int u+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+hd\mu}{\int\psi d\mu}$ $\displaystyle=\frac{\int
u_{K}+\varepsilon d^{\alpha}(\cdot,\mathcal{O})+hd\mu}{\int\psi_{K}d\mu}$
$\displaystyle=\frac{\int\bar{u}+\varepsilon
d^{\alpha}(\cdot,\mathcal{O})+hd\mu}{\int\psi_{K}d\mu}+\beta(u;\psi,{X},T)$
$\displaystyle=\frac{\int Gd\mu}{\int\psi
d\mu}+a_{\mathcal{O}}+\beta(u;\psi,{X},T),$

where we recall that $\beta(u;\psi,{X},T)$ is the minimum ergodic average
given by (2.1).

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

$\int Gd\mu>0\text{ for all
}\mu\in\mathcal{M}^{e}({X},T)\setminus\\{\mu_{\mathcal{O}}\\}.$ (3.8)

Denote that

$L_{\mathcal{O}}=\frac{D^{\alpha}(\mathcal{O})}{d_{\alpha,Z_{u,\psi}}(\mathcal{O})}.$

Thus one has an equivalent statement of Proposition 3.2 under the same setting
as the following:

######  Lemma 3.8.

There exist $\hat{L},\hat{\delta}>0$ which depend on
$\varepsilon,\alpha,u,\psi$ and system constants only such that if
$L_{\mathcal{O}}>\hat{L}$, $\|h\|_{\alpha}\leq 10\epsilon$ and
$\|h\|_{0}<\frac{D^{\alpha}(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}$,
then (3.8) holds.

###### Proof.

Put
$Area_{1}:=\left\\{x\in{X}:d(x,\mathcal{O})\leq\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}\right\\}.$
Note that

$\displaystyle\begin{split}|a_{\mathcal{O}}|&=\left|\frac{\sum_{y\in\mathcal{O}}\left(\bar{u}(y)+h(y)\right)}{\sum_{y\in\mathcal{O}}\psi_{K}(y)}\right|\\\
&\leq\frac{\sum_{y\in\mathcal{O}}\left(\|\bar{u}\|_{\alpha}d^{\alpha}(y,Z_{u,\psi})+\|h\|_{0}\right)}{\sum_{y\in\mathcal{O}}\psi_{min}}\\\
&=\frac{\|\bar{u}\|_{\alpha}d_{\alpha,Z_{u,\psi}}(\mathcal{O})}{\sharp\mathcal{O}\psi_{min}}+\frac{\|h\|_{0}}{\psi_{min}}.\end{split}$

Hence

$\displaystyle\frac{\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}}{\frac{D(\mathcal{O})}{2}}$
$\displaystyle\leq
2\left(\frac{\|\bar{u}\|_{\alpha}}{\epsilon\sharp\mathcal{O}\psi_{min}}\frac{1}{L_{\mathcal{O}}}+\frac{\|h\|_{0}}{\epsilon\psi_{min}D^{\alpha}(\mathcal{O})}\right)^{\frac{1}{\alpha}}$

Particularly, when
$L_{\mathcal{O}}>\frac{2(2Lip_{T})^{\alpha}\|\bar{u}\|_{\alpha}}{\epsilon\sharp\mathcal{O}\psi_{min}}$
and
$\|h\|_{0}<\frac{\epsilon\psi_{min}}{2(2Lip_{T})^{\alpha}}D^{\alpha}(\mathcal{O})$,
one has

$\displaystyle\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}<\frac{D(\mathcal{O})}{2Lip_{T}},$

where

$\displaystyle Lip_{T}=\left\\{\begin{array}[]{ll}1,&\text{ if }\sharp
T(X)=1,\\\ \max\left\\{1,\sup_{x\neq
y}\frac{d(Tx,Ty)}{d(x,y)}\right\\},&\text{ if }\sharp
T(X)>1.\end{array}\right.$ (3.11)

Thus in the choice of $\hat{L}$ and $\hat{\delta}$, we will require
$\hat{L}\geq\frac{2(2Lip_{T})^{\alpha}\|\bar{u}\|_{\alpha}}{\epsilon\psi_{min}}$
and $\hat{\delta}\leq\frac{\epsilon\psi_{min}}{2(2Lip_{T})^{\alpha}}$. Then
when $L_{\mathcal{O}}>\hat{L}$, and
$\|h\|_{0}<\frac{D^{\alpha}(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}$,
one has

$\displaystyle\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}<\frac{D(\mathcal{O})}{2Lip_{T}}.$
(3.12)

Firstly, we show that $Area_{1}$ contains all $x\in{X}$ with $G(x)\leq 0$.

Given $x\notin Area_{1}$ when $Area_{1}\neq X$, we are to show that $G(x)>0$.
There exists $y\in\mathcal{O}$ such that

$d(x,y)=d(x,\mathcal{O})>\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}.$

Note that

$\displaystyle\begin{split}\bar{u}+h-a_{\mathcal{O}}\psi_{K}\geq
h-|a_{\mathcal{O}}|\psi_{K}\geq-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}\end{split}$
(3.13)

since $\bar{u}\geq 0$ and $\|\psi_{K}\|_{0}\leq\|\psi\|_{0}$. Then

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

Secondly, we will show that by choosing $L_{\mathcal{O}}$, $\|h\|_{\alpha}$
and $\|h\|_{0}$ properly, for any $z\in{X}$ which is not a generic point of
$\mu_{\mathcal{O}}$, there is an $m\in\mathbb{N}\cup\\{0\\}$ such that
$\sum_{i=0}^{m}G(T^{i}z)>0$. The conditions proposed for $L_{\mathcal{O}}$,
$\|h\|_{\alpha}$ and $\|h\|_{0}$ will provide the existence of the constants
$\hat{L}$ and $\hat{\delta}$ being requested by Proposition 3.2.

Suppose that $z\in{X}$ is not a generic point of $\mu_{\mathcal{O}}$. There
are two cases. In the case $z\notin Area_{1}$ just note $m=0$ since $G(x)>0$
by claim 1.

In the case $z\in Area_{1}$, there is $y_{0}\in\mathcal{O}$ such that

$d(z,y_{0})=d(z,\mathcal{O})\leq\left(\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\right)^{\frac{1}{\alpha}}<\frac{D(\mathcal{O})}{2Lip_{T}}$

by (3.12). If $d(T^{k}z,T^{k}y_{0})\leq\delta$ for all $k\in\mathbb{N}$, by
ASP, we have

$d(T^{k}z,T^{k}y_{0})\leq Ce^{-\lambda
k}(d(z,y_{0})+d(T^{2k}z,T^{2k}y_{0}))\leq 2Ce^{-\lambda k}\delta\to 0\text{ as
}k\to+\infty.$

Then $z$ must be a generic point of $\mu_{\mathcal{O}}$ which is impossible by
our assumption. Hence, there must be some $m_{0}\in\mathbb{N}$ such that

$d(T^{m_{0}}z,T^{m_{0}}y_{0})>\delta>\frac{D(\mathcal{O})}{2}.$

Let $m_{1}\in\mathbb{N}$ be the smallest time such that

$\displaystyle\begin{split}\frac{D(\mathcal{O})}{2Lip_{T}}\leq
d(T^{m_{1}}z,T^{m_{1}}y_{0})\leq\frac{D(\mathcal{O})}{2}<\delta.\end{split}$
(3.14)

Then, we have

$d(T^{m_{1}}z,\mathcal{O})=d(T^{m_{1}}z,T^{m_{1}}y_{0})\geq\frac{D(\mathcal{O})}{2Lip_{T}}.$

Hence, by (3.13), we have

$\displaystyle\begin{split}G(T^{m_{1}}z)&=\bar{u}(T^{m_{1}}z)+\varepsilon
d^{\alpha}(T^{m_{1}}z,\mathcal{O})+h(T^{m_{1}}z)-a_{\mathcal{O}}\psi_{K}(T^{m_{1}}z)\\\
&\geq\varepsilon
d^{\alpha}(T^{m_{1}}z,\mathcal{O})-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}\\\
&\geq\varepsilon\cdot\left(\frac{D(\mathcal{O})}{2Lip_{T}}\right)^{\alpha}-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}.\\\
\end{split}$ (3.15)

Let $m_{2}\in\mathbb{N}$ the largest time with $0\leq m_{2}<m_{1}$ such that

$T^{m_{2}}z\in Area_{1}.$

Then for all $m_{2}<n<m_{1}$

$\displaystyle G(T^{n}z)>0.$ (3.16)

On the other hand, since $m_{2}<m_{1}$, by (3.14), one has that for all $0\leq
n\leq m_{2}$

$d(T^{n}z,T^{n}y_{0})\leq\frac{D(\mathcal{O})}{2}<\delta.$

Then by ASP, one has that for all $0\leq n\leq m_{2}$

$\displaystyle d^{\alpha}(T^{n}z,T^{n}y_{0})$ $\displaystyle\leq
C^{\alpha}e^{-\lambda\alpha\min(n,m_{2}-n)}\cdot(d(z,y_{0})+d(T^{m_{2}}(z),T^{m_{2}}y_{0}))^{\alpha}$
$\displaystyle\leq 2C^{\alpha}(e^{-\lambda\alpha
n}+e^{-\lambda\alpha(m_{2}-n)})\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}$

where we used the assumption that $z,T^{m_{2}}z\in Area_{1}$. Therefore,

$\displaystyle\begin{split}\sum_{n=0}^{m_{2}}d^{\alpha}(T^{n}z,T^{n}y_{0})\leq\frac{4C^{\alpha}}{1-e^{-\lambda\alpha}}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}.\end{split}$

Thus, one has that

$\displaystyle\begin{split}&\sum_{n=0}^{m_{2}}\left(G(T^{n}z)-G(T^{n}y_{0})\right)\\\
=&\sum_{n=0}^{m_{2}}\left(\bar{u}(T^{n}z)+\varepsilon
d^{\alpha}(T^{n}z,\mathcal{O})+h(T^{n}z)-\bar{u}(T^{n}y_{0})-\varepsilon
d^{\alpha}(T^{n}y_{0},\mathcal{O})-h(T^{n}y_{0})\right)\\\
&+a_{\mathcal{O}}\sum_{n=0}^{m_{2}}\left(\psi_{K}(T^{n}y_{0})-\psi_{K}(T^{n}z)\right)\\\
\geq&\sum_{n=0}^{m_{2}}\left(\bar{u}(T^{n}z)-\bar{u}(T^{n}y_{0})+h(T^{n}z)-h(T^{n}y_{0})+a_{\mathcal{O}}(\psi_{K}(T^{n}y_{0})-\psi_{K}(T^{n}z))\right)\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+|a_{\mathcal{O}}|\|\psi_{K}\|_{\alpha})\sum_{n=0}^{m_{2}}d^{\alpha}(T^{n}z,T^{n}y_{0})\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+|a_{\mathcal{O}}|\|\psi_{K}\|_{\alpha})\cdot\frac{4C^{\alpha}}{1-e^{-\lambda\alpha}}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon},\end{split}$
(3.17)

where we used the fact $d^{\alpha}(\cdot,\mathcal{O})\geq 0$ and
$d^{\alpha}(T^{n}y_{0},\mathcal{O})=0$. Also note that

$\displaystyle|a_{\mathcal{O}}|$
$\displaystyle=\left|\frac{\sum_{y\in\mathcal{O}}\left(\bar{u}(y)+h(y)\right)}{\sum_{y\in\mathcal{O}}\psi_{K}(y)}\right|\leq\frac{\|\bar{u}\|_{0}+\|h\|_{0}}{\psi_{min}}.$

Thus, one has that

$\displaystyle\begin{split}&\sum_{n=0}^{m_{2}}\left(G(T^{n}z)-G(T^{n}y_{0})\right)\\\
\geq&-(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+\|h\|_{0}}{\psi_{min}}\|\psi_{K}\|_{\alpha})\cdot\frac{4C^{\alpha}}{1-e^{-\lambda\alpha}}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}\end{split}$
(3.18)

by (3.17).

Note that $m_{2}+1=p\sharp\mathcal{O}+r$ for some nonnegative integer $p$ and
$0\leq r\leq\sharp\mathcal{O}-1$, then by (3.13), one has that

$\displaystyle\begin{split}\sum_{n=0}^{m_{2}}G(T^{n}y_{0})=\sum_{n=p\sharp\mathcal{O}}^{p\sharp\mathcal{O}+r-1}G(T^{n}y_{0})\geq-\sharp\mathcal{O}\cdot(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0})\end{split}$
(3.19)

where we used $\int Gd\mu_{\mathcal{O}}=0$.

Now we are ready to estimate $\sum_{n=0}^{m_{1}}G(f^{n}z)$ as the following:

$\displaystyle\sum_{n=0}^{m_{1}}G(T^{n}z)$ $\displaystyle\geq$
$\displaystyle\sum_{n=0}^{m_{2}}G(T^{n}z)+G(T^{m_{1}}z)$ $\displaystyle=$
$\displaystyle\left(\sum_{n=0}^{m_{2}}\left(G(T^{n}z)-G(T^{n}y_{0})\right)\right)+\left(\sum_{n=0}^{m_{2}}G(T^{n}y_{0})\right)+G(T^{m_{1}}z)$
$\displaystyle\geq$
$\displaystyle-\left(\|\bar{u}\|_{\alpha}+\|h\|_{\alpha}+\frac{\|\bar{u}\|_{0}+\|h\|_{0}}{\psi_{min}}\|\psi_{K}\|_{\alpha}\right)\cdot\frac{4C^{\alpha}}{1-e^{-\lambda\alpha}}\cdot\frac{|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0}}{\varepsilon}$
$\displaystyle(\text{by }(\ref{X-6}))$
$\displaystyle-\sharp\mathcal{O}\cdot(|a_{\mathcal{O}}|\|\psi\|_{0}+\|h\|_{0})$
$\displaystyle(\text{by }(\ref{X-7}))$
$\displaystyle+\varepsilon\cdot\left(\frac{D(\mathcal{O})}{2Lip_{T}}\right)^{\alpha}-|a_{\mathcal{O}}|\|\psi\|_{0}-\|h\|_{0}$
$\displaystyle(\text{by }(\ref{X-5}))$ $\displaystyle\geq$ $\displaystyle
L_{1}D^{\alpha}(\mathcal{O})-L_{2}d_{\alpha,Z_{u,\psi}}(\mathcal{O})-L_{3}\sharp\mathcal{O}\|h\|_{0}$
$\displaystyle=$ $\displaystyle
L_{1}D^{\alpha}(\mathcal{O})\Big{(}1-\frac{L_{2}}{L_{1}}\frac{1}{L_{\mathcal{O}}}-\frac{L_{3}\sharp\mathcal{O}}{L_{1}D^{\alpha}(\mathcal{O})}\|h\|_{0}\Big{)},$

where we take $\|h\|_{\alpha}\leq 10\varepsilon$ and
$\|h\|_{0}\leq\delta^{\alpha}$, and let

$\displaystyle L_{1}$ $\displaystyle=\frac{\varepsilon}{(2Lip_{T})^{\alpha}},$
$\displaystyle L_{2}$
$\displaystyle=\left(\frac{4C^{\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+\delta^{\alpha}}{\psi_{min}}\|\psi_{K}\|_{\alpha})}{(1-e^{-\lambda\alpha})\psi_{min}\varepsilon}+\frac{2\|\psi\|_{0}}{\psi_{min}}\right)\|\bar{u}\|_{\alpha},$
$\displaystyle L_{3}$
$\displaystyle=\left(\frac{4C^{\alpha}(\|\bar{u}\|_{\alpha}+10\varepsilon+\frac{\|\bar{u}\|_{0}+\delta^{\alpha}}{\psi_{min}}\|\psi_{K}\|_{\alpha})}{(1-e^{-\lambda\alpha})\psi_{min}\varepsilon}+\frac{2\|\psi\|_{0}}{\psi_{min}}\right)(1+\psi_{min}).$

Note that $L_{1},L_{2},L_{3}$ are positive and depending on $\epsilon,u,\psi$
and system constants only. By taking

$L_{\mathcal{O}}>2\frac{L_{2}}{L_{1}},\
\|h\|_{0}<\frac{1}{2}\frac{L_{1}D^{\alpha}(\mathcal{O})}{L_{3}\sharp\mathcal{O}},\text{
and }m=m_{1},$

one has that $\sum_{i=0}^{m}G(T^{i}z)>0$ provided that $z$ is not a generic
point of $\mu_{\mathcal{O}}$. Therefore, one possible choice for $\hat{L}$ and
$\hat{\delta}$ is to let

$\hat{L}=\max\left\\{\frac{3L_{2}}{L_{1}},\frac{2(2Lip_{T})^{\alpha}\|\bar{u}\|_{\alpha}}{\epsilon\psi_{min}}\right\\}\text{
and
}\hat{\delta}=\min\left\\{1,\frac{L_{1}}{3L_{3}},\frac{\epsilon\psi_{min}}{2(2Lip_{T})^{\alpha}}\right\\}.$

Finally, we finish the proof by showing that when $L_{\mathcal{O}}>\hat{L}$,
$\|h\|_{\alpha}\leq 10\varepsilon$, and
$\|h\|_{0}\leq\frac{D^{\alpha}(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}$
with $\hat{L}$ and $\hat{\delta}$ given above, the following holds

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

Given a ergodic probability measure $\mu\in\mathcal{M}^{e}({X},T)$, in the
case $\mu=\mu_{\mathcal{O}}$, we have $\int Gd\mu_{\mathcal{O}}=0$. In the
case $\mu\neq\mu_{\mathcal{O}}$, let $z$ be a generic point of $\mu$. Note
that $z$ is not a generic point of $\mu_{\mathcal{O}}$. Thus there exists
$m_{1}\in\mathbb{N}$ such that

$\sum_{n=0}^{m_{1}}G(T^{n}z)>0.$

Note that $T^{m_{1}+1}z$ is also not a generic point of $\mu_{\mathcal{O}}$.
Thus we have $m_{1}+1\leq m_{2}\in\mathbb{N}$ such that

$\sum_{n=m_{1}+1}^{m_{2}}G(T^{n}z)>0.$

By repeating the above process, we have $0\leq m_{1}<m_{2}<m_{3}<\cdots$ such
that

$\sum_{n=m_{i}+1}^{m_{i+1}}G(T^{n}z)>0\text{ for }i=0,1,2,3,\cdots,$

where $m_{0}=-1$. Therefore

$\displaystyle\int Gd\mu$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{m_{i}+1}\sum_{n=0}^{m_{i}}G(T^{n}z)$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{m_{i}+1}\left(\sum_{n=0}^{m_{1}}G(T^{n}z)+\sum_{n=m_{1}+1}^{m_{2}}G(T^{n}z)+\cdots+\sum_{n=m_{i-1}}^{m_{i}}G(T^{n}z)\right)$
$\displaystyle\geq 0.$

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

In the end, (3.8) holds ($\varepsilon$ may need to be modified a bit) by
noting that for any $\varepsilon^{\prime}>\varepsilon_{0}$, the function
$G^{\prime}:=G+(\varepsilon^{\prime}-\varepsilon)d(\cdot,\mathcal{O})$
satisfies that

$\int(G^{\prime}-G)d\mu=\int(\varepsilon^{\prime}-\varepsilon)d(\cdot,\mathcal{O})d\mu>0\quad\forall\mu\in\mathcal{M}^{e}(X,T)\setminus\\{\mu_{\mathcal{O}}\\}.$

This ends the proof. ∎

So far, we have accomplished the proof of Part I) of Theorem 2.1.

###  3.2. Proof of Part II) of Theorem 2.1

We will prove the following technical proposition which together with
Proposition 3.1 imply the Part II) of Theorem 2.1.

######  Proposition 3.9.

Let $(M,f)$ be a dynamical system on a smooth compact manifold $M$. Assume
that $(X,T)$ is a subsystem of $(M,f)$, which satisfies ASP and MCGBP, and
$T:X\rightarrow X$ is Lipschitz continuous. Then for $0<\varepsilon<1$,
$u\in\mathcal{C}^{1,0}({M})$ and strictly positive
$\psi\in\mathcal{C}^{0,1}({M})$, there exist positive numbers
$\hat{L}_{1},\hat{\delta}_{1}>0$ depending on $\varepsilon,\psi,u$ and system
constants only, and $\hat{\delta}_{1}^{\prime}>0$ depending on $\psi,u$ and
system constants only (independent on $\varepsilon$) such that the following
hold: if a periodic orbit $\mathcal{O}$ of $(X,T)$ meets the following
comparison condition

$\displaystyle D(\mathcal{O})>\hat{L}_{1}d_{1,Z_{u,\psi,T}}(\mathcal{O}),$
(3.20)

then there is a $w\in\mathcal{C}^{\infty}(M)$ with

$\|w\|_{0}<\hat{\delta}_{1}^{\prime}\varepsilon\text{ and
}\|D_{x}w\|_{0}<2\varepsilon$

such that the probability measure

$\left\\{\mu_{\mathcal{O}}:=\frac{1}{\sharp\mathcal{O}}\sum_{x\in\mathcal{O}}\delta_{x}\right\\}=\mathcal{M}_{min}\left((u+w+h)|_{X};\psi|_{X},{X},T\right),$

whenever $h\in\mathcal{C}^{1,0}(M)$ satisfies $\|D_{x}h\|_{0}<5\varepsilon$
and $\|h\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}_{1}$.

Here $D_{x}$ is the derivative of a given function and $Z_{u,\psi,T}$ is same
as the $Z_{u,\psi}$ given by (3.1) with respect to system $(X,T)$. The reason
of adding subindex ”$T$” is to avoid confusion on notions with such invariant
set with respect to system $(M,f)$.

###### Proof.

The proof is based on Proposition 3.2 and the following approximation theorem
due to Greene and Wu [GW].

######  Theorem 3.10.

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

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 a
$g_{2}\in\mathcal{C}^{\infty}$ such that $\|g_{1}-g_{2}\|_{0}<\varepsilon$ and
$\|g_{2}\|_{1}<\varepsilon+\|g_{1}\|_{1}$. Especially,
$\|D_{x}g_{2}\|_{0}<\varepsilon+\|g_{1}\|_{1}$.

Fix $\varepsilon,\mathcal{O},\psi,u$ as in the Proposition 3.9. Note that
Proposition 3.2 in the case of $\alpha=1$ is applicable for the current
setting. Thus, by taking $\hat{L}_{1}=\hat{L}$ and $\mathcal{O}$ satisfying
(3.4) for $\alpha=1$, one has that for any $g\in\mathcal{C}^{0,1}(M)$
satisfying that $\|g\|_{1}\leq 10\varepsilon$ and
$\|g\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\hat{\delta}$

$\left\\{\mu_{\mathcal{O}}:=\frac{1}{\sharp\mathcal{O}}\sum_{x\in\mathcal{O}}\delta_{x}\right\\}=\mathcal{M}_{min}\left((u+\varepsilon
d(\cdot,\mathcal{O})+g)|_{X};\psi|_{X},{X},T\right),$

where $\hat{L}$, $\hat{\delta}$ and $\mathcal{O}$ are as in Proposition 3.2.
Denote that

$\mathcal{U}_{\mathcal{C}^{0,1}}(\varepsilon
d(\cdot,\mathcal{O})):=\left\\{u+\varepsilon d(\cdot,\mathcal{O})+g\Big{|}\
g\in\mathcal{C}^{0,1}(M),\ \|g\|_{1}\leq 10\varepsilon,\
\|g\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\hat{\delta}\right\\}.$

Note that the only obstacle prevent one to derive Proposition 3.9 from
Proposition 3.2 directly is that $d(\cdot,\mathcal{O})$ is only Lipschitz
rather than $\mathcal{C}^{1}$. A nature idea to overcome this is to find a
$w\in\mathcal{C}^{1,0}(M)$ close to $\varepsilon d(\cdot,\mathcal{O})$ in
$\mathcal{C}^{0,1}(M)$ such that an open neighborhood
$\mathcal{U}_{\mathcal{C}^{1,0}}(w)$ of $u+w$ in $\mathcal{C}^{1,0}(M)$ is a
subset of $\mathcal{U}_{\mathcal{C}^{0,1}}(\varepsilon d(\cdot,\mathcal{O}))$,
which is doable by applying Theorem 3.10.

Precisely, for any $\varepsilon_{1}>0$, by Theorem 3.10, there exists a
function $w\in\mathcal{C}^{\infty}(M)$ such that

$\|w\|_{1}\leq\|\varepsilon d(\cdot,\mathcal{O})\|_{1}+\varepsilon_{1}$

and

$\|w-\varepsilon d(\cdot,\mathcal{O})\|_{0}<\varepsilon_{1}.$

Therefore,

$\displaystyle\|D_{x}w\|_{0}\leq\varepsilon+\varepsilon_{1}\text{ and
}\|w\|_{0}\leq\|\varepsilon d(\cdot,\mathcal{O})\|_{0}+\varepsilon_{1}.$
(3.21)

Next, we choose proper $\varepsilon_{1}$, $\hat{\delta}_{1}$ and
$\hat{\delta}_{1}^{\prime}$ to meet the requirement of the proposition as
follows. For $h\in\mathcal{C}^{1,0}(M)$ we rewrite $u+w+h$ as $u+\varepsilon
d(\cdot,\mathcal{O})+(w-\varepsilon d(\cdot,\mathcal{O})+h)$ It remains to
make $w-\varepsilon d(\cdot,\mathcal{O})+h$ satisfying the conditions of $h$
as in Proposition 3.2 by adjusting $\varepsilon_{1}$. Note that

$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_{1}\leq\|w\|_{1}+\|\varepsilon
d(\cdot,\mathcal{O})\|_{1}+\|h\|_{1}\leq
2\varepsilon+\varepsilon_{1}+\|h\|_{1},$ (3.22)

and

$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_{0}\leq\|w-\varepsilon
d(\cdot,\mathcal{O})\|_{0}+\|h\|_{0}<\varepsilon_{1}+\|h\|_{0}.$ (3.23)

Take

$\varepsilon_{1}=\min\left\\{\varepsilon,\frac{D(\mathcal{O})}{2\sharp\mathcal{O}}\cdot\hat{\delta}\right\\},\
\hat{\delta}_{1}^{\prime}=diam(M)+1,\
\hat{\delta}_{1}=\frac{1}{2}\hat{\delta},$

and let $\|h\|_{1}<5\varepsilon$ together with
$\|h\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}_{1}$.
Then one has that

$\displaystyle\|D_{x}w\|_{0}\leq 2\varepsilon\text{ and
}\|w\|_{0}\leq\hat{\delta}_{1}^{\prime}\varepsilon$ $\displaystyle\text{by
}(\ref{C1-1})$ $\displaystyle\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_{1}\leq
8\varepsilon<10\varepsilon$ $\displaystyle\text{by }(\ref{C1-2})$
$\displaystyle\|w-\varepsilon
d(\cdot,\mathcal{O})+h\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}$
$\displaystyle\text{by }(\ref{C1-3}).$

Denote that

$\mathcal{U}_{\mathcal{C}^{1,0}}(w):=\left\\{u+w+h\Big{|}\
h\in\mathcal{C}^{1,0}(M),\
\|h\|_{1}<5\varepsilon,\|h\|_{0}<\frac{D(\mathcal{O})}{\sharp\mathcal{O}}\cdot\hat{\delta}_{1}\right\\}.$

Thus,
$\mathcal{U}_{\mathcal{C}^{1,0}}(w)\subset\mathcal{U}_{\mathcal{C}^{0,1}}(\varepsilon
d(\cdot,\mathcal{O}))$, which is simultaneously a non-empty open subset in
$\mathcal{C}^{1,0}(M)$. This complete the proof. ∎

##  4\. Discussions on the case of $\mathcal{C}^{s,\alpha}$ observables

In this section, we consider the case when the observable functions has higher
regularity. Unlike the case of $\mathcal{C}^{0,\alpha}$ and
$\mathcal{C}^{1,0}$ observables, only partial results are presented in this
paper. To avoid unnecessarily tedious discussions, we will consider the
following model which is relatively simple and illustrative.

Let $(M,f)$ be a dynamical system on a smooth compact manifold $M$ and
$\psi:M\rightarrow\mathbb{R}^{+}$ be a strictly positive continuous function.
Denote that $Per_{s,\alpha}({M},\psi,f)$ is the collection of function
$u\in\mathcal{C}^{s,\alpha}(M)$ such that $\mathcal{M}_{min}(u;\psi,{M},f)$
contains only one probability measure which is periodic. Now we define
$Per^{*}_{s,\alpha}({M},\psi,f)$ the collection of function
$u\in\mathcal{C}^{s,\alpha}(M)$ such that $\mathcal{M}_{min}(u;\psi,{M},f)$
contains at least one periodic probability measure. And
$Loc_{s,\alpha}({M},\psi,f)$ is defined by

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

In the case $s\geq 1$ and $\alpha>0$ or $s\geq 2$, we do not have result like
Theorem 2.1. But, we have the following weak version.

######  Proposition 4.1.

Let $f:M\rightarrow M$ be a Lipschitz continuous selfmap on a smooth compact
manifold $M$ and $(M,f)$ has ASP and MCGBP. Let $\psi\in\mathcal{C}^{0,1}(M)$
be strictly positive. If $u\in\mathcal{C}(M)$ with $u\geq 0$ and there is
periodic orbit $\mathcal{O}$ of $(M,f)$ such that $u|_{\mathcal{O}}=0,$ then
for all $\varepsilon>0$, $s\in\mathbb{N}$ and $0\leq\alpha\leq 1$, there is a
function $w\in\mathcal{C}^{\infty}(M)$ with $\|w\|_{s,\alpha}<\varepsilon$ and
a constant $\varrho>0$ such that the probability measure

$\mu_{\mathcal{O}}=\frac{1}{\sharp\mathcal{O}}\sum_{x\in\mathcal{O}}\delta_{x}\in\mathcal{M}_{min}(u+w+h;\psi,{M},f),$

whenever $h\in\mathcal{C}^{0,1}({M})$ with $\|h\|_{1}<\varrho$ and
$\|h\|_{0}<\varrho.$

By using proposition 4.1, we have the following result immediately.

######  Theorem 4.2.

Let $f:M\rightarrow M$ be a Lipschitz continuous selfmap on a smooth compact
manifold $M$. If $(M,f)$ has ASP and MCGBP, then $Loc_{s,\alpha}({M},\psi)$ is
an open dense subset of $Per^{*}_{s,\alpha}({M},\psi)$ w.r.t.
$\|\cdot\|_{s,\alpha}$ for integer $s\geq 1$, real number $0\leq\alpha\leq 1$
and $\psi\in\mathcal{C}^{0,1}(M)$ is a strictly positive continuous function.

###### Proof.

Immediately from Remark 3.7 and Proposition 4.1 . ∎

Without result like Proposition 3.2, we can not get the full result about the
generality of $\mathcal{C}^{s,\alpha}(M),s\geq 2$ or $s\geq 1$ and $\alpha>0$.
So we rise the following question:

######  Question 4.3.

By a expanding map $f:\mathbb{T}\to\mathbb{T}:x\to 2x$, is there a
$u\in\mathcal{C}^{s,\alpha}(\mathbb{T})$, $s\geq 1$ and $0<\alpha\leq 1$ or
$s\geq 2$ such that any function near $u$ w.r.t. $\|\cdot\|_{s,\alpha}$ has no
periodic minimizing measure?

At last, we complete the proof of proposition 4.1.

###### Proof of proposition 4.1.

Fix $\varepsilon,s,\alpha,\mathcal{O},\psi$ as in proposition. $C$ and
$\delta$ are the constants as in ASP and $Lip_{f}$ is definited as in (3.11).
Just take $w\in\mathcal{C}^{\infty}$ with $\|w\|_{s,\alpha}<\varepsilon$,
$w|_{\mathcal{O}}=0$ and $w|_{M\setminus\mathcal{O}}>0$. For $0\leq r\leq
D(\mathcal{O})$, we note

$\theta(r)=\min\\{w(x):d(x,\mathcal{O})\geq r,x\in{M}\\}.$

It is clear that $\theta(0)=0$, $\theta(r)>0$ for $r\neq 0$ and $\theta$ is
non-decreasing. Now we fix the constants

$0<\rho<\frac{D(\mathcal{O})}{2Lip_{f}}$

and positive $\varrho$ smaller than

$\small\min\left\\{\frac{\theta(\rho)\psi_{min}}{\psi_{min}+\|\psi\|_{0}},\theta\left(\frac{D(\mathcal{O})}{2Lip_{f}}\right)\cdot\frac{1-e^{-\lambda}}{4C\rho},\frac{1}{2}\cdot\frac{\theta\left(\frac{D(\mathcal{O})}{2Lip_{f}}\right)}{\frac{2C\rho\|\psi\|_{1}}{(1-e^{-\lambda})\psi_{min}}+\sharp\mathcal{O}+\sharp\mathcal{O}\frac{\|\psi\|_{0}}{\psi_{min}}+\frac{\|\psi\|_{0}}{\psi_{min}}+1}\right\\}.$

By fixing $h\in\mathcal{C}^{0,1}({M})$ with $\|h\|_{1}<\varrho$ and
$\|h\|_{0}<\varrho$, we are to show that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(w+h;\psi,{M},f)$ which implies that
$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+w+h;\psi,{M},f)$ since $u\geq 0$ and
$u|_{\mathcal{O}}=0$ by assumption.
Note that $G=w+h-a_{\mathcal{O}}\psi$, where
$a_{\mathcal{O}}:=\frac{\sum_{y\in\mathcal{O}}(w+h)(y)}{\sum_{y\in\mathcal{O}}\psi(y)}$.
It is straightforward to see that

$\displaystyle\begin{split}|a_{\mathcal{O}}|&\leq\frac{\|h\|_{0}}{\psi_{min}},\end{split}$
(4.1)

where we used $w|_{\mathcal{O}}=0$. Then $\frac{\int Gd\mu}{\int\psi
d\mu}=\frac{\int w+hd\mu}{\int\psi d\mu}-a_{\mathcal{O}}.$ Therefore, to show
that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(w+h;\psi,{M},f)$, it is enough to
show that

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

where we used the assumption $\psi$ is strictly positive and the fact $\int
Gd\mu_{\mathcal{O}}=0$.
Claim 1. Put $Area_{1}=\\{y\in{M}:d(y,\mathcal{O})\leq\rho\\}$, then
$Area_{1}$ contains all $x\in{M}$ with $G(x)\leq 0$.

###### Proof of claim 1..

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

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

This ends the proof of claim 1.∎

Claim 2. If $z\in{M}$ is not a generic point of $\mu_{\mathcal{O}}$, then
there is $m\in\mathbb{N}\cup\\{0\\}$ such that $\sum_{i=0}^{m}G(f^{i}z)>0$.

###### Proof of claim 2..

If $z\notin Area_{1}$, just note $m=0$, we have nothing to prove.

Now we assume that $z\in Area_{1}$. There is $y_{0}\in\mathcal{O}$ such that

$d(z,y_{0})=d(z,\mathcal{O})\leq\rho<\frac{D(\mathcal{O})}{2Lip_{f}}<\delta.$

If $d(f^{k}z,f^{k}y_{0})\leq\delta$ for all $k\geq 0$, by ASP, we have

$d(f^{k}z,f^{k}y_{0})\leq Ce^{-\lambda
k}(d(z,y_{0})+d(f^{2k}z,f^{2k}y_{0}))\leq 2Ce^{-\lambda k}\delta\to 0\text{ as
}k\to+\infty.$

Hence, $z$ is a generic point of $\mu_{\mathcal{O}}$ which is impossible by
our assumption. Therefore, there must be some $m_{1}>0$ such that
$d(f^{m_{1}}z,f^{m_{1}}y_{0})\geq\delta.$ There exists $m_{2}>0$ the smallest
time such that

$\displaystyle\frac{D(\mathcal{O})}{2Lip_{f}}\leq
d(f^{m_{2}}z,f^{m_{2}}y_{0})\leq\frac{D(\mathcal{O})}{2},$ (4.2)

where we used the assumption $f$ is Lipschitz with Lipschitz constant
$Lip_{f}$. Then we have
$d(f^{m_{2}}z,\mathcal{O})=d(f^{m_{2}}z,f^{m_{2}}y_{0})\geq\frac{D(\mathcal{O})}{2Lip_{f}}$
and

$\displaystyle\begin{split}G(f^{m_{2}}z)=(w+h-a_{\mathcal{O}}\psi)(f^{m_{2}}z)&\geq\theta\left(\frac{D(\mathcal{O})}{2Lip_{f}}\right)-\|h\|_{0}-|a_{\mathcal{O}}|\|\psi\|_{0}.\\\
\end{split}$ (4.3)

where we used the definition of $\theta(\cdot)$. On the other hand,
$\frac{D(\mathcal{O})}{2Lip_{f}}>\rho$ by assumption which implies that

$\displaystyle f^{m_{2}}z\notin Area_{1}.$ (4.4)

We take $m_{3}$ the largest time with $0\leq m_{3}\leq m_{2}$ such that

$f^{m_{3}}z\in Area_{1},$

where we use the assumption $z\in Area_{1}$. By (4.4), it is clear that
$m_{3}<m_{2}$ since $m_{2}$ is the smallest time meets (4.2). Then by claim 1,

$\displaystyle G(f^{n}z)>0\text{ for all }m_{3}<n<m_{2}.$ (4.5)

Additionally, by the choice of $m_{2}$ and (4.2) one has that

$d(f^{n}z,f^{n}y_{0})\leq\frac{D(\mathcal{O})}{2}\leq\delta\text{ for all
}0\leq n\leq m_{3}.$

Therefore, by ASP, we have for all $0\leq n\leq m_{3}$,

$\displaystyle d(f^{n}z,f^{n}y_{0})\leq C\rho(e^{-\lambda
n}+e^{-\lambda(m_{3}-n)}),$

where we used $z,f^{m_{3}}z\in Area_{1}$. Hence,

$\displaystyle\sum_{n=0}^{m_{3}}d(f^{n}z,f^{n}y_{0})\leq\frac{2C\rho}{1-e^{-\lambda}}.$

Since $w\geq 0$ and $w|_{\mathcal{O}}=0$, one has

$\displaystyle\begin{split}&\sum_{n=0}^{m_{3}}\left(G(f^{n}z)-G(f^{n}y_{0})\right)\\\
=&\sum_{n=0}^{m_{3}}\left(w(f^{n}z)-w(f^{n}y_{0})+h(f^{n}z)-h(f^{n}y_{0})+a_{\mathcal{O}}\psi(f^{n}y_{0})-a_{\mathcal{O}}\psi(f^{n}z)\right)\\\
\geq&-(\|h\|_{1}+|a_{\mathcal{O}}|\|\psi\|_{1})\sum_{n=0}^{m_{3}}d(f^{n}z,f^{n}y_{0})\\\
\geq&-(\|h\|_{1}+|a_{\mathcal{O}}|\|\psi\|_{1})\cdot\frac{2C\rho}{1-e^{-\lambda}}.\end{split}$
(4.6)

By assuming that $m_{3}=p\sharp\mathcal{O}+q$ for some nonnegative integer $p$
and $0\leq q\leq\sharp\mathcal{O}-1$, one has

$\displaystyle\begin{split}\sum_{n=0}^{m_{3}}G(f^{n}y_{0})&=\sum_{m_{3}-q-1}^{m_{3}}G(f^{n}y_{0})\geq-\sharp\mathcal{O}\cdot(\|h\|_{0}+|a_{\mathcal{O}}|\|\psi\|_{0}).\\\
\end{split}$ (4.7)

where we used the facts $\int Gd\mu_{\mathcal{O}}=0$ and
$G\geq-\|h\|_{0}+|a_{\mathcal{O}}|\|\psi\|_{0}$. Combining (4.1),(4.3), (4.5),
(4.6) and (4.7), we have

$\displaystyle\sum_{n=0}^{m_{2}}G(f^{n}z)\geq$
$\displaystyle\sum_{n=0}^{m_{3}}G(f^{n}z)+G(f^{m_{2}}z)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{m_{3}}\left(G(f^{n}z)-G(f^{n}y_{0})\right)+\sum_{n=0}^{m_{3}}G(f^{n}y_{0})+G(f^{m_{2}}z)$
$\displaystyle\geq$
$\displaystyle-(\|h\|_{1}+|a_{\mathcal{O}}|\|\psi\|_{1})\cdot\frac{2C\rho}{1-e^{-\lambda}}-\sharp\mathcal{O}\cdot(\|h\|_{0}+|a_{\mathcal{O}}|\|\psi\|_{0})$
$\displaystyle+\theta\left(\frac{D(\mathcal{O})}{2C}\right)-\|h\|_{0}-|a_{\mathcal{O}}|\|\psi\|_{0}$
$\displaystyle=$
$\displaystyle\theta\left(\frac{D(\mathcal{O})}{2C}\right)-\|h\|_{1}\cdot\frac{2C\rho}{1-e^{-\lambda}}$
$\displaystyle-\left(\frac{2C\rho\|\psi\|_{1}}{(1-e^{-\lambda})\psi_{min}}+\sharp\mathcal{O}+\sharp\mathcal{O}\frac{\|\psi\|_{0}}{\psi_{min}}+\frac{\|\psi\|_{0}}{\psi_{min}}+1\right)\|h\|_{0}$
$\displaystyle>$ $\displaystyle 0,$

where we used the assumption of $h$. Therefore, $m=m_{2}$ is the time we need.
This ends the proof of claim 2. ∎

Now we end the proof. It is enough to show that for all
$\mu\in\mathcal{M}^{e}({M},f)$

$\int Gd\mu\geq 0.$

Given $\mu\in\mathcal{M}^{e}(f)$, 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 2, we have $m_{1}\in\mathbb{N}$ such that

$\sum_{n=0}^{m_{1}}G(f^{n}z)>0$

Note that $f^{m_{1}+1}z$ is also not a generic point of $\mu_{\mathcal{O}}$.
By claim 2, we have $m_{2}\geq m_{1}+1$ such that

$\sum_{n=m_{1}+1}^{m_{2}}G(f^{n}z)>0.$

By repeating the above process, we have $0\leq m_{1}<m_{2}<m_{3}<\cdots$ such
that

$\sum_{n=m_{i}+1}^{m_{i+1}}G(f^{n}z)>0,i=0,1,2,3,\cdots,$

where $m_{0}$ is noted by $-1$. Therefore

$\displaystyle\int Gd\mu$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{m_{i}+1}\sum_{n=0}^{m_{i}}G(f^{n}z)$
$\displaystyle=\lim_{i\to+\infty}\frac{1}{m_{i}+1}\left(\sum_{n=0}^{m_{1}}G(f^{n}z)+\sum_{n=m_{1}+1}^{m_{2}}G(f^{n}z)+\cdots+\sum_{n=m_{i-1}}^{m_{i}}G(f^{n}z)\right)$
$\displaystyle\geq 0.$

That is, we have $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,{M},f)$ by
our beginning discussion. This ends the proof.∎

##  Appendix A Mañé-Conze-Guivarc’h-Bousch’s Property

In this section, we mainly present Bousch’s work (see [Bousch_Mane] for
detail) to show that uniformly hyperbolic diffeomorphism on a smooth compact
manifold has MCGBP. The same argument shows that the Axiom A attractor also
has MCGBP.

Let $f:M\rightarrow M$ be a diffeomorphism on a smooth compact manifold $M$.
By a function $u:{M}\to\mathbb{R}$ and an integer $K\geq 1$, note
$u_{K}=\frac{1}{K}\sum_{i=0}^{K-1}u\circ f^{i}$. Since $f$ is assumed
Lipschitz, $u\in\mathcal{C}^{0,\alpha}(M)$ for some $0<\alpha\leq 1$ implies
that $u_{K}\in\mathcal{C}^{0,\alpha}(M)$. Additionally, one has that

$\displaystyle\int ud\mu=\int u_{K}d\mu\text{ for all
}\mu\in\mathcal{M}({M},f).$ (A.1)

Therefore,

$\beta(u;{M},f)=\beta(u_{K};{M},f)\text{ and
}\mathcal{M}_{min}(u;{M},f)=\mathcal{M}_{min}(u_{K};{M},f).$

######  Theorem A.1.

Let $f:M\rightarrow M$ be an Anosov diffeomorphism on a smooth compact
manifold $M$. Then for $0<\alpha\leq 1$, there is an integer $K=K(\alpha)$
such that for all $u\in\mathcal{C}^{0,\alpha}({M})$ with $\beta(u;{M},f)\geq
0$, there is a function $v\in\mathcal{C}^{0,\alpha}({M})$ such that $u_{K}\geq
v\circ f^{K}-v$.

###### Proof.

The proof mainly follows Bousch’s work in [Bousch_Mane], to which we refer
readers for detailed proof. It is worth to point out that the only difference
is that, in our setting, one needs a large integer $K=K(\alpha)$ to grantee
that $(M,f^{K})$ meets the condition in [Bousch_Mane], and to replace $u$ by
$u_{K}$ as $\beta(u;{M},f)=\beta(u_{K};{M},f)$. ∎

## 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.

## References

  * [1]
  * [2] [[BLL]cite.BLL ] Baraviera, A.; Leplaideur, R.; Lopes, A.O. Ergodic optimization, zero temperature limits and the max-plus algebra. IMPA, Rio de Janeiro, 2013.
  * [B] Bochi, J. Ergodic optimization of Birkhoff averages and Lyapunov exponents. Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, vol. 2, pp. 1821–1842.
  * [3] [[BZ]cite.BZ ] Bochi, J.; Zhang, Y. Ergodic optimization of prevalent super-continuous functions. Int. Math. Res. Not. IMRN 19 (2016), 5988–6017.
  * [4] [[Bo1]cite.Bousch_poisson ]Bousch, T. Le poisson n’a pas d’arêtes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 489–508.
  * [5] [[Bo2]cite.Bousch_Walters ]Bousch, T. La condition de Walters. Ann. Sci. École Norm. Sup. 34 (2001), no. 2, 287–311.
  * [6] [[Bo3]cite.Bousch_Mane ]Bousch, T. Le lemme de Mañé–Conze–Guivarc’h pour les systèmes amphidynamiques rectifiables. Ann. Fac. Sci. Toulouse Math.  20 (2011), no. 1, 1–14.
  * [Bo4] Bousch, T. Generity of minimizing periodic orbits after Contreras. British Math. Colloquium talk. QUML. April, 2014.
  * [7] [[BQ]cite.BQ ] Bressaud, X; Quas, A. Rate of approximation of minimizing measures. Nonlinearity, 20 (2007), no. 845-853.
  * [8] [[BF]cite.BF ]Barral, J.; Feng, D. Weighted thermodynamic formalism on subshifts and applications. Asian J. Math. 16 (2012), no. 2, 319–352.
  * [9] [[BCW]cite.BCW ]Barreira, L.; Cao, Y.; Wang, J. Multifractal analysis of asymptotically additive sequences J. Stat. Phys. 153 (2013), no. 5, 888–910.
  * [10] [[Co1]cite.Contreras_EO ] Contreras, G. Ground states are generically a periodic orbit. Invent. math. 205 (2016), 383–412
  * [Co2] Contreras, G. Generic Mañe sets. arxiv1410.7141.
  * [11] [[CLT]cite.CLT ] Contreras, G.; Lopes, A.O.; Thieullen, P. Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1379–1409.
  * [12] [[FH]cite.FH ] Feng, D.; Huang, W. Lyapunov spectrum of asymptotically sub-additive potentials. Comm. Math. Phys. 297 (2010), no. 1, 1–43.
  * [13] [[Je1]cite.Jenkinson_survey ] Jenkinson, O. Ergodic optimization. Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 197–224.
  * [Je2] Jenkinson, O. Ergodic optimization in dynamical systems. Ergodic Theory Dynam. Systems 2018, to appear.
  * [14] [[Ma]cite.Mane ] Mañé, R. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996), no. 2, 273–310.
  * [15] [[Mo]cite.Morris_entropy ] Morris, I.D. Maximizing measures of generic Hölder functions have zero entropy. Nonlinearity 21 (2008), no. 5, 993–1000.
  * [GW] Greene, R.E.; Wu, H. $C^{\infty}$ approximations of convex, subharmonic and plurisubharmonic functions.  Ann. Sci. École Norm. Sup.  12 (1979), no. 4, 47–84.
  * [16] [[QS]cite.QS ] Quas, A.; Siefken, J. Ergodic optimization of super-continuous functions on shift spaces. Ergodic Theory Dynam. Systems 32 (2012), no. 6, 2071–2082.
  * [SGOY] Shinbort, T.; Grebogi, C.; Ott, E.; Yorke, J. Using small perturbation to control chaos. Nature 363 (1993), 411–7.
  * [17] [[YH]cite.YH ] Yuan, G.; Hunt, B.R. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), no. 4, 1207–1224.
  * [18] [[OGY]cite.OGY90 ] Ott, E.; Grebogi, C.; Yorke, J. Controlling chaos. Phys. Rev. Lett. 64 (1990), 11–96.