Yangyang CHEN（陳陽洋） Yun ZHAO（趙云）

Department of Mathematics，Soochow University，Suzhou 215006，China

Wen-Chiao CHENG（鄭文巧）?

Department of Applied Mathematics，Chinese Culture University Yangmingshan，Taipei 11114，China

SUB-ADDITIVE PRESSURE ON A BOREL SET?

Yangyang CHEN（陳陽洋） Yun ZHAO（趙云）

Department of Mathematics，Soochow University，Suzhou 215006，China

E-mail：oufei861155909@163.com；zhaoyun@suda.edu.cn

Wen-Chiao CHENG（鄭文巧）?

Department of Applied Mathematics，Chinese Culture University Yangmingshan，Taipei 11114，China

E-mail：zwq2@faculty.pccu.edu.tw

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials.This study presents a variational inequality for sub-additive topological conditional pressure on a closed subset，which is the other form of the variational principle for the sub-additive topological pressure presented by Cao，F(xiàn)eng，and Huang in［9］.Moreover，under some additional assumptions，this result can be generalized to the non-compact case.

sub-additive potentials；topological pressure；conditional entropy；variational inequality

2010 MR Subject Classification 37A30；37L40

1 Introduction

In this study，（X，T）denotes a topological dynamical system（TDS for short）in the sense that T:X→X is a continuous map on a compact metric space X with the metric d.

Entropy is a critical factor in dynamical systems and the study of ergodic theory.Metric entropy was defined by Kolomogorov and Sinai from Shannon's information theory in 1959，while Adler，Konheim and McAndrew［1］used the concept of open covers to introduce topological entropy in 1965，and Bowen［7］later defined topological entropy on a metric space by using generating and separating sets，respectively.These three different definitions of topological entropy was proved equivalent provided the compactness of the space，see［23］for the details. Metric or measure-theoretic entropy measures the maximal loss of information in the iteration of finite partitions in a measure-preserving transformation，whereas topological entropy measures the maximal exponential growth rate of orbits.However，these concepts are not isolated. These two notions are connected by a famous variational principle stating that the topologicalentropy is the supremum of the metric entropies for all invariant probability measures of a given topological system and is described in the following:

where h（T）denotes the topological entropy for T and hμ（T）is the metric entropy.The term M（X，T）denotes all the T-invariant Borel probability measures on X.

As a natural generalization of topological entropy，topological pressure is a quantity which belongs to one of the concepts in the thermodynamic formalism.Topological pressure contains information about the dynamics of the system，which can be extracted by varying the potential energy function.It is well known that the topological pressure with a potential function plays a fundamental role in the study of the Hausdorff dimension of repellers and the hyperbolic set.Related studies include［4，7，13，14，17，19，21-23］.The relationship among topological pressure，potential function and metric entropy are formulated by a variational principle，which states that if T:X→X is a continuous map，f:X→R is a continuous function，and P（T，f）denotes the topological pressure of T with respect to f，then

In［11］，F(xiàn)alconer considered the topological pressure for sub-additive potentials，and proved the variational principle for sub-additive topological pressure under some Lipschitz conditions and bounded distortion assumptions on the sub-additive potentials.For an arbitrary subset and any non-additive potentials on the compact metric space，Barreira［3］defined non-additive topological pressure and also showed the variational principle under particular convergence conditions on the potentials.Recently，Cao，F(xiàn)eng and Huang［9］generalized the results of Rulle and Walters to the sub-additive potentials on the compact metric space.It deserves to mention that they didn't put any other assumptions on the potentials and their results had relevant applications in dimension theory，see［2］.Feng and Huang in［12］introduced the notion of asymptotically additive/subadditive potentials，and proved the variational principle for topological pressure with these two potential.The main goal of demonstration is to study the multifractal analysis of these two potentials.Note that Barreira［5，6］and Mummert［18］dealt with variational principle for topological pressure with almost additive potentials.Huang，Yi［16］and Zhang［24］considered the variational principle for the local topological pressure.For more information，see［25］and［26］for variational principle of conditional topological pressure and coset pressure with sub-additive potentials，respectively.

Furthermore，Li，Chen and Cheng［15］generalized the results of Walters regarding the variational principle for topological pressure and their results can be stated precisely as follows. Let T:X → X be a continuous map，and f:X → R be a continuous function.Given a T-invariant closed subset G of X，i.e.，T-1G=G，and consider the pressure PG（T，f）of T on G with respect to f，then the following variational inequality is obtained

wheredenotes the partition｛G，XG｝，and hμ（T|）denotes the conditional entropy of T with respect toμ，see［23］for the details.This formula can be used to generalize the results of Cheng［10］regarding topological entropy.Inspired by［15］and［9］，this paper considers thetopological pressure PG（T，F(xiàn)）of a sub-additive potential F on a T-invariant closed subset G of X.Inequality（1.1）is also generalized to the sub-additive case.Moreover，relying on more techniques of the proof of variational principle for sub-additive topological pressure built by Cao，F(xiàn)eng and Huang［9］leads to the following theorem.

Theorem 1.1 Assume T:X→X is a continuous map of the compact metric space X，and F=｛gn｝∞n=1is a sub-additive potential on X.If G is a T-invariant closed subset of X，then

It is of interest to note that Theorem 1.1 is also true if the potential F is asymptotically sub-additive by using Feng and Huang's result［12］.Roughly speaking，an asymptotically subadditive potential is a family of continuous functions which can be approximated by sub-additive potentials，see［12］for the precise definition.

The purpose of this paper is to demonstrate Theorem 1.1 and is organized as follows.Using the similar notion of topological pressure，Section 2 establishes conditional entropy，conditional sub-additive topological pressure focused on the T-invariant Borel set and states the variational principle revealed by［9］.Section 3 shows how to estimate the supremum of a special type of conditional entropy with a sub-additive potential by calculating the topological conditional pressure.The resulting variational inequality is based on the results presented by Cao，F(xiàn)eng and Huang'work［9］.Finally，Section 4 shows that the variational inequality is still true for the non-compact case if some assumptions are added.

2 Notations and Related Propositions

A reasonable measure-theoretic or topological entropy should be a measure of the uncertainty of the system.These entropies should be invariant under measurable or topological changes of coordinates，respectively.Topological pressure，which is an extension of topological entropy，is a rich source of dynamical systems.This value roughly measures the orbit complexity of the iterated map on the potential functions.This section first reviews the concept of conditional metric entropy on a probability space and sub-additive potentials.Then the topological pressure concentrated on a Borel set is given in order to present the variational inequality among these invariant values.

The conditional entropy of an ergodic theory is usually defined as follows.Let（X，B，μ）be a probability space.The terms α=｛A1，A2，···，Am｝and β=｛B1，B2，···，Bn｝denote the finite partitions of X.The conditional entropy of α with respect to β is defined as

and is called the metric entropy of T with respect to α.The metric entropy of T is given by the value，where α is any finite partition of X.

The following discussion is based on the assumption that T:X→X is a continuous map of the compact metric space X，and G is a T-invariant closed subset of X.Letdenote the partition｛G，XG｝，where XG is the complement of G.Let M（X，T）denote the set of all T-invariant Borel probability measures on（X，B（X）），where B（X）represents the Borel σ-algebra of X.Given a T-invariant measureμ∈M（X，T），define

Remark 2.1 If G=X orμ（G）=1，then actually hμ（T）=hμ（T|）.

This conditional metric entropy hμ（T|）has some basic properties，including power rule，product rule and affinity and is stated as follows.The main idea of these proofs come from Li，Chen and Cheng's approximations，see［10，15］.A simple example is the sub-shift of finite type on symbolic dynamics with two-sided shift.

Lemma 2.2（see［10］） The conditional entropy hμ（T|）is a measure-theoretic conjugacy invariant.

Lemma 2.3（see［10］） For each positive integer r，hμ（Tr|）=r·hμ（T|）.

Lemma 2.4（see［15］）Let（X1，B1，m1）and（X2，B2，m2）be probability spaces and let T1:X1→X1，T2:X2→X2be measure-preserving maps.Then

whereμ=m1×m2，Giis a Ti-invariant subspace of Xi，i=1，2.

Lemma 2.5（see［15］）Let T be a measure-preserving map of the probability space（X，B，μ）and let G be a T-invariant subset of X.Then the mapμ→ hμ（T|，α）is affine where α is any finite partition of X.Hence，so is the mapμ→hμ（T|）.In other words，for all 0＜λ＜1，and whenμ1，μ2are both invariant measures，we have

For convenience，notations and definitions of the sub-additive potential is adopted by Cao，F(xiàn)eng and Huang's recent work，［9］and can be described in the following.

For a sub-additive potentialandμ∈M（X，T），define

and

Then，define

Due to PG（T，F(xiàn)，?）is a decreasing function of ?，the limiting process ?→ 0 is reasonable. PG（T，F(xiàn)）is said to be the topological pressure of T restricted on G with respect to the subadditive potential F.

Remark 2.8 If G=X，for simplicity，we just note PX（T，F(xiàn)）as P（T，F(xiàn)）.Set PG（T，F(xiàn)）= -∞by convention when G=?.If F=｛gn｝n≥1is additive，i.e.，there exists a ?:X →R such that gn（x）=?+??T+···+??Tn-1，then the above definition reduces to the additive topological pressure.

The theory about the topological pressure，variational principle and equilibrium states plays a fundamental role in statistical mechanics，ergodic theory and dynamical systems.The variational principle of the sub-additive topological pressure was shown by Cao，F(xiàn)eng and Huang's work on the whole space X in［9］.

Theorem 2.9（see［9，Theorem 1.1］）Let F=｛gn｝∞n=1be a sub-additive potential on a TDS（X，T）.This leads to

3 Main Proof

For a continuous function T of a compact metric space（X，d），the well-known variational principle of topological pressure shows the relationships among pressure，entropy invariants，and potential energy from probabilistic and topological perspectives.This section provides a detailed proof of the variational principle for the topological conditional pressure with a subadditive potential.

Unavoidable，we will face the special case that F?（μ）=-∞，?μ∈M（X，T），while it doesn't exist in the additive case，since if F=｛gn｝n≥1is additive，that means the existence of a continuous function ?:X → R such that gn（x）=?+??T+···+??Tn-1，then |F?（μ）|=|R?dμ|≤||?||∞is always finite.It is worthwhile to point out that Li，Chen and Cheng used the standard technique of variational principle developed by P.Walters to prove the first inequality in their Theorem 1.1，see［15］.Following previous research by Cao，F(xiàn)eng and Huang's work［9］，this study derives the proof of the presented variational inequality by using a series of modifications.

Proof of Theorem 1.1 We divide the proof into three small steps.

Part I If?μ∈M（X，T），F(xiàn)?（μ）=-∞，i.e.，P（T，F(xiàn)）=-∞.For G is a closed T-invariant subset of X，it is not hard to show thatThus，the topological pressure on the compact subsetis well-defined.Since M（G，T）?M（X，T），andwe haveby an application of Theorem 2.9 on these subsystems（G，T|G）andDefine sup?=-∞，then Theorem 1.1 still holds.Therefore， we always assume that there existsμ∈M（X，T）such that F?（μ）/=-∞in the following two parts.

Part II This part will prove

If PG（T，F(xiàn)）=-∞，then the above inequality holds obviously.If PG（T，F(xiàn)）/=-∞，Theorem 2.9 on the subsystem（G，T|G）ensures that there existsμ∈M（G，T）?M（X，T）such that.By Theorem 2.9，one has

where the second equality follows from Remark 2.1，sinceμ（G）=1，?μ∈M（G，T）.

Part III Finally，we show that for eachμ∈M（X，T）satisfying F?（μ）/=-∞，one has

Given aμ∈M（X，T）with F?（μ）/=-∞.Ifμ（G）=1，by Remark 2.1 we have hμ（T|）= hμ（T）.Using Theorem 2.9，we have

Using a similar argument，ifμ（G）=0 then we have

Hence，in the following we assume that 0＜μ（G）＜1.Let η be a finite partition of the space X，then

where Gc=XG，μGandμGcdenotes the conditional probability measures induced byμon G and Gc，respectively.Dividing by n on both sides of the above equality and let n→∞，one has

Moreover，

On the other hand，we first assume that hμG（T）and hμG（T）are finite.Then for any ?＞0 there exists finite partitions η，ξ of X such that

Set ζ=η∨ξ，using the property of refinement of entropy，which implies

Multiply the above two inequalities byμ（G）andμ（Gc），respectively，together with（3.1）one has

Since ?＞0 is arbitrary，we obtain

Finally，if hμG（T）=+∞or hμGc（T）=+∞，modifying slightly the above arguments we have hμ（T|）=+∞.The above arguments show that

Furthermore，

and F?（μ）/=-∞，which implies

Note thatμG∈M（G，T），μGc∈M（Gc，T）?M（Gc，T），the application of Theorem 2.9 on subsystems（G，T）and（Gc，T）gives

and

Multiply the above two inequalities byμ（G）andμ（Gc），respectively，adding them，one easily obtains

as desired.□

If this closed T-invariant subset G represents the whole space X，then XG=?.This allows Theorem 2.1，which is the variational principle presented by Cao，F(xiàn)eng and Huang［9］.

4 Non-Compact Case

This section will consider the non-compact case，i.e.，the T-invariant subset G of X can be not closed.Different definitions of topological pressure can be used to obtain an even more general variational inequality.First，define the topological pressure of any subset.The following notations are adopted from Barreira［3］.

Let T:X → X be a continuous map on the compact metric space X.If U is a finite open cover of X and n≥1，denote by Wn（U）the collection of strings U=U1U2···Unwith U1，U2，···，Un∈U.For each U∈Wn（U），call the integer m（U）=n the length of U，and define the open set

This section is based on the assumption that the following property holds:

Given any subset Z?X，sub-additive potentials F=｛gn｝n≥1and open cover U of X，for α∈R，define

the reasonablenessof the above definition can be ensured by the structure theory of Carath′eodory（see［20］）.Moreover，the topological pressure of T restricted on Z with respect to F is defined by

The following paragraphs present some lemmas.

Lemma 4.1（see［3，Corollary1.8］）Let F=｛gn｝n≥1be a sub-additive potential on a TDS（X，T），and G a T-invariant Borel subset of X.If there exists a continuous function ψ:X→R，such that gn+1-gn?T→ψ uniformly on G as n→∞，and V（x）∩M（G）/=?，?x∈G，then

where M（G）denotes all the invariant measures that satisfyμ（G）=1 and T|G denotes the map T restricted on G.

Remark 4.2（see［3，Lemma 2］）In the above lemma，without the condition V（x）∩M（G）/=?，?x∈G，we can obtain

Lemma 4.3（see［15，Lemma 4.1］）Let（X，T）be a TDS，and G a T-invariant Borel subset.For eachμ∈M（G），we have

We still need the following lemma.

Lemma 4.4 Let F=｛gn｝n≥1be a sub-additive potential on a TDS（X，T），and G a T-invariant Borel subset of X.If there exists a continuous function ψ:X → R，such that gn+1-gn?T→ψ uniformly on G as n→∞，then for eachμ∈M（G）one has

Since gn+1-gn?T→ψ uniformly on G as n→∞，set an=G（gn+1-gn?T）dμ.In this case，an→RGψdμas n→∞.Letting n→∞on both sides of the above formula，then the desired result is obtained.

Theorem 4.5 Let F=｛gn｝n≥1be a sub-additive potential on X，and G a T-invariant Borel subset of X.If there exists a continuous function ψ:X→R，such that gn+1-gn?T→ψ uniformly on X as n→∞，in addition，we assume V（x）∩M（G）/=?，?x∈G，then

Proof Using Lemmas 4.1，4.3 and 4.4，we have

On the other hand，for eachμ∈M（X，T），in Section 3，we have established

and

whereμGandμGcdenotes the conditional probability induced byμon G and Gc，respectively. Obviously，μG∈M（G），μGc∈M（Gc），By Remark 4.2，we have

and

Multiply the above two inequalities byμGandμGcrespectively，and add them，we have

It follows from Lemma 4.4 that

Hence，

as desired.

At last，the following question is proposed:

Question Given a TDS（X，T）and a sub-additive potential F=｛gn｝n≥1on X.Let G be a T-invariant Borel subset of X.If we only assume that V（x）∩M（G）/=?，?x∈G，does the following variational inequality still hold？

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?Received July 23，2013；revised October 24，2014.For this research，Chen was partially supported by National University Student Innovation Program（111028508）.Cheng was supported by NSC Grant NSC 101-2115-M-034-001.Zhao was partially supported by NSFC（11371271）.This work was partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

?Corresponding author:Wen-Chiao CHENG.