# Differentiability Properties of the Pre-Image Pressure

Authors: Kesong Yan; Fanping Zeng; Gengrong Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We study the differentiability properties of the pre-image pressure. For a TDS(X,T) with finite topological pre-image entropy, we prove the pre-image pressure function Ppre(T,•) is Gateaux differentiable at f∈C(X,ℝ) if and only if Ppre(T,•) has a unique tangent functional at f. Also, we obtain some equivalent conditions for Ppre(T,•) to be Fréchet differentiable at f.


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## Body

## 1. Introduction

By a topological dynamical system (for short TDS), we mean a pair(X,T) where X is a compact metric space and T:X→X is a continuous surjection from X to itself. Entropies are fundamental to our current understanding of dynamical systems. The classical measure-theoretic entropy for an invariant measure and the topological entropy were introduced in [1, 2], respectively, and the classical variational principle was completed in [3, 4]. Topological entropy measures the maximal exponential growth rate of orbits for arbitrary topological dynamical systems, and measure-theoretic entropy measures the maximal loss of information for the iteration of finite partitions in a measure-preserving transformation.Topological pressure is a generalization of topological entropy for a dynamical system. The notion was first introduced by Ruelle [5] in 1973 for an expansive dynamical system and later by Walters [6] for the general case. The theory related to the topological pressure, variational principle, and equilibrium states plays a fundamental role in statistical mechanics, ergodic theory, and dynamical systems (see, e.g., the books [7–12]). Since the works of Bowen [13] and Ruelle [14], the topological pressure has become a basic tool in the dimension theory related to dynamical systems. One of the basic questions of physical interest in that of differentiability of the pressure. The differentiability of the pressure was considered by many people (see, e.g., [15–18]).Recently, the pre-image structure of maps has become deeply characterized via entropies. In several papers (see [19–24]), some important pre-image entropy invariants of dynamical systems have been introduced and their relationships with topological entropy have been established. In a certain sense, these new invariants give a quantitative estimate of how “noninvertible” a system is. In [25], we defined the topological pre-image pressure of topological dynamical systems, which is a generalization of the Cheng-Newhouse pre-image entropy (see [19]), and proved a variational principle for it. We gave some applications of the pre-image pressure to equilibrium states (see [25, 26]). Under the assumption that hpre(T)<∞ and the metric pre-image entropy function h{pre,•}(T) is upper semicontinuous, we obtained a way to describe a kind of continuous dependence of equilibrium states. Also, we proved that the set of all continuous functions with unique equilibrium states is a dense Gδ-set of C(X,ℝ), and for any finite collection of ergodic measures, we can find some continuous function such that its set of equilibrium states contains the given set (see [26]).The purpose of this paper is to study the differentiability properties of the pre-image pressure of the TDS(X,T) with finite topological pre-image entropy. In Section 2, we concentrate on reviewing some basic definitions and give some basic properties of tangent functionals to the pre-image pressure.In Section3, the Gateaux differentiability of the pre-image pressure is discussed. We show that the pre-image pressure function Ppre(T,•) is Gateaux differentiable at f∈C(X,ℝ) if and only if it has a unique tangent functional at f.In Section4, we discuss the Fréchet differentiability of the pre-image pressure. We obtain some equivalent conditions for Ppre(T,•) to be Fréchet differentiable at f. Also, we show that the pre-image function Ppre(T,•) is Fréchet differentiable if and only if (X,T) is uniquely ergodic, and hence the pre-pressure is linear.


## 2. Preliminaries

Throughout the paper, let(X,T) be a TDS with finite topological pre-image entropy hpre(T) (see [19] for definition). In this section, we will recall some basic definitions and give some useful properties.Let(X,T) be a TDS and let ℬ(X) be the collection of all Borel subsets of X. Recall that a cover of X is a family of Borel subsets of X whose union is X. An open cover is one that consists of open sets. A partition of X is a cover of X consisting of pairwise disjoint sets. We denote the set of finite covers, finite open covers, and finite partition of X by 𝒞X, 𝒞Xo, and 𝒫X, respectively. Given two covers 𝒰,𝒱, 𝒰 is said to be finer than 𝒱 (denoted by 𝒰≽𝒱) if each element of 𝒰 is contained in some element of 𝒱. We set 𝒰∨𝒱={U∩V:U∈𝒰,V∈𝒱}.Denote byC(X,ℝ) the Banach space of all continuous, real-valued functions on X endowed with the supremum norm. For f∈C(X,R) and n∈ℕ, we denote ∑i=0n-1f(Ti(x)) by (Snf)(x).


### 2.1. Topological Pre-Image Pressure

In an early paper with Zeng et al. [25], following the idea of topological pressure (see Chapter 9, [12]), we defined a new notion of pre-image pressure, which extends the Cheng-Newhouse pre-image entropy hpre(T) [19]. For a given TDS (X,T), the pre-image pressure of T is a map Ppre(T,•):C(X,ℝ)→ℝ which is convex, Lipschitz continuous, increasing, with Ppre(T,0)=hpre(T). More precisely, let 𝒰∈𝒞Xo. For x∈X and k∈ℕ, we put(2.1)Pn(T,f,U,T-k(x)):=inf⁡V∑B∈Vsup⁡y∈Be(Snf)(y),
where the infimum is taken over all finite subcovers 𝒱 of ⋁i=0n-1T-i𝒰 with respect to T-k(x). We define the pre-image pressure ofTrelated to𝒰atf as(2.2)Ppre(T,f,U):=lim⁡n→∞1nlog⁡Ppre,n(T,f,U),
where Ppre,n(T,f,𝒰):=supx∈X,k⩾nPn(T,f,𝒰,T-k(x)). The pre-image pressure ofTatf is defined by(2.3)Ppre(T,f):=sup⁡U∈CXoPpre(T,f,U).
It is clear that Ppre(T,f)⩽P(T,f) (topological pressure, see [12]) and Ppre(T,0)=hpre(T). Ppre(T,f)⩽∥f∥ if T is a homeomorphism.


### 2.2. Measure-Theoretic Pre-Image Entropy

Denote byℳ(X), ℳ(X,T), and ℳe(X,T) the set of all Borel probability measures, T-invariant Borel probability measures and T-invariant ergodic measures, on X, respectively. Note that ℳ(X,T)⊇ℳe(X,T)≠∅, and both ℳ(X) and ℳ(X,T) are convex compact metric spaces when endowed with the weak*-topology; ℳe(X,T) is a Gδ subset of ℳ(X,T) (see [12, Chapter 6]). Beside the weak*-topology on ℳ(X,T), we also have the norm topology arising from the metric:(2.4)‖μ-ν‖:=sup⁡{|∫gdμ-∫gdν|:g∈C(X,R),‖g‖⩽1}.
Note that ∥μ-ν∥=2 if μ≠ν∈ℳe(X,T).Givenα∈𝒫X, μ∈ℳ(X) and a sub-σ-algebra 𝒜⊆ℬ(X), define(2.5)Hμ(α∣A)≔∑A∈α∫X-E(1A∣A)log⁡E(1A∣A)dμ,
where 𝔼(1A∣𝒜) is the conditional expectation of 1A with respect to 𝒜. It is a standard fact that Hμ(α∣𝒜) increases with respect to α and decreases with respect to 𝒜. Note that β∈𝒫X naturally generates a sub-σ-algebra ℱ(β) of ℬ(X); where there is no ambiguity, we write ℱ(β) as β. It is easy to check, for α,β∈𝒫X, that Hμ(α∣β)=Hμ(α∨β)-Hμ(β). More generally, for a sub-σ-algebra 𝒜⊆B(X), we have(2.6)Hμ(α∨β∣A)=Hμ(β∣A)+Hμ(α∣β∨A).Whenμ∈ℳ(X,T) and 𝒜⊆ℬ(X) is a T-invariant sub-σ-algebra, that is, T-1𝒜=𝒜 (up to μ-null sets), it is not hard to see that an=Hμ(⋁i=0n-1T-iα∣𝒜) is a nonnegative subadditive sequence for a given α∈𝒫X, that is, an+m≤an+am for all positive integers n and m. It is well known that (2.7)lim⁡n→∞ann=inf⁡n≥1ann.
The conditional entropy ofαwith respect to𝒜 is then defined by(2.8)hμ(T,α∣A):=lim⁡n→∞1nHμ(⋁i=0n-1T-iα∣A)=inf⁡n≥11nHμ(⋁i=0n-1T-iα∣A).
Moreover, the metric conditional entropy of(X,T)with respect to𝒜 is defined by(2.9)hμ(T,X∣A)=sup⁡α∈PXhμ(T,α∣A).
Note that if 𝒩={∅,X} is a trivial sub-σ-algebra, we recover the metric entropy, and we write hμ(T,α∣𝒩) and hμ(T,X∣𝒩) simple by hμ(T,α) and hμ(T).Particularly, setℬ-=⋂n=0∞T-nℬ(X), then ℬ- is a T-invariant sub-σ algebra. We call ℬ- the infinite pastσ-algebra related toℬ(X). We define the measure-theoretic (or metric) pre-image entropy ofαwith respect to(X,T) by(2.10)hpre,μ(T,α):=hμ(T,α∣B-)=lim⁡n→∞1nHμ(⋁i=0n-1T-iα∣B-).
Moreover, we define the metric pre-image entropy of(X,T) by(2.11)hpre,μ(T):=sup⁡α∈PXhpre,μ(T,α).


### 2.3. A Variational Principle for Pre-Image Pressure

The following variational relationship for topological pre-image pressure and measure-theoretic pre-image entropy is established in [25].Theorem 2.1.
Let(X,T) be a TDS and f∈C(X,ℝ). Then,
(2.12)Ppre(T,f)=sup⁡μ∈M(X,T){hpre,μ(T)+∫Xfdμ}.We also have (see, e.g., [26]) the following proposition.Proposition 2.2.
Let(X,T) be a TDS, α∈𝒫X and μ∈ℳ(X,T). Then, μ↦hpre,μ(T,α) and μ↦hpre,μ(T) are both affine functions on ℳ(X,T). Moreover, if the ergodic decomposition of μ is μ=∫ℳe(X,T)θdλ(θ), then
(2.13)hpre,μ(T,α)=∫Me(X,T)hpre,θ(T,α)dλ(θ),hpre,μ(T)=∫Me(X,T)hpre,θ(T)dλ(θ).


### 2.4. Equilibrium States and Tangent Functionals to Pre-Image Pressure

Givenf∈C(X,ℝ). A finite signed Borel measure μ on (X,ℬ(X)) is called a tangent functional to Ppre(T,•) at f if (2.14)Ppre(T,f+g)-Ppre(T,f)⩾∫gdμ,∀g∈C(X,R).
Let 𝒯f(X,T) denote the collection of all tangent functionals to Ppre(T,•) at f. An application of the Hahn-Banach theorem gives 𝒯f(X,T)≠∅. It is easy to see that μ∈𝒯f(X,T) if and only if (2.15)Ppre(T,f)-∫fdμ=inf⁡{Ppre(T,g)-∫gdμ:g∈C(X,R)}.
Also, we have 𝒯f(X,T)⊆ℳ(X,T) (see [25] for details).Theorem 2.3.
The following holds.(1)
Forf∈C(X,ℝ),
(2.16)Tf(X,T)=⋂n=1∞{μ∈M(X,T):hpre,μ(T)+∫fdμ>Ppre(T,f)-1n}¯.(2)
Iff1,f2∈C(X,ℝ) and μ∈𝒯f1(X,T)∩𝒯f2(X,T), then
(2.17)Ppre(T,pf1+(1-p)f2)=Ppre(T,f1)+(1-p)∫(f1-f1)dμ,∀p∈[0,1],
and 𝒯pf1+(1-p)f2(X,T)⊆𝒯f1(X,T)∩𝒯f1(X,T).Proof.
(1) Letμ∈If≡⋂n=1∞{μ∈ℳ(X,T):hpre,μ(T)+∫fdμ>Ppre(T,f)-1/n}¯. By Theorem 2.1, there is μn∈ℳ(X,T) with μn→μ and hpre,μn(T)+∫fdμn→Ppre(T,f). Hence, for each g∈C(X,ℝ),
(2.18)Ppre(T,f+g)-Ppre(T,f)⩾lim⁡n→∞(hpre,μn(T)+∫(f+g)dμn-Ppre(T,f))=∫gdμ,
which follows that μ∈𝒯f(X,T). Now suppose there is μ0∈𝒯f(X,T)∖If. Since If is convex, the standard separation theorem [27, page 417] follows that there exists g∈C(X,ℝ) with
(2.19)∫gdμ0>sup⁡{∫gdμ:μ∈If}.
By Theorem 2.1, we can choose μn∈ℳ(X,T) such that
(2.20)hpre,μn(T)+∫(f+gn)dμn>Ppre(T,f+gn)-1n2.
Without loss of generality, we can assume μn→μ*. Then,
(2.21)∫gdμ0=n∫gndμ0⩽n(Ppre(T,f+gn)-Ppre(T,f))<n(hpre,μn(T)+∫(f+gn)dμn+1n2-hpre,μn(T)-∫fdμn)=∫gdμn+1n→∫gdμ*.
However, μ*∈If follows from the fact:
(2.22)hpre,μn(T)+∫fdμn>Ppre(T,f+gn)-∫gndμn-1n2(by(2.20))>Ppre(T,f)-2⋅‖g‖n-1n2,(by[25,Lemma4.1(3)]),
which is a contradiction.
(2) If0<p<1, then
(2.23)pPpre(T,f1)+(1-p)Ppre(T,f2)⩾Ppre(T,pf1+(1-p)f2)(by [25, Lemma 4.1(3)])=Ppre(T,f1+(1-p)(f2-f1))⩾Ppre(T,f1)+(1-p)∫(f2-f1)dμ(sinceμ∈Tf1(X,T)).
Hence,
(2.24)Ppre(T,f2)-∫f2dμ⩾Ppre(T,f1)-∫f1dμ.
By symmetry, Ppre(T,f2)-∫f2dμ=Ppre(T,f1)-∫f1dμ, which implies
(2.25)Ppre(T,f1)+(1-p)∫(f2-f1)dμ=pPpre(T,f1)+(1-p)Ppre(T,f2)⩾Ppre(T,pf1+(1-p)f2).A memberμ∈ℳ(X,T) is called an equilibrium state for Ppre(T,•) at f if(2.26)Ppre(T,f)=hpre,μ(T)+∫fdμ.
Let ℳf(X,T) denote the collection of all equilibrium states for Ppre(T,•) at f.The set𝒯f(X,T) is convex and compact in the weak*-topology. The set ℳf(X,T) is convex but it may be not closed in the weak*-topology. Note that ℳf(X,T)⊆𝒯f(X,T)⊆ℳ(X,T) and ℳf(X,T) could be empty (see Example 5.1, [25]). We also have ℳf(X,T)=𝒯f(X,T) if and only if the metric pre-image entropy map hpre,•(T) is upper semicontinuous at every element of 𝒯f(X,T), Theorem  5.2 [25]. The extreme points of ℳf(X,T) are precisely the ergodic members of ℳf(X,T) and if μ∈ℳf(X,T), then almost every measure in the ergodic decomposition of μ is a member of ℳf(X,T) (see Proposition 2.1, [26]). When the metric pre-image entropy map hpre,•(T) is upper semicontinuous on ℳ(X,T), then ⋃f∈C(X,ℝ)ℳf(X,T) is dense in ℳ(X,T) in the norm topology, and given any finite collection of ergodic measures {μ1,…,μn}, there is some f∈C(X,ℝ) such that {μ1,…,μn}⊆ℳf(X,T) [26, Theorem  4.2].The following theorem shows when tangent functionals to pre-image pressure are not equilibrium states.Theorem 2.4.
Let(X,T) be a TDS and f∈C(X,ℝ). The following statements are mutually equivalent:(1)
μ∈𝒯f(X,T)∖ℳf(X,T);(2)
hpre,μ(T)+∫fdμ<Ppre(T,f) and there exist {μn}n=1∞⊆ℳ(X,T) with μn→μ and hpre,μn(T)+∫fdμn→Ppre(T,f);(3)
μ∈𝒯f(X,T) and hpre,•(T) is not upper semicontinuous at μ.Proof.
(1)⇒(2)
Follows from the variational principle and Theorem2.3(1).(2)⇒(3)
By Theorem2.3(1), μ∈𝒯f(X,T). If hpre,•(T) is upper semicontinuous at μ, then(2.27)Ppre(T,f)=lim⁡n→∞(hpre,μn(T)+∫fdμn)⩽hpre,μ(T)+∫fdμ.
Hence, μ∈ℳf(X,T) by the variational principle.(3)⇒(1)
If (3) holds, then there areμn∈ℳ(X,T) with μn→μ and limn→∞hpre,μn(T)>hpre,μ(T). Hence,
(2.28)Ppre(T,f)⩾lim⁡n→∞(hpre,μn(T)+∫fdμn)>hpre,μ(T)+∫fdμ. Therefore, μ∈𝒯f(X,T)∖ℳf(X,T).


## 2.1. Topological Pre-Image Pressure

In an early paper with Zeng et al. [25], following the idea of topological pressure (see Chapter 9, [12]), we defined a new notion of pre-image pressure, which extends the Cheng-Newhouse pre-image entropy hpre(T) [19]. For a given TDS (X,T), the pre-image pressure of T is a map Ppre(T,•):C(X,ℝ)→ℝ which is convex, Lipschitz continuous, increasing, with Ppre(T,0)=hpre(T). More precisely, let 𝒰∈𝒞Xo. For x∈X and k∈ℕ, we put(2.1)Pn(T,f,U,T-k(x)):=inf⁡V∑B∈Vsup⁡y∈Be(Snf)(y),
where the infimum is taken over all finite subcovers 𝒱 of ⋁i=0n-1T-i𝒰 with respect to T-k(x). We define the pre-image pressure ofTrelated to𝒰atf as(2.2)Ppre(T,f,U):=lim⁡n→∞1nlog⁡Ppre,n(T,f,U),
where Ppre,n(T,f,𝒰):=supx∈X,k⩾nPn(T,f,𝒰,T-k(x)). The pre-image pressure ofTatf is defined by(2.3)Ppre(T,f):=sup⁡U∈CXoPpre(T,f,U).
It is clear that Ppre(T,f)⩽P(T,f) (topological pressure, see [12]) and Ppre(T,0)=hpre(T). Ppre(T,f)⩽∥f∥ if T is a homeomorphism.


## 2.2. Measure-Theoretic Pre-Image Entropy

Denote byℳ(X), ℳ(X,T), and ℳe(X,T) the set of all Borel probability measures, T-invariant Borel probability measures and T-invariant ergodic measures, on X, respectively. Note that ℳ(X,T)⊇ℳe(X,T)≠∅, and both ℳ(X) and ℳ(X,T) are convex compact metric spaces when endowed with the weak*-topology; ℳe(X,T) is a Gδ subset of ℳ(X,T) (see [12, Chapter 6]). Beside the weak*-topology on ℳ(X,T), we also have the norm topology arising from the metric:(2.4)‖μ-ν‖:=sup⁡{|∫gdμ-∫gdν|:g∈C(X,R),‖g‖⩽1}.
Note that ∥μ-ν∥=2 if μ≠ν∈ℳe(X,T).Givenα∈𝒫X, μ∈ℳ(X) and a sub-σ-algebra 𝒜⊆ℬ(X), define(2.5)Hμ(α∣A)≔∑A∈α∫X-E(1A∣A)log⁡E(1A∣A)dμ,
where 𝔼(1A∣𝒜) is the conditional expectation of 1A with respect to 𝒜. It is a standard fact that Hμ(α∣𝒜) increases with respect to α and decreases with respect to 𝒜. Note that β∈𝒫X naturally generates a sub-σ-algebra ℱ(β) of ℬ(X); where there is no ambiguity, we write ℱ(β) as β. It is easy to check, for α,β∈𝒫X, that Hμ(α∣β)=Hμ(α∨β)-Hμ(β). More generally, for a sub-σ-algebra 𝒜⊆B(X), we have(2.6)Hμ(α∨β∣A)=Hμ(β∣A)+Hμ(α∣β∨A).Whenμ∈ℳ(X,T) and 𝒜⊆ℬ(X) is a T-invariant sub-σ-algebra, that is, T-1𝒜=𝒜 (up to μ-null sets), it is not hard to see that an=Hμ(⋁i=0n-1T-iα∣𝒜) is a nonnegative subadditive sequence for a given α∈𝒫X, that is, an+m≤an+am for all positive integers n and m. It is well known that (2.7)lim⁡n→∞ann=inf⁡n≥1ann.
The conditional entropy ofαwith respect to𝒜 is then defined by(2.8)hμ(T,α∣A):=lim⁡n→∞1nHμ(⋁i=0n-1T-iα∣A)=inf⁡n≥11nHμ(⋁i=0n-1T-iα∣A).
Moreover, the metric conditional entropy of(X,T)with respect to𝒜 is defined by(2.9)hμ(T,X∣A)=sup⁡α∈PXhμ(T,α∣A).
Note that if 𝒩={∅,X} is a trivial sub-σ-algebra, we recover the metric entropy, and we write hμ(T,α∣𝒩) and hμ(T,X∣𝒩) simple by hμ(T,α) and hμ(T).Particularly, setℬ-=⋂n=0∞T-nℬ(X), then ℬ- is a T-invariant sub-σ algebra. We call ℬ- the infinite pastσ-algebra related toℬ(X). We define the measure-theoretic (or metric) pre-image entropy ofαwith respect to(X,T) by(2.10)hpre,μ(T,α):=hμ(T,α∣B-)=lim⁡n→∞1nHμ(⋁i=0n-1T-iα∣B-).
Moreover, we define the metric pre-image entropy of(X,T) by(2.11)hpre,μ(T):=sup⁡α∈PXhpre,μ(T,α).


## 2.3. A Variational Principle for Pre-Image Pressure

The following variational relationship for topological pre-image pressure and measure-theoretic pre-image entropy is established in [25].Theorem 2.1.
Let(X,T) be a TDS and f∈C(X,ℝ). Then,
(2.12)Ppre(T,f)=sup⁡μ∈M(X,T){hpre,μ(T)+∫Xfdμ}.We also have (see, e.g., [26]) the following proposition.Proposition 2.2.
Let(X,T) be a TDS, α∈𝒫X and μ∈ℳ(X,T). Then, μ↦hpre,μ(T,α) and μ↦hpre,μ(T) are both affine functions on ℳ(X,T). Moreover, if the ergodic decomposition of μ is μ=∫ℳe(X,T)θdλ(θ), then
(2.13)hpre,μ(T,α)=∫Me(X,T)hpre,θ(T,α)dλ(θ),hpre,μ(T)=∫Me(X,T)hpre,θ(T)dλ(θ).


## 2.4. Equilibrium States and Tangent Functionals to Pre-Image Pressure

Givenf∈C(X,ℝ). A finite signed Borel measure μ on (X,ℬ(X)) is called a tangent functional to Ppre(T,•) at f if (2.14)Ppre(T,f+g)-Ppre(T,f)⩾∫gdμ,∀g∈C(X,R).
Let 𝒯f(X,T) denote the collection of all tangent functionals to Ppre(T,•) at f. An application of the Hahn-Banach theorem gives 𝒯f(X,T)≠∅. It is easy to see that μ∈𝒯f(X,T) if and only if (2.15)Ppre(T,f)-∫fdμ=inf⁡{Ppre(T,g)-∫gdμ:g∈C(X,R)}.
Also, we have 𝒯f(X,T)⊆ℳ(X,T) (see [25] for details).Theorem 2.3.
The following holds.(1)
Forf∈C(X,ℝ),
(2.16)Tf(X,T)=⋂n=1∞{μ∈M(X,T):hpre,μ(T)+∫fdμ>Ppre(T,f)-1n}¯.(2)
Iff1,f2∈C(X,ℝ) and μ∈𝒯f1(X,T)∩𝒯f2(X,T), then
(2.17)Ppre(T,pf1+(1-p)f2)=Ppre(T,f1)+(1-p)∫(f1-f1)dμ,∀p∈[0,1],
and 𝒯pf1+(1-p)f2(X,T)⊆𝒯f1(X,T)∩𝒯f1(X,T).Proof.
(1) Letμ∈If≡⋂n=1∞{μ∈ℳ(X,T):hpre,μ(T)+∫fdμ>Ppre(T,f)-1/n}¯. By Theorem 2.1, there is μn∈ℳ(X,T) with μn→μ and hpre,μn(T)+∫fdμn→Ppre(T,f). Hence, for each g∈C(X,ℝ),
(2.18)Ppre(T,f+g)-Ppre(T,f)⩾lim⁡n→∞(hpre,μn(T)+∫(f+g)dμn-Ppre(T,f))=∫gdμ,
which follows that μ∈𝒯f(X,T). Now suppose there is μ0∈𝒯f(X,T)∖If. Since If is convex, the standard separation theorem [27, page 417] follows that there exists g∈C(X,ℝ) with
(2.19)∫gdμ0>sup⁡{∫gdμ:μ∈If}.
By Theorem 2.1, we can choose μn∈ℳ(X,T) such that
(2.20)hpre,μn(T)+∫(f+gn)dμn>Ppre(T,f+gn)-1n2.
Without loss of generality, we can assume μn→μ*. Then,
(2.21)∫gdμ0=n∫gndμ0⩽n(Ppre(T,f+gn)-Ppre(T,f))<n(hpre,μn(T)+∫(f+gn)dμn+1n2-hpre,μn(T)-∫fdμn)=∫gdμn+1n→∫gdμ*.
However, μ*∈If follows from the fact:
(2.22)hpre,μn(T)+∫fdμn>Ppre(T,f+gn)-∫gndμn-1n2(by(2.20))>Ppre(T,f)-2⋅‖g‖n-1n2,(by[25,Lemma4.1(3)]),
which is a contradiction.
(2) If0<p<1, then
(2.23)pPpre(T,f1)+(1-p)Ppre(T,f2)⩾Ppre(T,pf1+(1-p)f2)(by [25, Lemma 4.1(3)])=Ppre(T,f1+(1-p)(f2-f1))⩾Ppre(T,f1)+(1-p)∫(f2-f1)dμ(sinceμ∈Tf1(X,T)).
Hence,
(2.24)Ppre(T,f2)-∫f2dμ⩾Ppre(T,f1)-∫f1dμ.
By symmetry, Ppre(T,f2)-∫f2dμ=Ppre(T,f1)-∫f1dμ, which implies
(2.25)Ppre(T,f1)+(1-p)∫(f2-f1)dμ=pPpre(T,f1)+(1-p)Ppre(T,f2)⩾Ppre(T,pf1+(1-p)f2).A memberμ∈ℳ(X,T) is called an equilibrium state for Ppre(T,•) at f if(2.26)Ppre(T,f)=hpre,μ(T)+∫fdμ.
Let ℳf(X,T) denote the collection of all equilibrium states for Ppre(T,•) at f.The set𝒯f(X,T) is convex and compact in the weak*-topology. The set ℳf(X,T) is convex but it may be not closed in the weak*-topology. Note that ℳf(X,T)⊆𝒯f(X,T)⊆ℳ(X,T) and ℳf(X,T) could be empty (see Example 5.1, [25]). We also have ℳf(X,T)=𝒯f(X,T) if and only if the metric pre-image entropy map hpre,•(T) is upper semicontinuous at every element of 𝒯f(X,T), Theorem  5.2 [25]. The extreme points of ℳf(X,T) are precisely the ergodic members of ℳf(X,T) and if μ∈ℳf(X,T), then almost every measure in the ergodic decomposition of μ is a member of ℳf(X,T) (see Proposition 2.1, [26]). When the metric pre-image entropy map hpre,•(T) is upper semicontinuous on ℳ(X,T), then ⋃f∈C(X,ℝ)ℳf(X,T) is dense in ℳ(X,T) in the norm topology, and given any finite collection of ergodic measures {μ1,…,μn}, there is some f∈C(X,ℝ) such that {μ1,…,μn}⊆ℳf(X,T) [26, Theorem  4.2].The following theorem shows when tangent functionals to pre-image pressure are not equilibrium states.Theorem 2.4.
Let(X,T) be a TDS and f∈C(X,ℝ). The following statements are mutually equivalent:(1)
μ∈𝒯f(X,T)∖ℳf(X,T);(2)
hpre,μ(T)+∫fdμ<Ppre(T,f) and there exist {μn}n=1∞⊆ℳ(X,T) with μn→μ and hpre,μn(T)+∫fdμn→Ppre(T,f);(3)
μ∈𝒯f(X,T) and hpre,•(T) is not upper semicontinuous at μ.Proof.
(1)⇒(2)
Follows from the variational principle and Theorem2.3(1).(2)⇒(3)
By Theorem2.3(1), μ∈𝒯f(X,T). If hpre,•(T) is upper semicontinuous at μ, then(2.27)Ppre(T,f)=lim⁡n→∞(hpre,μn(T)+∫fdμn)⩽hpre,μ(T)+∫fdμ.
Hence, μ∈ℳf(X,T) by the variational principle.(3)⇒(1)
If (3) holds, then there areμn∈ℳ(X,T) with μn→μ and limn→∞hpre,μn(T)>hpre,μ(T). Hence,
(2.28)Ppre(T,f)⩾lim⁡n→∞(hpre,μn(T)+∫fdμn)>hpre,μ(T)+∫fdμ. Therefore, μ∈𝒯f(X,T)∖ℳf(X,T).


## 3. Gateaux Differentiability of the Pre-Image Pressure

In [26], we studied the uniqueness of the equilibrium state for the pre-image pressure. We showed that when the metric pre-image entropy map hpre,•(T) is upper semicontinuous on ℳ(X,T), then the set of all functions with unique equilibrium state is dense in C(X,ℝ). Without the upper semicontinuity assumption, one can show that all functions with unique tangent functional are dense in C(X,ℝ) (can see [27, page 450] or [11, Appendix A.3.6]). In this section, we will show a continuous function with unique tangent functional to pre-image pressure if and only if it is Gateaux differentiable.Givenf,g∈C(X,ℝ). Since Ppre(T,•) is convex, the map t↦(Ppre(T,f+tg)-Ppre(T,f))/t is increasing and hence(3.1)d+Ppre(T,f)(g)=lim⁡t→0+Ppre(T,f+tg)-Ppre(T,f)t,d-Ppre(T,f)(g)=lim⁡t→0-Ppre(T,f+tg)-Ppre(T,f)t
exist. Note that d+Ppre(T,f)(g)=-d-Ppre(T,f)(-g). The pre-image pressure function Ppre(T,•) is said to be Gateaux differentiable atf if, for all g∈C(X,ℝ),(3.2)lim⁡t→0Ppre(T,f+tg)-Ppre(T,f)t
exist. It is easy to check that Ppre(T,•) is Gateaux differentiable at f if and only if g↦d+Ppre(T,f)(g) is linear.Lemma 3.1.
Let(X,T) be a TDS and f,g∈C(X,ℝ). Then,
(3.3)d+Ppre(T,f)(g)=sup⁡{∫gdμ:μ∈Tf(X,T)}.Proof.
Ifμ∈𝒯f(X,T), then, for g∈C(X,ℝ),
(3.4)∫gdμ⩽Ppre(T,f+tg)-Ppre(T,f)t,∀t>0.
Hence,
(3.5)sup⁡{∫gdμ:μ∈Tf(X,T)}⩽d+Ppre(T,f)(g).
Next, we prove the converse inequality. Set a=d+Ppre(T,f)(g). Define a continuous linear functional γ:{tg:t∈ℝ}→ℝ by
(3.6)γ(tg)=ta,t∈R.
The convexity of Ppre(T,•) implies
(3.7)γ(tg)=t⋅d+Ppre(T,f)(g)⩽Ppre(T,f+tg)-Ppre(T,f),∀t∈R.
By the Hahn-Banach theorem, γ can be extended to a continuous linear functional on C(X,ℝ) such that
(3.8)γ(h)⩽Ppre(T,f+h)-Ppre(T,f),∀h∈C(X,R).
By the Riesz representation theorem, there is μ∈ℳ(X) with
(3.9)γ(h)=∫hdμ,∀g∈C(X,R).
Combining (3.6), (3.8), and (3.9), we have μ∈𝒯f(X,T), and
(3.10)∫gdμ=γ(g)=a=d+Ppre(T,f)(g).
The lemma is proved.Theorem 3.2 (Uniqueness of tangent functional and Gateaux differentiability).
The following statements are mutually equivalent:(1)
the pre-image pressure functionPpre(T,•) is Gateaux differentiable at f∈C(X,ℝ);(2)
the unique tangent functional toPpre(T,•) at f is μ;(3)
for eachg∈C(X,ℝ),(3.11)lim⁡t→0Ppre(T,f+tg)-Ppre(T,f)t=∫gdμ.Proof.
(1)⇒(2)
If the pre-image pressure functionPpre(T,•) is Gateaux differentiable at f, then the function g↦d+Ppre(T,f)(g) is linear. By Lemma 3.1,(3.12)sup⁡{∫gdμ:μ∈Tf(X,T)}=d+Ppre(T,f)(g)=-d+Ppre(T,f)(-g)=-sup⁡{∫-gdμ:μ∈Tf(X,T)}=inf⁡{∫gdμ:μ∈Tf(X,T)}
for each g∈C(X,ℝ). This implies there is a unique tangent functional to Ppre(T,•) at f.
(2)⇒(3)
It directly follows from Lemma3.1.(3)⇒(1)
It follows from the definition.


## 4. Fréchet Differentiability of the Pre-Image Pressure

In this section, we will study the Fréchet differentiability of pre-image pressure. The pre-image pressure functionPpre(T,•) is said to be Fréchet differentiable atf if there is γ∈C(X,ℝ)* such that(4.1)lim⁡g→0|Ppre(T,f+g)-Ppre(T,f)-γ(g)|‖g‖=0.
The pre-image pressure function Ppre(T,•) is said to be Fréchet differentiable if it is Fréchet differentiable at each f∈C(X,ℝ).Note that ifPpre(T,•) is Fréchet differentiable at f, then it is Gateaux differentiable at f and γ(g)=∫gdμ, where μ is the unique tangent functional to Ppre(T,•) at f.Theorem 4.1.
The following conditions are mutually equivalent:(1)
Ppre(T,•) if Fréchet differentiable at f;(2)
𝒯f(X,T)={μf} and ∥μn-μf∥→0 for each {μn}⊆ℳ(X,T) with hpre,μn(T)+∫fdμn→Ppre(T,f);(3)
Ppre(T,•) is locally affine at f;(4)
𝒯f(X,T)={μf} and
(4.2)lim⁡g→0sup⁡{‖μ-μf‖:μ∈Tf+g(X,T)}=0.Proof.
(1)⇒(2)
SupposePpre(T,•) is Fréchet differentiable at f. Then, f has a unique tangent functional μf to Ppre(T,•) at f. Let {μn}n=1∞⊆ℳ(X,T) with hpre,μn(T)+∫fdμn→Ppre(T,f). Given ϵ>0, there are N∈ℕ and δ>0 such that(4.3)Ppre(T,f)⩽hpre,μn(T)+∫fdμn+ϵδ,∀n⩾N,(4.4)0⩽Ppre(T,f+g)-Ppre(T,f)-∫gdμf⩽ϵ⋅‖g‖whenever‖g‖<δ.
Hence, if n⩾N and ∥g∥<δ, then
(4.5)∫gdμn-∫gdμ=Ppre(T,f)+∫gdμn-Ppre(T,f)-∫gdμf⩽hpre,μn(T)+∫(f+g)dμn+ϵδ-Ppre(T,f)-∫gdμf(by 4.4)⩽Ppre(T,f+g)-Ppre(T,f)-∫gdμf+ϵδ(byTheorem2.1)⩽ϵ⋅(‖g‖+δ)⩽2ϵδ(by 4.4).
Note that (4.5) is also true when -g instead of g. So,
(4.6)|∫gdμn-∫gdμ|⩽2ϵδwhenever‖g‖<δ,n⩾N.
Therefore,
(4.7)‖μn-μf‖=sup⁡{|∫gdμn-∫gdμf|:‖g‖<1}=1δsup⁡{|∫gdμn-∫gdμf|:‖g‖<δ}⩽2ϵ(by 4.6).
By arbitrary of ϵ, ∥μn-μf∥→0.(2)⇒(3)
By the variational principle of pre-image pressure, we can pick ergodic measuresμn such that hpre,μn(T)+∫fdμn→Ppre(T,f), Then ∥μn-μf∥→0. Note that two distinct ergodic measures have norm-distance 2. So there is N such that μn=μf for all n⩾N. Hence, μf∈ℳe(X,T) and(4.8)ϵ:=Ppre(T,f)-sup⁡{hpre,μ(T)+∫fdμ:μ∈Me(X,T),μ≠μf}>0.
This implies for each g∈C(X,ℝ), with ∥f-g∥<ϵ/2, that
(4.9)sup⁡{hpre,μ(T)+∫gdμ:μ∈Me(X,T),μ≠μf}⩽Ppre(T,f)-ϵ+‖f-g‖⩽Ppre(T,g)-ϵ+2‖f-g‖<Ppre(T,g).
By the variational principle of pre-image pressure again, we have
(4.10)Ppre(T,g)=hpre,μf(T)+∫gdμfwhenever‖f-g‖<ϵ2.
Hence, Ppre(T,•) is affine on the neighborhood B(f,ϵ/2)={g∈C(X,ℝ):∥f-g∥<ϵ/2} of f.(3)⇒(4)
Is obvious.(4)⇒(1)
Letg∈C(X,ℝ) and μ∈𝒯f+g(X,T). By definition, we have(4.11)Ppre(T,f)-Ppre(T,f+g)⩾-∫gdμ.
Hence,
(4.12)0⩽Ppre(T,f+g)-Ppre(T,f)-∫gdμf⩽∫gdμ-∫gdμf(by(4.11))⩽‖g‖⋅‖μ-μf‖.
Therefore,
(4.13)0⩽Ppre(T,f+g)-Ppre(T,f)-∫gdμf‖g‖⩽sup⁡{‖μ-μf‖:μ∈Tf+g(X,T)}⟶0
as g→0. That is Ppre(T,•) is Fréchet differentiable at f.Corollary 4.2.
Let(X,T) be a TDS with finite pre-image entropy. Then, Ppre(T,•) is Fréchet differentiable if and only if T is uniquely ergodic.Proof.
Using Theorem4.1, Ppre(T,•) is locally affine whenever Ppre(T,•) is Fréchet differentiable. Hence, the map f∈C(X,ℝ)↦μf∈ℳ(X,T) is locally constant, where 𝒯f(X,T)={μf} for each f∈C(X,ℝ). Since C(X,ℝ) is connected, the map is constant. So μf=μ0 for all f∈C(X,ℝ). If μ∈ℳ(X,T)∖{μ0}, then we can choose f∈C(X,ℝ) such that ∫fdμ>∫fdμ0. Then for sufficiently large k, we have
(4.14)Ppre(T,kf)⩾hpre,μ(T)+∫kfdμ>hpre,μ0(T)+∫kfdμ0=Ppre(T,kf),
which is a contradiction. Therefore, ℳ(X,T)={μ0}.Remark 4.3.
In the situation of Corollary4.2, there is only one invariant measure, and the pre-image pressure is the expectation with respect to this measure, hence, it is linear.



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*Source: 951691-2012-05-27.xml*