Topological Pressure of Discontinuous Potentials and Variational Principle for Flows¹¹footnotemark: 1

Ruolan Xiong

{}^{\href mailto:[email protected]}

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Abstract

Let $X$ be a compact metric space and $\Phi=\{\varphi_{t}\}_{t\in\mathbb{R}}$ be a continuous flow on $X$ . We introduce two types of topological pressure for family of discontinuous potentials $a=\{a_{t}\}_{t>0}$ . First, define the topological pressure of family of measurable potentials $a=\{a_{t}\}_{t>0}$ on a subset $Z$ for flow and proof its invariant principle. The second topological pressure is defined on a invariant subset having a nested family of subsets, we also proof its invariant principle.

keywords:

dynamical system, topological pressure, flow, discontinuous potential, variational principle

MSC:

[2020]37D35, 26A18

1 Introduction

Topological pressure emerged in the 1970s as an extension of topological entropy, with the purpose of measuring the complexity of motion in dynamical systems. Different potentials have varying effects on the system’s motion, which topological entropy fails to reflect. Therefore, topological pressure is defined to capture the relationship between the uncertainty in the system and the potentials.

The refined definition of topological pressure was initially formulated by David Ruelle, a prominent theoretical physicist and esteemed member of the French Academy of Sciences. Drawing inspiration from the contributions of Sinai and Bowen, Ruelle introduced the concept of topological pressure in 1973, building upon the notion of ”pressure” within statistical mechanics. His definition pertained specifically to $Z^{v}$ -actions that adhere to expansivity and specification on a compact metric space. He proved the following equation:

P(\varphi)=\mathop{\max}\limits_{\mu\in I}\{s(\mu)+\mu(\varphi)\},

which is the variational principle([21]). In 1984, Pesin and Pitskel further extended this condition, defining the topological pressure of continuous mappings on non-compact sets([17]). In the 1960s, the concept of topological pressure for sequences of subadditive potentials already emerged([14]). In 1988, Falconer studied the topological pressure of sequences of subadditive potentials on mixed repellers and provided a variational principle under Lipschitz conditions and bounded variation([10]). Subsequently, there have been numerous related studies on the topological pressure of sequences of subadditive potentials([7, 12, 23, 9]). In 1996, Barreira further relaxed the subadditive condition and defined the topological pressure of sequence of potentials (not necessarily subadditive) on a subsets of compact metric space([5]). Over the years, the concept of topological pressure has evolved beyond just continuous mappings in discrete dynamical systems. The topological pressure and variational principles for families of continuous potentials $a=\{a_{t}\}_{t>0}$ have also emerged([4]). Similar to the concept of metric entropy, there is also topological entropy in dynamical systems, which originates from topological pressure and is related to measures([13, 24, 8]).

The above conclusions are all about continuous potentials. When the potential does not satisfy continuity, does topological pressure still exist? In 2006, Mummert first defined the topological pressure of a discontinuous map $\lambda$ on a subset $\Lambda$ of a compact metric space $(X,T)$ ([16]). In Mummert’s study, the set $\Lambda$ is represented as the union of a nested sequence of sets: $\Lambda=\mathop{\bigcup}\limits_{l\geq 1}\Lambda_{l}$ . The potential only needs to be continuous on the closure of each subset, not necessarily on the entire set. Thus, the classical topological pressure $P_{\Lambda_{l}}$ $(\varphi)$ can be defined on each subset $\Lambda_{l}$ , and then taking the supremum over $l$ , we obtain the topological pressure on the subset $\Lambda$ : $P_{\Lambda}$ $(\varphi)=\mathop{\sup}\limits_{l\geq 1}P_{\Lambda_{l}}$ $(\varphi)$ . Subsequently, Ma Xianfeng et al. extended Mummert’s conclusion from a single potential to a subadditive potential sequence([15]). In 2012, J. Barral and D. J. Feng studied the topological pressure of upper semi-continuous subadditive potentials([2]). In 2016, Feng and Huang gave the definition of weighted topological pressure for upper semi-continuous entropy maps, along with a variational principle([11]). In 2017, Marc Rauch directly defined the topological pressure of measurable potentials in compact metric spaces using the Caratheodory structure theory and proved the variational principle([19]). In subsequent research, he also introduced the topological pressure and variational principle for subadditive potential sequences([20]). Additionally, topological pressure can be defined for systems with discontinuous semi-flow([1]).

There have been some results for the topological pressure of discontinuous potentials in discrete dynamical systems, but in continuous dynamical systems it remains to be studied. This paper provides two definitions of the topological pressure of discontinuous potentials with respect to the flow and introduces the corresponding variational principles for each.

First, the topological pressure of family of measurable potentials on a subset is introduced. In a continuous dynamical system $(X,\Phi)$ , given a nonempty subset $\emptyset\neq Z\subseteq X$ , take the real numbers $\epsilon>0$ , $\alpha\in\mathbb{R}$ , We define the topological pressure of the family of potential $a$ on the set $Z$ as :

P_{Z}(a)=\mathop{\lim}\limits_{\epsilon\to\ 0}P_{Z}(a,\epsilon)=\mathop{\lim}\limits_{\epsilon\to\ 0}\inf\{\alpha\in\mathbb{R}:M(Z,a,\alpha,\epsilon)=0\},

where

M(Z,a,\alpha,\epsilon)=\mathop{\lim}\limits_{T\to\ +\infty}M(Z,a,\alpha,\epsilon,T),

M(Z,a,\alpha,\epsilon,T)=\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)},

the lower bound is taken from all countably open covers $\Gamma\subseteq X\times[T,+\infty)$ that cover $Z$ and $a(x,t,\epsilon)=\sup\{a_{t}(y):y\in B_{t}(x,\epsilon)\}$ . In the third part we can see that such definition is well-defined, stemming from the Caratheodory dimension theory of Pesin([18]). Then we have the first main conclusion of this article.

Theorem 1.1: Let $(X,\Phi)$ be a DTS without fixed point and $\mathcal{G}(a,\lambda):=\{\mu\in\mathcal{A}(\lambda)\cap\mathcal{E}_{\Phi}(X)\cap\mathcal{\mathcal{M}}_{\Phi}(X):\mathop{\lim}\limits_{t\to\infty}\frac{1}{t}a_{t}(x)=\lambda(\mu)\,\,for\,\,\mu-$ almost $x\in X\}$ . For each subset $\mathcal{Y}\subseteq\mathcal{G}(a,\lambda)$ one has

P_{A(a,\lambda,\mathcal{Y})}(a)=\sup\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\}.

In particular, one can choose for each $\mu\in\mathcal{Y}$ a Borel set $B_{\mu}\subseteq A(a,\lambda,\mathcal{Y})$ such that $\mu(B_{\mu}=1)$ , and $P_{B}(a)=\sup\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\}$ .

For the second definition, consider a $\Phi$ -invariational subset $Z\subseteq X$ . $Z$ consists of a nested subset of $\{Z_{l}\}_{l\leq 1}$ , i.e.: $Z=\mathop{\bigcup}\limits_{l\geq 1}Z_{l}$ and $Z_{l}\subseteq Z_{l+1}$ for all $l\in\mathbb{N}$ . We require that $a$ be continuou on the closure of each subset $Z_{l}$ , but not necessarily on $Z$ . The topological pressure of $a$ on $Z$ with respect to the flow $\Phi$ is defined as:

P_{Z}(a)=\mathop{\sup}\limits_{l\geq 1}P_{Z_{l}}(a)

where $P_{Z_{l}}(a)$ is the classical topological pressure of $a$ on $Z_{l}$ . Since $a$ is continuous on the closure of every subset $Z_{l}$ , $P_{Z_{l}}(a)$ is well-defined. Then we have the second main conclusion of this article.

Theorem 1.2: Let $a$ be a family of functions $\{a_{t}\}_{t>0}$ with tempered variation such that $\mathop{\sup}\limits_{t\in[0,T]}||a_{t}||_{\infty}<+\infty$ for all $T>0$ . Let $Z=\mathop{\bigcup}\limits_{l\geq 1}Z_{l}\subseteq X$ be a Borel $\Phi-$ invariant set and $a$ is continuous with respect to the family of subsets $\{Z_{l}\}$ . If there exists a continuous function $b:X\to\mathbb{R}$ such that

a_{t+s}-a_{t}\circ\varphi(x)\to\int_{0}^{s}(b\circ\varphi_{u})\,du

(1.1)

uniformly on $Z$ when $t\to+\infty$ for some $s>0$ , then

P_{\mathcal{L}(Z)}(a)=\sup\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\}.

Although the above two topological pressures are defined using different methods, they both reflect the relationship between the potentials and the complexity of the system’s motion on a subset of a continuous dynamical system.

2 Prelimilaries

Let $(X,\mathcal{B})$ be a measurable space and $\Phi=(\varphi_{t})_{t\in\mathbb{R}}$ be a continuous flow on $X$ . This is, a family of heomorphisms $\varphi_{t}:X$ $\rightarrow$ $X$ such that $\varphi_{0}=id$ and $\varphi_{t}\circ\varphi_{s}=\varphi_{t+s}$ for all $t,s\in$ $\mathbb{R}$ .

Definition 2.1: A measure $\mu$ on $(X,\mathcal{B})$ is said to be $\Phi-$ invariant, if $\mu(\varphi^{-1}_{t}B)=\mu(B)$ for each $B\in\mathcal{B}$ and $t\in\mathbb{R}$ . A set $Z\subseteq$ $X$ is called $\Phi-$ invariant, if $\varphi^{-1}(Z)=Z$ for any $t\in\mathbb{R}$ . A measure $\mu$ on $(X,\mathcal{B})$ is said to be ergodic for the flow if every Borel $\Phi-$ invariant set $B\in\mathcal{B}$ satisfies $\mu(B)=0$ or $\mu(B)=1$ .

The set of all $\Phi-$ invariant probability measures on $(X,\mathcal{B})$ is denoted by $\mathcal{M}_{\Phi}(X)$ , and the set of the ergodic probability measures is denoted by $\mathcal{E}_{\Phi}(X)$ . For each measurable subset $Z\subseteq$ $X$ , we define $\mathcal{M}_{\Phi}(Z)=\{\mu\in\mathcal{M}_{\Phi}(X):\mu(Z)=1\}$ and $\mathcal{E}_{\Phi}(Z)=\{\mu\in\mathcal{E}_{\Phi}(X):\mu(Z)>0\}$ . For $\mu\in\mathcal{M}_{\Phi}(X)$ , the quantity $h_{\mu}(\Phi):=h_{\mu}(\varphi_{1})$ .

Let $(X,d)$ be a compact metric space. Given $x\in X$ and $t,\epsilon>0$ , we consider the set

B_{t}(x,\epsilon)=\{y\in X:d(\varphi_{s}(x),\varphi_{s}(y)<\epsilon,s\in[0,t]\}.

We call $\mathcal{U}\subseteq 2^{X}$ to be a finite open cover of $X$ if $\#(\mu)<\infty$ and $X\subseteq\mathop{\bigcup}\limits_{U\in\mathcal{U}}U$ , where every $U\in\mathcal{U}$ is open.

Given $x\in X$ and $t>0$ , define the probability measures

\delta_{x,t}=\frac{1}{t}\int_{0}^{t}\delta_{\varphi_{s}(x)}\,ds.

where $\delta_{y}$ is the probability measure concentrated on $\{y\}$ . Denote by $V_{\Phi}(x)\subseteq M_{\Phi}(X)$ the set of all $\Phi-$ invariant sublimits of $\{\delta_{x,t}\}_{t>0}$ in the weak*-topology. We can proof that $V_{\Phi}(x)\neq\emptyset$ for every $x\in X$ , and the $V_{\Phi}(x)$ is a compact metrizable space. The set $Z_{\mu}:=\{x\in X:\mathop{\lim}\limits_{t\to\infty}\delta_{x,t}=\mu\}$ is called the set of generic points of $\mu$ . Note that $\mu(Z_{\mu})=1$ if $\mu$ is ergodic.

Given $\epsilon>0$ , we say that a set $\Gamma\subseteq X\times\mathbb{R}_{0}^{+}$ covers a subset $Z\subseteq X$ if $Z\subseteq\mathop{\bigcup}\limits_{(x,t)\in\Gamma}B_{t}(x,\epsilon)$ . Let $\mathcal{G}_{Z}(\epsilon,T)$ be the set of all $\Gamma$ satisfying $t\geq T$ . Let $a=\{a_{t}\}_{t>0}$ be a family of function $:X\to\mathbb{R}$ with tempered variation, that is, such that $\mathop{\lim}\limits_{\epsilon\to\ 0}\overline{\mathop{\lim}\limits_{t\to\infty}}\frac{\gamma_{t}(a,\epsilon)}{t}=0$ where

\gamma_{t}(a,\epsilon)=\sup\{\mid a_{t}(y)-a_{t}(z)\mid:y,z\in B_{t}(x,\epsilon)\,for\,some\,x\in X\}.

We write $a(x,t,\epsilon)=\sup\{a_{t}(y):y\in B_{t}(x,\epsilon)\}$ for $(x,t)\in\Gamma$ .

Definition 2.2: Fix $\emptyset\neq Z\subseteq X$ , $\epsilon>0$ , $\alpha\in\mathbb{R}$ , Let

M(Z,a,\alpha,\epsilon,T)=\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}.

with the infimum taken over all countable sets $\Gamma\subseteq X\times[T,+\infty)$ covering $Z$ . And let

M(Z,a,\alpha,\epsilon)=\mathop{\lim}\limits_{T\to\ +\infty}M(Z,a,\alpha,\epsilon,T).

Clearly, $M(Z,a,\alpha,\epsilon,T)$ increases as $T$ increases.We have that

M(Z,a,\alpha,\epsilon)=\mathop{\sup}\limits_{T}M(Z,a,\alpha,\epsilon,T).

Lemma 2.3: Let $\beta\in\mathbb{R}$ and $Z\subseteq X$ . If $M(Z,a,\beta,\epsilon)<\infty$ , then $M(Z,a,\alpha,\epsilon)=0$ for all $\alpha>\beta$ and $T>0$ .

Proof.

The case $Z=\emptyset$ is clear. Choose some $A\in\mathbb{R}$ such that $M(Z,a,\beta,\epsilon)<A$ . Then $M(Z,a,\beta,\epsilon,T)<A$ for all $T>0$ . Hence

	$\displaystyle 0$	$\displaystyle\leq\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}$
		$\displaystyle=\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\beta t}\right)}\exp(t(\beta-\alpha))$
		$\displaystyle\leq A(\exp(\beta-\alpha))^{T}\to 0$

as $T\to\infty$ .The statement is proved. ∎

By Lemma 2.3, the following quantity is well defined:

P_{Z}(a,\epsilon)=\inf\{\alpha\in\mathbb{R}:M(Z,a,\alpha,\epsilon)=0\}.

For $0<\epsilon^{{}^{\prime}}<\epsilon$ , and $T>0$ , $B_{t}(x,\epsilon^{{}^{\prime}})\subseteq B_{t}(x,\epsilon)$ , if $Z\subseteq\mathop{\bigcup}\limits_{(x,t)\in\Gamma}B_{t}(x,\epsilon^{{}^{\prime}})$ one has $Z\subseteq\mathop{\bigcup}\limits_{(x,t)\in\Gamma}B_{t}(x,\epsilon)$ . This shows that $\mathcal{G}_{z}(\epsilon^{{}^{\prime}},T)\subseteq\mathcal{G}_{z}(\epsilon,T)$ . Hence the following limit is also well-defined:

P_{Z}(a)=\mathop{\lim}\limits_{\epsilon\to\ 0}P_{Z}(a,\epsilon)=\mathop{\sup}\limits_{\epsilon>0}P_{Z}(a,\epsilon).

Definition 2.4: The quantity $P_{Z}(a)$ is called topological pressure of $a$ on $Z$ with respected to $\Phi$ .

For $Z=\emptyset$ , we emphasize that $P_{\emptyset}(a)=-\infty$ for each potential $a$ . This follows from $M(\emptyset,a,\alpha,\epsilon,T)=0$ for every $\epsilon>0,\,\alpha\in\mathbb{R}$ and $T>0$ . Thus $P_{\emptyset}(a)\neq-\infty$ implies $Z\neq\emptyset$ .

Next we introduce some properties of topological pressure.

Lemma 2.5: For $\epsilon>0,\,\alpha\in\mathbb{R},\,T>0$ , and $Y\subseteq Z\subseteq X$ , The following inequalities hold:

M(Y,a,\alpha,\epsilon,T)\leq M(Z,a,\alpha,\epsilon,T).

In particular,

P_{Y}(a,\epsilon)\leq P_{Z}(a,\epsilon)

for all $\epsilon>0$ , and then

P_{Y}(a)\leq P_{Z}(a).

Proof.

For a cover $\Gamma\subseteq X\times\mathbb{R}_{0}^{+}$ , if $\Gamma$ covers $Z$ , then it can cover $Y$ . Hence $M(Y,a,\alpha,\epsilon,T)\leq M(Z,a,\alpha,\epsilon,T)$ and then $P_{Y}(a,\epsilon)\leq P_{Z}(a,\epsilon)$ , $P_{Y}(a)\leq P_{Z}(a)$ . ∎

Lemma 2.6: Given a set $Z\subseteq X$ , suppose $Z=\mathop{\bigcup}\limits_{i\in I}Z_{i}$ , where $I\subseteq\mathbb{N}$ and $Z_{i}\subseteq X$ for all $i\in I$ . then

P_{Z}(a)=\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a).

Proof.

Since $Z_{i}\subseteq Z$ , we have $P_{Z_{i}}(a)\leq P_{Z}(a)$ for each $i\in I$ and so

P_{Z}(a)\leq\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a).

Take $\alpha>\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a,\epsilon)$ . Then $M(Z_{i},a,\alpha,\epsilon)=0$ for each $i$ . Hence given $\delta>0$ and $T>0$ , for each $i$ there exists $\Gamma_{i}\subseteq X\times[T,\,+\infty)$ covering $Z_{i}$ such that

\sum\limits_{\left({x,t}\right)\in\Gamma_{i}}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}<\frac{\delta}{2^{i}}.

Then $\Gamma\subseteq\mathop{\bigcup}\limits_{i\in I}\Gamma_{i}$ covers $Z$ and

\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}\leq\sum\limits_{i\in I}\sum\limits_{\left({x,t}\right)\in\Gamma_{i}}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}\leq\sum\limits_{i\in I}\frac{\delta}{2^{i}}\leq\delta.

This gives $M(Z,a,\alpha,\epsilon)\leq\delta$ and then $M(Z,a,\alpha,\epsilon)=0$ since the arbitrariness of $\delta$ . Therefore, $\alpha\geq P_{Z}(a,\epsilon)$ and letting $\alpha\to\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a,\epsilon)$ gives

\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a,\epsilon)\geq P_{Z}(a,\epsilon).

Letting $\epsilon\to 0$ , we have that $\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a)\geq P_{Z}(a)$ . Hence

P_{Z}(a)=\mathop{\sup}\limits_{i\in I}P_{Z_{i}}(a).

∎

3 Variational principle

In this section, we will introduce the variatiional principle of above topological pressure.

Definition 3.1: A mapping $\lambda:\,\mathcal{M}_{\Phi}(X)\to[-\infty,\,+\infty]$ is called Lyapunov exponent.
The corresponding set $\mathcal{A}(\lambda):=\{\mu\in\mathcal{M}_{\Phi}(X):h_{\mu}(\Phi)<\infty\,or\,\lambda(\mu)>-\infty\}$ is called the set of all allowed $\Phi$ -invariant measures with respected to $\lambda$ . This means that for measures $\mu\in\mathcal{A}(\lambda)$ the quantity $h_{\mu}(\Phi)+\lambda(\mu)$ is well-defined. Fix some $\mu\in\mathcal{M}_{\Phi}(X)$ , a point $x\in X$ such that $\mu\in V_{\Phi}(X)$ is called allowed point with respected to $\lambda,\,a$ and $\mu$ if

\mathop{\limsup}\limits_{s\to\infty}\frac{1}{t_{s}}a_{t_{s}}(x)\leq\lambda(\mu)

(3.1)

for all sub-family $(t_{s})_{s>0}$ which satisfies $\delta_{x,\,t_{s}}\to\mu$ as $s\to\infty$ . The set of all those points $x$ is denoted by $A(a,\,\lambda,\,\mu)$ . For a subset $\mathcal{Y}\subseteq\mathcal{M}_{\Phi}(X)$ , denote in addition

A(a,\,\lambda,\,\mathcal{Y}):=\mathop{\bigcup}\limits_{\mu\in\mathcal{Y}}A(a,\,\lambda,\,\mu)

and

V(a,\,\lambda,\,\mathcal{Y}):=\mathop{\bigcap}\limits_{\mu\in\mathcal{Y}}A(a,\,\lambda,\,\mu).

Note that $A(a,\,\lambda,\,\mu)$ can be empty in the case of $\mu\notin V_{\Phi}(X)$ for each $x\in X$ .

Proposition 3.2: Let $f:\,X\to[-\infty,\,+\infty]$ be upper semi-continuous, $\lambda(\mu):=\int_{X}\,f\,d\mu$ and $a_{t}:=\int_{0}^{t}f\circ\varphi_{s}\,ds$ . Then one has

A(a,\,\lambda,\,\mathcal{Y})=\{x\in X:V_{\Phi}(x)\cap\mathcal{Y}\neq\emptyset\},

V(a,\,\lambda,\,\mathcal{Y})=\{x\in X:\mathcal{Y}\subseteq V_{\Phi}(x)\}.

That is both sets are independent of $f$ .

Proof.

Let $x\in X$ and $\mu\in V_{\Phi}(x)\cap\mathcal{Y}$ . If $(t_{s})_{s>0}$ is any sub-family such that $\mathop{\lim}\limits_{s\to\infty}\delta_{x,t_{s}}=\mu$ . Then

\mathop{\limsup}\limits_{s\to\infty}\frac{1}{t_{s}}a_{t_{s}}(x)=\mathop{\limsup}\limits_{s\to\infty}\frac{1}{t_{s}}\int_{0}^{t_{s}}f(\varphi_{s}(x))\,ds=\mathop{\limsup}\limits_{s\to\infty}\int_{X}f\,d\delta_{x,t_{s}}\leq\int_{x}f\,d\mu

as $f$ is upper semi-continuous (lemma A.2(d) in [20]). Thus $x\in A(a,\,\lambda,\,\mu)$ and $\{x\in X:V_{\Phi}(x)\cap\mathcal{Y}\neq\emptyset\}\subseteq A(a,\,\lambda,\,\mathcal{Y})$ . If $x\in A(a,\,\lambda,\,\mathcal{Y})$ , then there exists a $\mu\in\mathcal{Y}$ such that $x\in A(a,\,\lambda,\,\mu)$ . By definition this means $V_{\Phi}(x)\cap\mathcal{Y}\neq\emptyset$ , and hence $A(a,\,\lambda,\,\mathcal{Y})\subseteq\{x\in X:V_{\Phi}(x)\cap\mathcal{Y}\neq\emptyset\}$ . Let $x\in V(a,\,\lambda,\,\mathcal{Y})$ , then for every $\mu\in\mathcal{Y}$ and $x\in A(a,\,\lambda,\,\mu)$ , $\mu\in V_{\Phi}(x)$ and then $\mathcal{Y}\subseteq V_{\Phi}(x)$ . We have $V(a,\,\lambda,\,\mathcal{Y})\subseteq\{x\in X:\mathcal{Y}\subseteq V_{\Phi}(x)\}$ . On the other hand, for $x\in\{x\in X:\mathcal{Y}\subseteq V_{\Phi}(x)\}$ , one has $\mu\in V_{\Phi}(x)$ for every $\mu\in\mathcal{Y}$ . Then there exists a subsequence $(t_{l})_{l>o}$ such that $\mathop{\lim}\limits_{l\to\infty}\frac{1}{t_{l}}\delta_{x,t_{l}}=\mu$ . Semilarly,

\mathop{\limsup}\limits_{s\to\infty}\frac{1}{t_{s}}a_{t_{s}}(x)\leq\int_{x}f\,d\mu=\lambda(\mu).

Hence $x\in V(a,\,\lambda,\,\mu)$ for every $\mu\in\mathcal{Y}$ and then $x\in V(a,\,\lambda,\,\mathcal{Y})$ , so $V(a,\,\lambda,\,\mathcal{Y})\subseteq\{x\in X:\mathcal{Y}\subseteq V_{\Phi}(x)\}$ . The statement for $V(a,\,\lambda,\,\mathcal{Y})$ holds. ∎

Let $\sup\emptyset:-\infty$ , then for each $\mathcal{Y}\subseteq\mathcal{A}(\lambda)$ , the quantities

P_{\mathcal{Y}}(\lambda):=\sup\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\}

and

Q_{\mathcal{Y}}(\lambda):=\inf\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\}

are well-defined and called upper variational pressure and lower variational pressure of $\lambda$ over $\mathcal{Y}$ respectively.

Theorem 3.3: Let $\lambda$ be a Lyapunov exponent. If $\mathcal{Y}\subseteq\mathcal{A}(\lambda)$ , then one has

P_{A(a,\,\lambda,\,\mathcal{Y})}(a)\leq\sup\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\},

(3.2)

P_{V(\lambda,\,\lambda,\,\mathcal{Y})}(a)\leq\inf\{h_{\mu}(\Phi)+\lambda(\mu):\mu\in\mathcal{Y}\}.

(3.3)

Proof.

In case the $P_{\mathcal{Y}}(\lambda)=\infty$ we are done, now we assume $P_{\mathcal{Y}}(\lambda)<\infty$ . This implies $0\leq h_{\mu}(\Phi)<\infty$ and $\lambda(\mu)<\infty$ for each $\mu\in\mathcal{Y}$ . Thus we can divide $\mathcal{Y}$ into two parts:

\mathcal{Y}_{-\infty}:=\{\mu\in\mathcal{Y}:\lambda=-\infty\}

and

\mathcal{Y}^{{}^{\prime}}:=\{\mu\in\mathcal{Y}:\lambda>-\infty\}.

As a result we obtain by lemma 2.6 and $A(a,\,\lambda,\,\mathcal{Y})=A(a,\,\lambda,\,\mathcal{Y}_{-\infty})\,\cup\,A(a,\,\lambda,\,\mathcal{Y}^{{}^{\prime}})$

P_{A(a,\,\lambda,\,\mathcal{Y})}(a)\,=\,\max\{P_{A(a,\,\lambda,\,\mathcal{Y}_{-\infty})}(a),\,P_{A(a,\,\lambda,\,\mathcal{Y}^{{}^{\prime}})}(a)\}.

(3.4)

Now suppose we have already shown $(3.2)$ for each $\mathcal{F}\subseteq\mathcal{A}(\lambda)$ such that $-\infty<\lambda(\mu)<\infty$ for all $\mu\in\mathcal{F}$ . Define a sequence of Lyapunov exponents $\lambda_{N}(\mu):=-h_{\mu}(\Phi)-N$ for $N\in\mathbb{N}$ and $\mu\in\mathcal{M}_{\Phi}(X)$ . As the entropies are finite for all $\mu\in\mathcal{Y}_{-\infty}$ , one has $-\infty<\lambda_{N}(\mu)<\infty$ for all $\mu\in\mathcal{Y}_{-\infty}$ and $N\in\mathbb{N}$ . In addition

A(a,\,\lambda,\,\mathcal{Y}_{-\infty})\subseteq A(a,\,\lambda_{N},\,\mathcal{Y}_{-\infty})

holds for each $N\in\mathbb{N}$ . Thus by using lemma 2.5 and $(3.2)$ we obtain

	$\displaystyle P_{A(a,\,\lambda,\,\mathcal{Y}_{-\infty})}(a)$	$\displaystyle\leq P_{A(a,\,\lambda_{N},\,\mathcal{Y}_{-\infty})}(a)\leq P_{\mathcal{Y}_{-\infty}}(\lambda_{N})$
		$\displaystyle=\sup\{h_{\mu}(\Phi)-h_{\mu}(\Phi)-N:\mu\in\mathcal{Y}_{-\infty}\}$
		$\displaystyle\leq-N.$

Note that if $\mathcal{Y}_{-\infty}=\emptyset$ , we already have

P_{A(a,\,\lambda,\,\mathcal{Y}_{-\infty})}(a)=P_{\mathcal{Y}_{-\infty}}(\lambda_{N})=-\infty.

otherwise letting $N\to\infty$ yields

P_{A(a,\,\lambda,\,\mathcal{Y}_{-\infty})}(a)=-\infty.

And by $(3.4)$ we end at

P_{A(a,\,\lambda,\,\mathcal{Y})}(a)=P_{A(a,\,\lambda,\,\mathcal{Y}^{{}^{\prime}})}(a)\leq P_{\mathcal{Y}^{{}^{\prime}}}(\lambda)\leq P_{\mathcal{Y}}(\lambda).

Hence it remains to show $(3.2)$ for all subsets $\mathcal{Y}\subseteq\mathcal{A}(\lambda)$ satisfying $h_{\mu}(\Phi)<\infty$ and $-\infty<\lambda(\mu)<\infty$ for all $\mu\in\mathcal{Y}$ . Now pick such a set $\mathcal{Y}$ and suppose $P_{\mathcal{Y}}(\lambda)=-\infty$ . This implies $\mathcal{Y}=\emptyset$ ; Otherwise a measure $\mu\in\mathcal{Y}$ exists such that $P_{\mathcal{Y}}(\lambda)\neq h_{\mu}(\Phi)+\lambda(\mu)>-\infty$ , which is a contradiction. This means $A(a,\,\lambda,\,\mathcal{Y})=\emptyset$ . Thus $P_{A(a,\,\lambda,\,\mathcal{Y})}(a)=P_{\mathcal{Y}}(\lambda)=-\infty$ . Therefore, without restriction we may assume

-\infty<P_{\mathcal{Y}}(\lambda)<\infty.

(3.5)

To proceed we need some technical lemmas.

Lemma 3.4: Let $E$ be a finite set, given $q\in\mathbb{N}$ and $\alpha=(a_{1},\,a_{2},\,...\,a_{q})\in E^{q}$ . Define the probability measure $\nu_{\alpha}$ as

\nu_{\alpha}(e)=\frac{1}{q}\#\{j\in\mathbb{N}:a_{j}=e\}

for every $e\in E$ , and set the entropy of $\alpha$ to

H(\alpha):=-\mathop{\sum}\limits_{e\in E}\nu_{\alpha}(e)\log\nu_{\alpha}(e).

Then for $h\neq 0$ , one has

\mathop{\limsup}\limits_{q\to\infty}\frac{1}{q}\log\#\{\alpha\in E^{q}:H(\alpha)\leq h\}\leq h.

For a proof of this lemma, see lemma 2.16 in [6]

Lemma 3.5: Let $x\in X$ , and $\mu\in V_{\Phi}(X)$ such that $x$ is allowed with respected to $\lambda,\,a$ , and $\mu$ . Let $\delta>0$ and $\Gamma\subseteq X\times\{1\}$ be a finite cover of $X$ . For the open cover $\mathcal{V}=\{V_{1},\,V_{2},\,...V_{r}\}$ of $X$ , where $V_{j}=B_{1}(x_{j},\frac{\epsilon}{2})$ with $(x_{j},1)\in\Gamma$ , there exists $m,\,p\in\mathbb{N}$ with $p$ arbitrary large, and a sequence $U=V_{i_{1}}V_{i_{2}}...V_{i_{p}}$ such that
$(a)$ : $x\in\cap_{r=1}^{p}\varphi_{-r+1}(V_{i_{r}})$ .
$(b)$ : There exists a subset $V\in(\nu^{m})^{k}$ of $U$ of length $km\geq p-m$ satisfying inequality

H(V)\leq m(h_{\mu}(\Phi)+\delta).

$(c)$ : $a_{p}(x)\leq p(\lambda(\mu)+3\delta)$ .

Proof.

$(a)$ and $(b)$ are the statements of [4]. The statement $(c)$ can be proven like lemma 4.7 $(3)$ in [20], one can constructs an increasing sub-family $(t_{j}^{\prime})_{j\in\mathbb{R}}$ and corresponding vectors $\{V_{i_{j}}\}_{j=1}^{p}$ such that conditions $(a)$ and $(b)$ are satisfied, and $\delta_{x,t_{j}^{\prime}}\to\mu$ as $j\to\infty$ . Then by $(3.1)$ one has

\mathop{\limsup}\limits_{j\to\infty}\frac{1}{t_{j}^{\prime}}a_{t_{j}^{\prime}}(x)\leq\lambda(\mu).

Thus there is an $j_{0}\geq 0$ such that $a_{t_{j}^{\prime}}(x)\leq t_{j}^{\prime}(\lambda(\mu)+\delta)$ for all $j\leq j_{0}$ . Hence for each $j\geq 0$ the number $p:=[t_{j+j_{0}}^{\prime}]+1$ together with $v_{i_{j+j_{0}}}$ satisfies all the three conditions. ∎

Continuing the proof. The first goal is to cover $A(a,\,\lambda,\,\mathcal{Y})$ with countable many suitable subsets. we fix $\delta>0$ and a finite open cover $\mathcal{U}$ of $X$ such as lemma 3.5. In addition fix for each $x\in A(a,\,\lambda,\,\mathcal{Y})$ a measure $\mu_{x}\in\mathcal{Y}$ such that $x\in A(a,\,\lambda,\,\mu_{x})$ . Choose some $u_{1},\,u_{2},\,...\in\mathbb{R}$ such that for every $z\in\mathbb{R}$ there exists a $u_{i}$ satisfying $|u_{i}-z|<\delta$ . Now denote for $m,\,i\,\geq 1$ by $Z_{m,i}$ the set of points $x\in A(a,\,\lambda,\,\mathcal{Y})$ , which meet the following criteria:
$\bullet$ the measure $\mu_{x}$ fulfills $\lambda(\mu_{x})\in[u_{i}-\delta,\,u_{i}+\delta]$ .
$\bullet$ All three properties in lemma 4.5 are satisfied by $u_{x},\,\delta,\,\mathcal{U}$ and $m$ .
As $\{u_{i}\}_{i\in\mathbb{N}}$ is $\delta-$ dense in $\mathbb{R}$ and by $(3.5)$ one has $\lambda(\mu_{x})\in\mathbb{R}$ , lemma 3.5 ensures for every $x\in A(a,\,\lambda,\,\mathcal{Y})$ the existence of some corresponding $m,\,i\in\mathbb{N}$ . Hence we obtain

A(a,\,\lambda,\,\mathcal{Y})=\mathop{\bigcup}\limits_{m\in\mathbb{N}}\mathop{\bigcup}\limits_{i\in\mathbb{N}}Z_{m,i}.

For simplicity we may assume that all $Z_{m,i}$ are nonempty, else they can be called out of the union.

Now fix $Z_{m,i}\neq\emptyset$ and denote for each $q\geq 1$

R_{q}:=\{V\in(\mathcal{U}^{m})^{q}:H(V)\leq m(P_{\mathcal{Y}}(\lambda)-u_{i}+2\delta)\}

(3.6)

Pick some $x\in Z_{m,i}$ , then by lemma 4.5 one can finds arbitrary large $N\geq 1$ and corresponding $q\leq\frac{N}{m},\,U\in\mathcal{U}^{N},\,V\in(\mathcal{U}^{m})^{q}$ satisfying

0\leq\frac{1}{m}H(V)\leq h_{\mu_{x}}(\Phi)+\delta\leq h_{\mu_{x}}(\Phi)+\lambda(\mu_{x})-u_{i}+2\delta\leq P_{\mathcal{Y}}(\lambda)-u_{i}+2\delta.

This means $V\in R_{q}$ and especially

0\leq m(P_{\mathcal{Y}}(\lambda)-u_{i}+2\delta)

(3.7)

Applying lemma 3.4 to $(3.6)$ and $(3.7)$ , there exists a $q_{0}\in\mathbb{N}$ such that

\frac{1}{q}\log(\#R_{q})\leq m(P_{\mathcal{Y}}(\lambda)-u_{i}+3\delta)

for all $q\geq q_{0}$ . Fix $N\geq N_{0}:=q_{0}m$ , count all vectors $U$ which can appear in the above situation for any $x\in Z_{m,i}$ , and denote that number by $b_{N}$ , namely,

b_{N}=\#\mathop{\bigcup}\limits_{x\in Z_{m,i}}\{U\in\mathcal{U}^{N}:U\,satisfies\,(a)\,(b)\,and\,(c)\}.

Hence, as $q\geq q_{0}$ :

b_{N}\leq(\#\,\mathcal{U})^{m}(\#\,R_{q})\leq(\#\,\mathcal{U})^{m}\exp(qm(P_{\mathcal{Y}}(\lambda)-u_{i}+3\delta)).

This means, as $N=qm+r$ for some corresponding $0\leq r\leq m$ ,

\mathop{\limsup}\limits_{N\to\infty}\frac{1}{N}\log\,b_{N}\leq\mathop{\limsup}\limits_{N\to\infty}(m\,\log(\#\,\mathcal{U})+qm(P_{\mathcal{Y}}(\lambda)-u_{i}+3\delta))\leq P_{\mathcal{Y}}(\lambda)-u_{i}+3\delta.

As a result there exists some $N_{1}\geq N_{0}$ such that

b_{N}\leq\exp(N(P_{\mathcal{Y}}(\lambda)-u_{i}+4\delta))

(3.8)

for all $N\geq N_{1}$ .

For each $l\geq N_{1}$ we define the collection $\Gamma_{l}$ containing all $U\in\mathop{\bigcup}\limits_{N\geq l}\mathcal{U}^{N}$ which satisfying the properties of lemma 3.5 for some $x\in Z_{m,i}$ . This is a cover of $Z_{m,i}$ , denote as $\Gamma_{l}^{{}^{\prime}}$ . Note that by lemma 3.5 for each $U\in\Gamma_{l}$ one has

a_{N^{{}^{\prime}}}(x)\leq N^{{}^{\prime}}(u_{i}+3\delta)

(3.9)

where $N^{{}^{\prime}}=m(U)$ and $m(U)$ is the number of elements in $U$ . Hence we can estimate for $\alpha\in\mathbb{R}$ and $l\geq N_{1}$ :

	$\displaystyle M(Z_{m,i},\,a,\,\alpha,\,\delta,\,l)$	$\displaystyle\leq\sum_{(x,t)\in\Gamma_{l}^{\prime}}\exp(-\alpha t+t(u_{i}+3\delta))$
		$\displaystyle\leq\mathop{\sum}\limits_{N={l}}^{\infty}b_{N}\exp(-\alpha N+N(u_{i}+3\delta))$
		$\displaystyle\leq\sum\limits_{N={l}}^{\infty}(\exp(-\alpha+P_{\mathcal{Y}}(\lambda)+7\delta))^{N}$

Here the last step we used the estimate $(4.8)$ . Now for every $\alpha>P_{\mathcal{Y}}(\lambda)+7\delta$ , we obtain

\beta:=\exp(-\alpha+P_{\mathcal{Y}}(\lambda)+7\delta)<1

and hence

M(Z_{m,i},\,a,\,\alpha,\,\delta)\leq\mathop{\limsup}\limits_{l\to\infty}\sum\limits_{N={l}}^{\infty}\beta^{N}=0.

This means $P_{Z_{m,i}}(a,\delta)\leq P_{\mathcal{Y}}(\lambda)+7\delta$ for fixed $Z_{m,i}$ . To finish the proof we take the supreme for over all $m,\,i$ and apply that $A(a,\,\lambda,\,\mathcal{Y})=\mathop{\bigcup}\limits_{m\in\mathbb{N}}\mathop{\bigcup}\limits_{i\in\mathbb{N}}Z_{m,i}$ together with lemma 3.5:

P_{A(a,\,\lambda,\,\mathcal{Y})}(a,\delta)=\mathop{\sup}\limits_{m,i}P_{Z_{m,i}}(a,\delta)\leq P_{\mathcal{Y}}(\lambda)+7\delta.

Finally Letting $\delta\to 0$ results $P_{A(a,\,\lambda,\,\mathcal{Y})}(a)\leq P_{\mathcal{Y}}(\lambda)$
For the second statement, fix $\mu\in\mathcal{Y}$ . As $V(a,\,\lambda,\,\mathcal{Y})\subseteq A(a,\,\lambda,\,\mu)$ , one has by lemma 3.4 and $(3.2)$

P_{V(a,\,\lambda,\,\mathcal{Y})}(a)\leq P_{A(a,\,\lambda,\,\mathcal{Y})}(a)\leq h_{\mu}(\Phi)+\lambda(\mu).

Taking the infimum over all $\mu\in\mathcal{Y}$ , yields the result. ∎

Theorem 3.6: Let $(X,\Phi)$ be a compact metric space without fixed points. Fix $\mu\in\mathcal{\mathcal{E}}_{\Phi}(X)$ and let $(a_{t})_{t>0}$ be a Borel measurable potential on $(X,\Phi)$ . Suppose there exists a constant $b\in[-\infty,+\infty]$ and a Borel set $B\subseteq X$ satisfying $\mu(B)>0$ , such that

\mathop{\liminf}\limits_{t\to\infty}\frac{1}{t}a_{t}(x)\leq b

(3.10)

for each $x\in B$ . Then if $h_{\mu}(\Phi)+b$ is well-defined, one has

P_{B}(a)\geq h_{\mu}(\Phi)+b.

Proof.

we need following lemma.

Lemma 3.7[22]: Let $(X,\Phi)$ be a compact metric space without fixed points. For any $\mu\in\mathcal{\mathcal{E}}_{\Phi}(X)$ and define $h_{\mu}(x,\,\epsilon,\,t):=-\frac{1}{t}\log\mu(B_{t}(x,\epsilon))$ . Then one has

\mathop{\lim}\limits_{\epsilon\to o}\mathop{\liminf}\limits_{t\to\infty}h_{\mu}(x,\,\epsilon,\,t)=\mathop{lim}\limits_{\epsilon\to o}\mathop{\limsup}\limits_{t\to\infty}h_{\mu}(x,\,\epsilon,\,t)=h_{\mu}(\Phi)

(3.11)

for $\mu-$ almost $x\in X$ .
Let $G\subseteq B$ such that $(3.11)$ holds for each $x\in G$ . Note that $\mu(G)>0$ . Assume first $h_{\mu}(\Phi)+b$ is finite. Let $\epsilon>\epsilon^{\prime}>0$ and $\delta>0$ . Define the Borel sets

G^{\delta,\epsilon}:=\{x\in G:\mathop{\liminf}\limits_{t\to\infty}h_{\mu}(x,\,\epsilon,\,t)>h_{\mu}(\Phi)-\delta\}.

then $G^{\delta,\epsilon}\subseteq G^{\delta,\epsilon^{\prime}}$ and $G=\mathop{\cup}\limits_{\epsilon>0}G^{\delta,\epsilon}$ , hence: $0<\mu(G)=\mathop{\lim}\limits_{m\to\infty}\mu(G^{\delta,\frac{1}{m}})$ . This shows that there is an $\epsilon_{\delta}>0$ such that $0<\mu(G^{\delta,\epsilon_{\delta}}\leq\mu(G^{\delta,\epsilon})$ for all $0<\epsilon\leq\epsilon_{\delta}$ .For each $x\in G^{\delta,\epsilon_{\delta}}$ there exists a minimal $T(\delta,x)\in\mathbb{R}^{+}$ such that:

\exp(-t(h_{\mu}(\Phi)-\delta))\geq\mu(B_{t}(x,\epsilon_{\delta})),

(3.12)

\frac{1}{t}a_{t}(x)\geq b-\delta.

(3.13)

for all $t\geq T(\delta,x)$ . Define for each $T>0$ the Borel sets:

G^{\delta,\epsilon_{\delta},T}:=\{x\in G^{\delta,\epsilon_{\delta}}:x\,\,satisfies\,\,(3.12)\,\,and\,\,(3.13)\,\,for\,\,all\,\,t\geq T\}.

There exists an $M(\delta)\in\mathbb{R}^{+}$ such that $0<\mu(G^{\delta,\epsilon_{\delta},M(\delta)})\leq\mu(G^{\delta,\epsilon_{\delta},M})$ for all $M\geq M(\delta)$ . Now define $A_{\delta}:=G^{\delta,\epsilon_{\delta},M(\delta)}$ . If $\Gamma=\{(x_{l},t_{l})\}_{l\in L}$ is an cover of $A_{\delta}$ such that $t_{l}\geq M$ , then $\Gamma^{*}:=\{(x_{l},t_{l})\}_{l\in L^{\prime}}$ is also an cover of $A_{\delta}$ , where $L^{\prime}:=\{l\in L:B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}\neq\emptyset\}$ , and $B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}\subseteq B_{t_{l}}(x_{l},\epsilon)$ . Fix $0<\epsilon<\frac{\epsilon_{\delta}}{2}$ and $M\geq M(\delta)$ . Fix $y_{l}\in B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}$ and let $x\in B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}$ . Then $d(\varphi_{t}(y_{l}),\varphi_{t}(x))\leq d(\varphi_{t}(y_{l}),\varphi_{t}(x_{l}))+d(\varphi_{t}(x_{l}),\varphi_{t}(x))\leq 2\epsilon<\epsilon_{\delta}$ for all $0\leq t\leq t_{l}$ . Thus $B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}\subseteq B_{t_{l}}(y_{l},\epsilon_{\delta})$ for all $l\in L^{\prime}$ . Hence, as $y_{l}\in A_{\delta}$ for all $l\in L^{\prime}$

\exp(-t_{l}(h_{\mu}(\Phi)-\delta))\geq\mu(B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}).

(3.14)

In addition,one has by $(3.13)$

a_{t_{l}}(B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta})=\mathop{\sup}\limits_{x\in B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}}a_{t_{l}}(x)\geq t_{l}(b-\delta).

(3.15)

Hence,setting $\alpha_{\delta}:=h_{\mu}(\Phi)+b-2\delta$ , one has using $(3.14)$ and $(3.15)$

\exp(-t_{l}\alpha_{\delta}+a_{t_{l}}(B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta}))\geq\mu(B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta})

for all $l\in L^{\prime}$ . Thus, for each cover $\Gamma=\{(x_{l},t_{l})\}_{l\in L}$ of $A_{\delta}$ that $y_{l}\geq M$ , where $0<\delta<\frac{\epsilon_{\delta}}{2}$ and $M\geq M(\delta)$ . There is the estimate

	$\displaystyle\mathop{\sum}\limits_{(x,t)\in\Gamma}\exp(-\alpha_{\delta}t+a_{t}(B_{t}(x,\epsilon)\cap A_{\delta})$	$\displaystyle\geq\mathop{\sum}\limits_{(x,t)\in\Gamma^{*}}\exp(-\alpha_{\delta}t+a_{t}(B_{t}(x,\epsilon)\cap A_{\delta})$
		$\displaystyle\geq\mathop{\sum}\limits_{l\in L^{\prime}}\mu(B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta})$
		$\displaystyle\geq\mu(\mathop{\cup}\limits_{l\in L^{\prime}}B_{t_{l}}(x_{l},\epsilon)\cap A_{\delta})$
		$\displaystyle=\mu(A_{\delta}).$

This shows $M(A_{\delta},\,a,\,\epsilon,\,\alpha_{\delta})\geq\mu(A_{\delta})>0$ , and hence by lemma 2.5 and lemma 2.3

P_{B}(a,\epsilon)\geq P_{A_{\delta}}(a,\epsilon)\geq\alpha_{\delta}=h_{\mu}(\Phi)+b-2\delta

for all $0<\epsilon<\frac{\epsilon_{\delta}}{2}$ . Now letting $\epsilon\to 0$ and $\delta\to 0$ shows that $P_{B}(\Phi)\geq h_{\mu}(\Phi)+b$ , if $h_{\mu}(\Phi)$ is finite.
If $b=\infty$ , replace $(3.13)$ by $\frac{1}{t}a_{t}(x)\geq\frac{1}{\delta}$ and if $h_{\mu}(\Phi)=\infty$ , replace $(3.12)$ by $\exp(-\frac{t}{\delta})\geq\mu(B_{t}(x,\epsilon))$ and set $G^{\delta,\epsilon}:=\{x\in G:\mathop{\liminf}\limits_{t\to\infty}h_{\mu}(x,\epsilon,t)>\frac{1}{\delta}\}$ . Then the proof works in the same way. ∎

Next we proof the theorem 1.1.

Proof.

By theorem 3.3, $P_{A(a,\lambda,\mathcal{Y})}(a)\leq P_{\mathcal{Y}}(\lambda)$ . For each $\mu\in\mathcal{Y}$ , there is a Borel set $B_{\mu}\subseteq Z_{\mu}=\{x\in X:\delta_{t,x}\to\mu\}$ such that $\mu(B_{\mu})=1$ and $\mathop{\lim}\limits_{t\to\infty}\frac{1}{t}a_{t}(x)=\lambda(\mu)$ for all $x\in B_{\mu}$ . Hence $B_{\mu}\subseteq A(a,\lambda,\ \mu)\subseteq A(a,\lambda,\mathcal{Y})$ , and this shows by theorem 3.6 and lemma 2.6 that

h_{\mu}(\Phi)+\lambda(\mu)=P_{\{\mu\}}(\lambda)\leq P_{B_{\mu}}(a)\leq P_{A(a,\lambda,\mathcal{Y})}(a).

Taking the supremum on the left side yields the result. ∎

4 Topological pressure based on nested set strings

In this section we will introduce other definition of topology pressure for discontinuous potential $(a_{t})_{t>o}$ . Before that, let’s review the definition of topological pressure for continuous potentials on compact metric space $(X,\Phi)$ .

Let $\Phi$ be a continuous flow on $(X,d)$ and $a=(a_{t})_{t>0}$ be a family of continuous function $a_{t}$ : $X\to\mathbb{R}$ with tempered variation. Given $\epsilon>0$ , for each $Z\subseteq X$ and $\alpha\in\mathbb{R}$ , let

M(Z,a,\alpha,\epsilon)=\mathop{\lim}\limits_{T\to\ +\infty}\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}

(4.1)

with the infimum taken over all countable sets $\Gamma\subseteq X\times[T,+\infty)$ covering $Z$ . and let

\underline{M}(Z,a,\alpha,\epsilon)=\mathop{\underline{\lim}}\limits_{T\to\ +\infty}\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}

(4.2)

and

\overline{M}(Z,a,\alpha,\epsilon)=\mathop{\overline{\lim}}\limits_{T\to\ +\infty}\mathop{\inf}\limits_{\Gamma}\sum\limits_{\left({x,t}\right)\in\Gamma}{\exp\left({a\left({x,t,\varepsilon}\right)-\alpha t}\right)}

(4.3)

with the infimum taken over all countable sets $\Gamma\subseteq X\times\{T\}$ covering $Z$ . When $\alpha$ does from $-\infty$ to $+\infty$ , the above quantities $(4.1),\,(4.2)$ and $(4.3)$ jump from $+\infty$ to $0$ at unique values and so one can define

P_{Z}(a,\epsilon)=\inf\{\alpha\in\mathbb{R}:M(Z,a,\alpha,\epsilon)=0\},

\underline{P}_{Z}(a,\epsilon)=\inf\{\alpha\in\mathbb{R}:\underline{M}(Z,a,\alpha,\epsilon)=0\},

\overline{P}_{Z}(a,\epsilon)=\inf\{\alpha\in\mathbb{R}:\overline{M}(Z,a,\alpha,\epsilon)=0\}.

Theorem 4.1: For any family of continuous functions $a$ with tempered variation and any set $Z\subseteq X$ , the limits

P_{Z}(a)=\mathop{\lim}\limits_{\epsilon\to\ 0}P_{Z}(a,\epsilon),

(4.4)

\underline{P}_{Z}(a)=\mathop{\lim}\limits_{\epsilon\to\ 0}\underline{P}_{Z}(a,\epsilon)

and

\overline{P}_{Z}(a)=\mathop{\lim}\limits_{\epsilon\to\ 0}\overline{P}_{Z}(a,\epsilon)

exists.

Proof.

Take $\delta\in(0,\epsilon)$ and $\Gamma\subseteq X\times\mathbb{R}_{0}^{+}$ with $Z\subseteq\mathop{\bigcup}\limits_{(x,t)\in\Gamma}B_{t}(x,\delta)$ . Since $B_{t}(x,\delta)\subseteq B_{t}(x,\epsilon)$ , one has $Z\subseteq\mathop{\bigcup}\limits_{(x,t)\in\Gamma}B_{t}(x,\epsilon)$ . Let

\gamma(\epsilon)=\overline{\mathop{\lim}\limits_{t\to\infty}}\frac{\gamma_{t}(a,\epsilon)}{t}.

Given $\eta>0,\,Z\in B_{t}(x,\epsilon)$ , we have

a_{t}(y)-a_{t}(z)\leq|a_{t}(y)-a_{t}(z)|\leq\gamma_{t}(a,\epsilon)\leq t(\gamma(\epsilon)+\eta)

for any large $t$ . Thus,

a_{t}(y)\leq\mathop{\sup}\limits_{(x,y)\in B_{t}(x,\delta)}[a_{t}(z)+t(\gamma(\epsilon)+\eta)]\leq a(x,t,\delta)+t(\gamma(\epsilon)+\eta).

and

a(x,t,\epsilon)\leq a(x,t,\delta)+t(\gamma(\epsilon)+\eta)

for any large $t$ . Therefore,

M(Z,a,\alpha,\epsilon)\leq M(Z,a,\alpha-\gamma(\epsilon)-t,\delta),

and so

P_{z}(a,\epsilon)\leq\inf\{\alpha\in\mathbb{R}:M(Z,a,\alpha-\gamma(\epsilon)-t,\delta)=0\}=P_{z}(a,\delta)+\gamma(\epsilon)+\eta.

Letting $\delta\to 0$ we have

P_{Z}(a,\epsilon)-\gamma(\epsilon)-\eta\leq\mathop{\underline{\lim}}\limits_{\delta\to 0}P_{Z}(a,\delta).

Since $a$ is tempered variational we have $\gamma(\epsilon)\to 0$ when $\epsilon\to 0$ , which together with the arbitrariness of $\eta$ yields the inequality

\mathop{\overline{\lim}}\limits_{\delta\to 0}P_{Z}(a,\epsilon)\leq\mathop{\underline{\lim}}\limits_{\delta\to 0}P_{Z}(a,\delta).

This shows that $P_{Z}(a)$ is well-defined. The existence of the other two limits can be established in a similar way. ∎

The number $P_{Z}(a)$ is called non-additive topological pressure of the family $a$ on $Z$ , while $\underline{P}_{Z}(a)$ and $\overline{P}_{Z}(a)$ are called: respectively, the non-additive lower and upper capacity topological pressure of $a$ on $Z$ . Clearly:

{P_{Z}}(a)\leq\underline{P}_{Z}(a)\leq\overline{P}_{Z}(a).

If $Z_{1}\subseteq Z_{2}$ , then: $P_{Z_{1}}(a)\leq P_{Z_{2}}(a),\,\overline{P}_{Z_{1}}(a)\leq\overline{P}_{Z_{2}}(a),\,\underline{P}_{Z_{1}}(a)\leq\underline{P}_{Z_{2}}(a)$

Next we introduce the topological pressure of discontinuous potentials.

Let $X$ be a compact metric space, and $\Phi:X\to X$ a continuous flow. Consider any $\Phi-$ invariant subset $Z\subseteq X$ possessing a nested family of subsets $\{Z_{l}\}_{l\leq 1}$ . The $Z$ and the $Z_{l}$ are not required to be compact; the $Z_{l}$ are not required to be $\Phi$ -invariant. Consider a family of measurable functions $a=(a_{t})_{t>0}:X\to\mathbb{R}$ , we say that $a$ is continuous with respected to the family of subsets $\{Z_{l}\}$ if $a_{t}$ is continuous on the closure of each $Z_{l}$ for all $t>0$ . The potential function $a$ is not necessarily continuous on $Z$ . we define the topological pressure of $a$ on $Z$ , with respect to $\Phi$ as:

P_{Z}(a)=\mathop{\sup}\limits_{l\geq 1}P_{Z_{l}}(a),

where $P_{Z_{l}}(a)$ is the topological pressure of $a$ on $Z_{l}$ as defined above. We show that the topological pressure does not depend on the choice of the family of sets $\{Z_{l}\}$ .

Theorem 4.2: Assume that an $\Phi-$ invariant subset $Z\subseteq X$ has two nested families of subsets $\{A_{l}\}$ and $\{B_{l}\}$ which exhaust $Z$ , let $\{a_{t}:X\to\mathbb{R}\}_{t>0}$ be continuous with respect to both $\{A_{l}\}$ and $\{B_{l}\}$ . Then

P_{Z}(a)=\mathop{\sup}\limits_{l\geq 1}P_{A_{l}}(a)=\mathop{\sup}\limits_{l\geq 1}P_{B_{l}}(a).

Proof.

Set $P_{Z}^{\prime}(a)=\mathop{\sup}\limits_{l\geq 1}P_{A_{l}}(a)$ , and $P_{Z}^{\prime\prime}(a)=\mathop{\sup}\limits_{l\geq 1}P_{B_{l}}(a)$ . For every $\epsilon>0$ , there exists an $n$ such that $P_{(}A_{n})\geq P_{Z}^{\prime}(a)-\epsilon$ , as the $B_{l}$ exhaust $Z$ , we can write $A_{n}=\mathop{\bigcup}\limits_{m\geq 1}(A_{n}\cap B_{m})$ . As $a$ is continuous on the closure of $A_{n}$ and each $B_{m}$ ,we have

P_{A_{n}}(a)=\mathop{\sup}\limits_{m\geq 1}P_{A_{n}\cap B_{m}}(a)\leq\mathop{\sup}\limits_{l\geq 1}P_{B_{m}}(a)=P_{Z}^{\prime\prime}(a).

Thus $P_{Z}^{\prime\prime}(a)\geq P_{Z}^{\prime}(a)-\epsilon$ for every $\epsilon$ . Reversing the roles of $P_{Z}^{\prime}(a)$ and $P_{Z}^{\prime\prime}(a)$ gives the result. ∎

We continue to assume that $\Phi$ is a continuous flow on a compact metric space $X$ . Let $\mathcal{M}_{\Phi}(X)$ be the set of $\Phi-$ invariant probability measures on $X$ and $\mathcal{E}_{\Phi}(X)$ be the set of ergodic probability measures on $x$ . Given a Borel $\Phi-$ invariant set $Z\subseteq X$ , For convenience, let $\mathcal{M}_{Z}:=\mathcal{M}_{\Phi}(Z)$ . Given $x\in X$ and $t>0$ , consider the Borel $\Phi-$ invariant set

\mathcal{L}(Z)=\{x\in Z:V_{\Phi}(x)\cap M_{Z}\neq\emptyset\}

and

Z_{\mu}=\{x\in Z:V_{\Phi}(x)=\{\mu\}\}.

For each $\mu\in M_{\Phi}(X)$ , let $h_{\mu}(\Phi)=h_{\mu}(\varphi_{1})$ .

Now we proof theorem 1.2.

Proof.

We will divide the proof process into three steps.

Step 1:Some auxiliary content.

Take $x\in\mathcal{L}(Z)$ and $\mu\in V_{\Phi}(X)\cap M_{Z}$ , given $\delta>0$ , there exists an increasing sequence $\{t_{j}\}_{j\in\mathbb{N}}$ in $\mathbb{R}_{0}^{+}$ such that

\left|\frac{1}{t_{j}}\int_{0}^{t_{j}}b(\varphi_{s}(x))\,ds-\int_{Z}b\,d\mu\right|<\delta

for all $j\in\mathbb{N}$ . This implies that

\left|\frac{a_{t_{j}}(x)}{t_{j}}-\int_{Z}b\,d\mu\right|\leq\left|\frac{a_{t_{j}}(x)}{t_{j}}-\frac{1}{t_{j}}\int_{0}^{t_{j}}b(\varphi_{s}(x))\,ds\right|+\delta

(4.5)

Moreover, let $b_{t}=a_{t+s}-a_{t}\circ\varphi_{s}-\int_{o}^{s}(b\circ\varphi_{u})\,du$ .

For each $n\in\mathbb{N}$ with $t-ns\geq 0$ we have

	$\displaystyle a_{t}-\int_{0}^{t}(b\circ\varphi_{u})\,du$	$\displaystyle=a_{t}-a_{t-s}\circ\varphi_{s}-\int_{0}^{s}(b\circ\varphi_{u})\,du+a_{t-s}\circ\varphi_{s}-\int_{t}^{s}(b\circ\varphi_{u})\,du$
		$\displaystyle=b_{t-s}+[a_{t-s}-\int_{0}^{t-s}(b\circ\varphi_{u})\,du]\circ\varphi_{s}$
		$\displaystyle=b_{t-s}+b_{t-2s}\circ\varphi_{s}+[a_{t-2s}-\int_{0}^{t-2s}(b\circ\varphi_{u})\,du]\circ\varphi_{2s}.$

and so, proceeding inductively,

a_{t}-\int_{0}^{t}(b\circ\varphi_{u})\,du=\sum\limits_{k=0}^{n}{{b_{t-ks}}\circ{\varphi_{\left({k-1}\right)s}}+{a_{t-ns}}\circ\varphi{}_{ns}-\int_{0}^{t-ns}{\left({b\circ{\varphi_{u}}}\right)du}}.

(4.6)

Hence, it follows from $(4.5)$ that

	$\displaystyle\left\|\frac{a_{t_{j}}(x)}{t_{j}}-\int_{Z}b\,d\mu\right\|$	$\displaystyle\leq\left\|\frac{a_{t_{j}}(x)}{t_{j}}-\frac{1}{t_{j}}\int_{0}^{t_{j}}b(\varphi_{s}(x))\,ds\right\|+\delta$

		$\displaystyle\leq\frac{1}{t_{j}}\sum\limits_{k=1}^{n}\\|b_{t_{j}-ks}\\|_{\infty}+\frac{\\|a_{t_{j}-ns}\\|_{\infty}+(t_{j}-ns)\\|b\\|_{\infty}}{t_{j}}+\delta.$

Now let $n_{j}=[\frac{t_{j}}{s}]$ , then $t_{j}-n_{j}s\leq s$ and since $\mathop{\sup}\limits_{t\in[0,s]}\|a_{t}\|_{\infty}\leq+\infty$ , we have

\frac{\|a_{t_{j}-n_{j}s}\|_{\infty}+(t_{j}-n_{j}s)\|b\|_{\infty}}{t_{j}}<\delta

for any sufficiently large $j$ . Hence, by $(1.1)$ and since $\mathop{\sup}\limits_{t\in[0,T]}\|a_{t}\|_{\infty}\leq+\infty$ for all $T>0$ , taking $n=n_{j}$ , we obtain

\left|\frac{a_{t_{j}}(x)}{t_{j}}-\int_{Z}b\,d\mu\right|\leq\frac{1}{t_{j}}\sum\limits_{k=1}^{n_{j}}\|b_{t_{j}-ks}\|_{\infty}+2\delta\leq 3\delta

again for any sufficiently large $j$ .

Now let $E$ be a finite set. Given $k\in\mathbb{N}$ and $c=(c_{1},\,c_{2},\,...,\,c_{k})\in E^{k}$ , we define a probability measure $\mu$ on $E$ by

\mu_{c}(e)=\frac{1}{k}\#\{j:c_{j}=e\}

for $e\in E$ . moreover, let: $H(c)=-\mathop{\sum}\limits_{e\in E}\mu_{c}(e)\log\mu_{c}(e)$ .

Step 2: proof that $P_{\mathcal{L}(Z)}(a)\leq\sup\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\}$ .

As $\mathcal{L}(Z)=\mathop{\bigcup}\limits_{l\geq 1}(\mathcal{L}(Z))\cap Z_{l}$ , we have that

P_{\mathcal{L}(Z)}(a)=\mathop{\sup}\limits_{l\geq 1}P_{\mathcal{L}(Z))\cap Z_{l}}(a).

We show that for every $l\geq 1$ , $P_{\mathcal{L}(Z)}(a)\leq\sup\left\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\right\}$ .

let the $\lambda(\mu)$ in lemma 3.5 be $\int_{Z}b\,d\mu$ , then we obtain following lemma.

Lemma 4.4: Given $x\in\mathcal{L}(Z)\cap Z_{l}$ , and $\mu\in V_{\Phi}(X)\cap M_{Z}$ , let $\Gamma\subseteq X\times\{1\}$ be a finite cover of $X$ for the open cover $\mathcal{V}=\{V_{1},\,V_{2},\,...,\,V_{r}\}$ of $X$ , where $V_{j}=B_{1}(x_{j},\frac{\epsilon}{2})$ with $(x_{j},1)\in\Gamma$ , there exists $m,\,p\in\mathbb{N}$ with $p$ arbitrary large, and a sequence $U=V_{i_{1}}V_{i_{2}}...V_{i_{p}}$ such that
(a): $x\in\mathop{\cap}\limits_{r=1}^{p}\varphi_{-r+1}V_{i_{r}}$ and $a_{p}(x)\leq p(\int_{Z}b\,d\mu+3\delta)$ .
(b): there exists a subset $V\in(\mathcal{V}^{m})^{k}$ of $U$ of length $km\geq p-m$ satisfying the inequality

H(V)\leq m(h_{\mu}(\Phi)+\delta).

Given $m\in\mathbb{N}$ and $u\in\mathbb{R}$ , let $Z_{m,u}$ be the set of points $x\in\mathcal{L}(Z)\cap Z_{l}$ such that the two properties in lemma 4.4 hold for some $\mu\in V_{\Phi}(X)\cap M_{Z}$ with: $\int_{Z}b\,d\mu\in[u-\delta,\,u+\delta]$ . Moreover, let $n_{p}$ be the number of all sequences $U\in\mathcal{V}^{p}$ satisfying the same two properties for some $x\in Z_{m,u}$ . this means that

n_{p}=\#\mathop{\bigcup}\limits_{x\in Z_{m,u}}\{U\in\mathcal{V}^{p}:U\,satisfies\,(a),(b)\}.

Proceeding as lemma 5.3 in [3] one can show that

n_{p}\leq\exp[p(h_{\mu}(\Phi|_{Z})+2\delta)]=\exp[p(h_{\mu}(\Phi)+2\delta)]

for any sufficiently large $p$ (since $\mu(Z)=1$ ).

For each $\tau\in\mathbb{N}$ , the collection of all sequences $U\in\mathcal{V}^{p}$ satisfying the two properties in lemma 4.4 for some $x\in Z_{m,u}$ and $p\geq\tau$ cover the set $Z_{m,u}$ , therefore,

	$\displaystyle M(z_{m,u},a,\alpha,\epsilon)$	$\displaystyle=\mathop{\lim}\limits_{T\to+\infty}\mathop{\inf}\limits_{\Gamma}\mathop{\sum}\limits_{(x,t)\in\Gamma}\exp(a_{t}(x,t,\epsilon)-\alpha t)$
		$\displaystyle\leq\mathop{\overline{\lim}}\limits_{\tau\to+\infty}\sum\limits_{p=\tau}^{+\infty}n_{P}\exp[-\alpha p+p(\int_{Z}b\,d\mu+3\delta)+\gamma_{P}(a,\epsilon)]$
		$\displaystyle\leq\mathop{\overline{\lim}}\limits_{\tau\to+\infty}\sum\limits_{p=\tau}^{+\infty}\exp[p(h_{\mu}(\Phi)+\int_{Z}b\,d\mu+5\delta-\alpha+\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\epsilon)}{t})]$
		$\displaystyle\leq\mathop{\overline{\lim}}\limits_{\tau\to+\infty}\sum\limits_{p=\tau}^{+\infty}\beta^{p}$

where $\beta=\exp(-\alpha+c+5\delta+\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\epsilon)}{t})$ and $c=\sup\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\}$ .
thus, we obtain

M(z_{m,u},a,\alpha,\epsilon)\leq\mathop{\overline{\lim}}\limits_{\tau\to+\infty}\sum\limits_{p=\tau}^{+\infty}\beta^{p}.

(4.7)

For

\alpha>c+5\delta+\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\epsilon)}{t}.

(4.8)

we have $\beta<1$ and so it from $(4.7)$ that

M(z_{m,u},a,\alpha,\epsilon)\leq\mathop{\overline{\lim}}\limits_{\tau\to+\infty}\sum\limits_{p=\tau}^{+\infty}\beta^{p}=0\,\,\,and\,\,\alpha>P_{Z_{m,u}}(a,\epsilon).

(4.9)

Now take points $u_{1},u_{2},...,u_{r}$ , such that for each $u\in[min\,b,max\,a]$ there exists $j\in\{1,2,...,r\}$ with $|u-u_{j}|<\delta$ . Then: $\mathcal{L}(Z)\cap Z_{l}=\mathop{\bigcup}\limits_{m\in\mathbb{N}}\mathop{\bigcup}\limits_{i=1}^{r}Z_{m,u_{i}}$ and so it follows from $(4.9)$ and $(4.10)$ together with the lemma 2.6 that

	$\displaystyle c+5\delta+\mathop{\overline{\lim}}\limits_{\epsilon\to 0}\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\epsilon)}{t}$	$\displaystyle\geq\mathop{\overline{\lim}}\limits_{\epsilon\to 0}\mathop{\sup}\limits_{m,u_{i}}P_{Z_{m,u}}(a,\epsilon)$
		$\displaystyle=\mathop{\overline{\lim}}\limits_{\epsilon\to 0}P_{\mathcal{L}(Z)\cap Z_{l}}(a,\epsilon)$
		$\displaystyle=P_{\mathcal{L}(Z)\cap Z_{l}}(a).$

Since the arbitrariness of $\delta$ and $l$ and $a$ has tempered variation, we find that

P_{\mathcal{L}(Z)}(a)\leq c=\sup\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\}.

Step 3: proof $P_{\mathcal{L}(Z)}(a)\geq\sup\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu:\mu\in M_{Z}\}$ .

Lemma 4.5: For each $\mu\in M_{Z}$ there exists a $\Phi-$ invariant function $\overline{b}\in\mathbf{L}^{1}(x,\mu)$ such that

\mathop{\lim}\limits_{t\to\infty}\frac{a_{t}}{t}=\mathop{\lim}\limits_{t\to\infty}\frac{1}{t}\int_{0}^{t}(b\circ\varphi_{u})\,du=\overline{b}.

Proof.

it follows from $(4.6)$ that

\left|\frac{a_{t}(x)}{t}-\frac{1}{t}\int_{0}^{t}b(\varphi_{u}(x))\,du\right|\leq\frac{1}{t}\sum\limits_{k=1}^{n}\|b_{t-ks}\|_{\infty}+\frac{\|a_{t-ns}\|_{\infty}+(t-ns)\|b\|_{\infty}}{t}.

Let $n=[\frac{t}{s}]$ , then $t-ns\leq s$ and since $\sup\limits_{t\in[0,s]}\|a_{t}\|_{\infty}<+\infty$ , we have

\sup\limits_{t\geq 0}(\|a_{t-ns}\|_{\infty}+(t-ns)\|b\|_{\infty})<\infty.

Since $\sup\limits_{t\in[0,t]}\|a_{t}\|_{\infty}<+\infty$ for all $T>0$ , it follows from $(1.1)$ that $\frac{1}{t}(a_{t}-\int_{0}^{t}b\circ\varphi_{u}\,du)\to 0$ uniformly on $Z$ when $t\to\infty$ . On the other hand, since $b\in\mathbf{L}^{1}(x,\mu)$ , by Birkhoff’s ergodic theorem for flows there exists a $\Phi$ -invariant function $\overline{b}\in\mathbf{L}^{1}(x,\mu)$ such that

\lim\limits_{t\to\infty}\frac{1}{t}\int_{0}^{t}b\circ\varphi_{u}\,du=\overline{b}

$\mu-$ almost everywhere and in $\mathbf{L}^{1}(x,\mu)$ . This yields the desired statement. ∎

Lemma 4.6: For each ergodic measure $\mu\in M_{Z}$ , we have

P_{Z}(a)\geq h_{\mu}(\Phi)+\int_{Z}b\,d\mu.

Proof.

we will show that there exists an $l$ so that: $P_{Z_{l}}(a)\geq h_{\mu}(\Phi)+\int_{Z}b\,d\mu$ . Given $\epsilon>0$ , there exists $\delta\in(0,\epsilon)$ , a measurable partition $\xi=\{c_{1},c_{2},...,c_{m}\}$ of $X$ and an open cover $\mathcal{V}=\{v_{1},v_{2},...,v_{k}\}$ of $X$ for some $k\geq m$ such that:
(a): $diam\,c_{j}\leq\epsilon,\overline{v_{i}}\subseteq c_{i}$ and $\mu(c_{i}\backslash v_{i})<\delta^{2}$ for $i=1,2,...,m$ .
(b): The set $E=\bigcup\limits_{i=m+1}^{k}v_{i}$ has measure $\mu(E)<\delta^{2}$ .
Now we consider a measure $\nu$ in the ergodic decomposition of $\mu$ with respect to the time-1 map $\varphi_{1}$ . The later is described by a measure $\tau-$ in the space $\mathcal{M}^{\prime}$ of $\varphi_{1}$ -invariant probability measure that is concentrated on the ergodic measure (with respect to $\varphi_{1}$ ). Note that $\nu(E)<\delta$ for $\nu$ in a set $\mathcal{M}_{\delta}\subseteq\mathcal{M}^{\prime}$ of positive $\tau-$ measure such that $\tau(\mathcal{M}_{\delta})\to 1$ when $\delta\to 0$ since

\delta^{2}>\mu(E)=\int_{\mathcal{M}^{\prime}}\nu(E)\,d\tau(\nu)\geq\int_{\mathcal{M}^{\prime}|_{\mathcal{M}_{\delta}}}\nu(E)\,d\tau(\nu)\geq\delta\tau(\mathcal{M}^{\prime}\backslash\mathcal{M}_{\delta}).

For each $x\in Z$ and $n\in\mathbb{N}$ , let $t_{n}(x)$ be the number of integers $l\in[0,n)$ such that $\varphi_{1}^{l}(x)\in E$ . By Birkhoff’s ergodic theorem, since $\nu$ is ergodic for $\varphi_{1}$ we have

\lim\limits_{n\to\infty}\frac{t_{n}(x)}{n}=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{j=0}^{n-1}\chi_{E}(\varphi_{1}^{j}(x))=\int_{X}\,d\nu=\nu(E)

(4.10)

for $\nu-$ almost every $x\in X$ . On the other hand, by lemma 4.5 and Birkhoff’s ergodic theorem we have

\lim\limits_{t\to+\infty}\frac{a_{t}(x)}{t}=\lim\limits_{t\to+\infty}\frac{1}{t}\int_{o}^{t}(b\circ\varphi_{u})\,du=\int_{Z}b\,d\mu

(4.11)

for $\mu-$ almost every $x\in X$ . By $(4.10)$ and $(4.11)$ and Egrov’s theorem, there exists $\nu\in\mathcal{M}_{\delta}$ , $n_{1}\in\mathbb{N}$ and a measurable set $A_{1}\subseteq Z$ with $\nu(A_{1})\geq 1-\delta$ such that

\frac{t_{n}(x)}{n}<2\delta\,\,and\,\,\left|\frac{a_{n}(x)}{n}-\int_{Z}b\,d\mu\right|<\delta

(4.12)

for every $x\in A_{1}$ and $n>n_{1}$ .
Moreover, let $\xi_{n}=\bigcap\limits_{j=0}^{n}\varphi_{1}^{-j}(\xi|_{Z_{l}})$ where $\xi|_{Z_{l}}$ is the partition induced by $\xi$ on $Z_{l}$ . It from the Shannon-Mcmillian-Breiman theorem and Egorov’s theorem that there exists $n_{2}\in\mathbb{N}$ and a measurable set $A_{2}\subseteq Z$ with $\nu(A_{2})\geq 1-\delta$ such that

\nu(\xi_{n}(x))\geq\exp[(-h_{\nu}(\varphi_{1},\xi)+\delta)n]

(4.13)

for every $x\in A_{2}$ and $n>n_{2}$ . Take $p=max\{n_{1},n_{2}\}$ , and $A=A_{1}\cap A_{2}$ . Note that $\nu(A)\geq 1-2\delta$ . By construction,properties $(4.12)$ and $(4.13)$ holds for every $x\in A$ and $n>p$ .

Since the $\{Z_{i}\}$ are nested and exhaust $Z$ , we can choose $l$ so that $\nu(Z_{l})>1-\delta$ . We have that: $\nu(Z_{l})\cap A>1-3\delta$ . Now let $\Delta$ be a Lebesgue number of of the cover $\mathcal{V}$ and $\overline{\epsilon}>0$ such that $2\overline{\epsilon}<\Delta$ . Given $\alpha\in\mathbb{R}$ , take $q\geq p$ such that for each $n\geq q$ there exists a set $\Gamma\subseteq X\times[n,+\infty)$ covering $z$ with

\left|\sum\limits_{(x,t)\in\gamma}\exp(a(x,t.\overline{\epsilon})-\alpha t)-M(Z_{l},a,\alpha,\overline{\epsilon})\right|<\delta

(4.14)

Given $b\in\mathbb{N}$ , let $\Gamma_{b}=\{(x,b)\in\Gamma:B_{b}(x,\overline{\epsilon})\cap A\neq\emptyset\}$ and define $B_{b}=\bigcup\limits_{(x,t)\in\Gamma_{b}}B_{t}(x,\overline{\epsilon})$ . One can proceed as in the proof of lemma 2 in [17] to show that

\#\,\Gamma_{b}\geq\nu(B_{b}\cap A)\exp[h_{\nu}(\varphi_{1},\xi)l-(1+2\log\#\,\xi)l\delta]

(4.15)

for each $b\in\mathbb{N}$ . Indeed, let $\textit{L}_{b}$ be the number of elements $c$ of $\xi_{b}$ such that $c\cap B_{b}\cap A\neq\emptyset$ . It follows from $(4.13)$ that

\nu(B_{b}\cap A)\leq\sum\limits_{c\cap B_{b}\cap A\neq\emptyset}\nu(c)\leq\textit{L}_{b}\exp[(-h_{\nu}(\varphi_{1},\xi)+\delta)b].

(4.16)

Note that by eventually making $\overline{\epsilon}$ sufficiently small, for each $x\in Z$ there exists $i_{1},i_{2},...,i_{b}\in\{1,2,...,k\}$ such that $B_{b}(x,\overline{\epsilon})\subseteq V$ , where $V=\bigcap\limits_{j=1}^{b}\varphi_{1}^{-b+1}v_{i_{j}}$ (this follows readily from the uniform continuity of the map $(t,x)\mapsto\varphi_{t}(x)$ on the compact set $[0,1]\times X$ ). Given $(x,b)\in\Gamma_{b}$ , we have $B_{b}(x,\overline{(}\epsilon))\cap A_{1}\neq\emptyset$ . Hence, it follows from the first inequality in $(4.12)$ that the number $S_{(x,b)}$ of elements $c$ of the partition $\xi_{b}$ such that $c\cap B_{b}(x,\overline{\epsilon})\cap A\neq\emptyset$ satisfies $S_{(x,b)}\leq m^{2\delta b}=\exp(2\delta b\log m)$ . Therefore,

\textit{L}_{b}\leq\sum\limits_{(x,b)\in\Gamma_{b}}S_{(x,b)}\leq\#\,\Gamma_{b}\exp(2\delta b\log m).

(4.17)

Inequality $(4.15)$ follows readily from $(4.16)$ and $(4.17)$ .

Observe that by the second inequality in $(4.12)$ we have

\sup\limits_{B_{b}(x,\overline{\epsilon})}a_{b}\geq b(\int_{Z}b\,d\mu-\delta)-\gamma_{b}(a,\overline{\epsilon})

for all $b\geq q$ and $(x,b)\in\gamma_{b}(a,\overline{\epsilon})$ . Therefore,

		$\displaystyle\sum\limits_{(x,t)\in\Gamma}\exp(a(x,t,\overline{\epsilon})-\alpha t)\geq\sum\limits_{b=q}^{+\infty}\sum\limits_{(x,t)\in\Gamma_{b}}\exp(\sup\limits_{B_{b}(x,\overline{\epsilon})}a_{b}-\alpha b)$
		$\displaystyle\geq\sum\limits_{b=q}^{+\infty}\#\,\Gamma_{b}\exp[(-\alpha+\int_{Z}b\,d\mu-\delta)b-\gamma_{b}(a,\overline{\epsilon})]$
		$\displaystyle\geq\sum\limits_{b=q}^{+\infty}\nu(B_{b}\cap A)\exp[(h_{\nu}(\varphi_{1},\xi)+\int_{Z}b\,d\mu-\frac{\gamma_{b}(a,\overline{\epsilon})}{b}-\alpha)b-2(1+\log\#\,\xi)b\delta].$

Without loss of generality one can also assume that $\delta$ is sufficiently small such that

\alpha<h_{\nu}(\varphi_{1},\xi)+\int_{Z}b\,d\mu-\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\overline{\epsilon})}{t}-2(1+\log\#\,\xi)\delta-\delta,

then

\mathop{\sum}\limits_{(x,t)\in\Gamma}\exp(a(x,t,\overline{\epsilon})-\alpha t)\geq\mathop{\sum}\limits_{b=t}^{+\infty}\nu(B_{b}\cap A)\geq 1-2\delta,

and so it follows from $(4.14)$ that $M(Z_{l},a,\alpha,\overline{\epsilon})\geq 1-3\delta>0$ . Therefore, $P_{Z_{l}}(a,\epsilon)\geq\alpha$ , which implies that

P_{Z_{l}}(a,\epsilon)\geq h_{\nu}(\varphi_{1},\xi)+\int_{Z}b\,d\mu-\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\overline{\epsilon})}{t}.

Finally, we consider measurable partition $\xi_{b}$ and open covers $\mathcal{V}_{b}$ as before with $\epsilon=\frac{1}{b}$ . For each $b$ take $\overline{\epsilon_{b}}>0$ such that $2\overline{\epsilon_{b}}<\frac{1}{b}$ is a Lebesgue number of the cover $\nu_{b}$ . Since diam $\xi_{b}\to 0$ when $b\to+\infty$ , it follows that

\mathop{lim}\limits_{b\to+\infty}h_{\nu}(\varphi_{1},\xi_{b})=h_{\nu}(\varphi_{1}).

Moreover, since the family $a$ has tempered variation property, we obtain

	$\displaystyle P_{Z_{l}}(a)=\mathop{\lim}\limits_{b\to+\infty}P_{Z_{l}}(a,\overline{\epsilon_{b}})$	$\displaystyle\geq\mathop{\lim}\limits_{b\to+\infty}h_{\nu}(\varphi_{1},\xi_{b})+\int_{Z}b\,d\mu-\mathop{\lim}\limits_{b\to+\infty}\mathop{\overline{\lim}}\limits_{t\to+\infty}\frac{\gamma_{t}(a,\overline{\epsilon_{b}})}{t}$
		$\displaystyle=h_{\nu}(\varphi_{1})+\int_{Z}b\,d\mu.$

Integrating with respect to $\nu$ gives

P_{Z_{l}}(a)\geq\int_{\mathcal{M}_{\delta}}h_{\nu}(\varphi_{1})\,d\tau(\nu)+\int_{Z}b\,d\nu.

and letting $\delta\to 0$ yields the inequality:

P_{Z_{l}}(a)\geq\int_{\mathcal{M}^{\prime}}h_{\nu}(\varphi_{1})\,d\tau(\nu)+\int_{Z}b\,d\nu=h_{\nu}(\Phi)+\int_{Z}b\,d\nu.

This completes the proof of the lemma. ∎

When $\mu\in\mathcal{M}_{Z}$ is ergodic, $Z_{\mu}$ is a nonempty $\Phi-$ invariant subset of $\mathcal{L(Z)}$ with $\mu(Z_{\mu})=1$ . Hence, it follows from lemma 4.6 that

P_{\mathcal{L}(Z)}(a)\geq h_{\mu}(\Phi)+\int_{Z_{\mu}}b\,d\mu=h_{\mu}(\Phi)+\int_{Z}b\,d\mu.

When $\mu\in\mathcal{M}_{Z}$ is arbitrary, one can decompose $X$ into ergodic components and the previous argument shows that

P_{\mathcal{L}(Z)}(a)\geq\mathop{\sup}\limits_{\mu\in\mathcal{M}_{Z}}\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu\}.

This completes the proof of the theorem. ∎

It follows from theorem 4.3 that if $V_{\Phi}(X)\cap M_{Z}\neq\emptyset$ for each $x\in Z$ , and so in particular if $Z$ is compact and $\Phi-$ invariant, then

P_{Z}(a)=\mathop{\sup}\limits_{\mu\in\mathcal{M}_{Z}}\{h_{\mu}(\Phi)+\int_{Z}b\,d\mu\}.

References

[1] L. Backes and F. Rodrigues, Topological pressure for discontinuous semiflows and a variational principle for impulsive dynamical systems, Topol. Methods Nonlinear Anal. 59:303–330, 2022
[2] J. Barral and D. Feng, Weighted thermodynamic formalism on subshifts and applications, Asian J. Math. 16: 319–352, 2012
[3] L. Barreira, Thermodynamic formalism and applications to dimension theory, Progress in Mathematics. 294:xii+297, 2011
[4] L. Barreira and C. Holanda, Nonadditive topological pressure for flows, Nonlinearity. 33: 3370–3394, 2020
[5] L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergodic Theory Dynam. Systems. 16: 871–927, 1996
[6] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics. 470:viii+75, 2008
[7] Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst. 20:639–657, 2008
[8] Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Ergodic Theory Dynam. Systems. 33:831–850, 2013
[9] W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function, Discrete Contin. Dyn. Syst. 32:487–497, 2012
[10] K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A. 21: L737–L742, 1988
[11] D. Feng, and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Comm. Math. Phys. 297: 1–43, 2010
[12] D. Feng, and W. Huang, Variational principle for weighted topological pressure, J. Math. Pures Appl. (9). 106:411–452, 2016
[13] L. He, J. Lv and L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sin. (Engl. Ser.). 20:709–718, 2004
[14] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B. 30:499–510, 1968
[15] X. Ma, E. Chen and Y. Pei, Thermodynamic formalism for subadditive sequence of discontinuous functions, Anal. Theory Appl. 26: 174–185, 2010
[16] A. Mummert, A variational principle for discontinuous potentials, Ergodic Theory Dynam. Systems. 27: 583–594, 2007
[17] Y. B. Pesin and B. S. Pitskel , Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18: 50–63, 96, 1984
[18] Y. B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics: Contemporary views and applications. xii+304, 1997
[19] M. Rauch, Variational principles for the topological pressure of measurable potentials, Discrete Contin. Dyn. Syst. Ser. S. 10: 367–394, 2017
[20] M. Rauch, Variational principles for topological pressure, issertation, Jena, Friedrich-Schiller-University Jena, 2018
[21] D. Ruelle, Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187: 237–251, 1973
[22] Y. Wang, E. Chen, Z. Lin, and T. Wu, Bowen entropy of sets of generic points for fixed-point free flows, J. Differ. Equations. 269:9846–9867, 2020
[23] Y. Zhao and Y. Cao, On the topological pressure of random bundle transformations in sub-additive case, J. Math. Anal. Appl. 342:715–725, 2008
[24] Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear Anal. 70:2237–2247, 2009

Topological Pressure of Discontinuous Potentials and Variational Principle for Flows11footnotemark: 1

Abstract

keywords:

MSC:

1 Introduction

2 Prelimilaries

Proof.

Proof.

Proof.

3 Variational principle

Proof.

Proof.

Proof.

Proof.

Proof.

4 Topological pressure based on nested set strings

Proof.

Proof.

Proof.

Proof.

Proof.

References

Topological Pressure of Discontinuous Potentials and Variational Principle for Flows¹¹footnotemark: 1