Differentiability Properties of the Pre-Image Pressure

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 𝑃_pre(𝑇,•) is Gateaux differentiable at 𝑓∈𝐶(𝑋,ℝ) if and only if 𝑃_pre(𝑇,•) has a unique tangent functional at 𝑓. Also, we obtain some equivalent conditions for 𝑃_pre(𝑇,•) to be Fréchet differentiable at 𝑓.

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 ℎ_pre(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 𝐺_𝛿-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 Section 3, the Gateaux differentiability of the pre-image pressure is discussed. We show that the pre-image pressure function 𝑃_pre(𝑇,•) is Gateaux differentiable at 𝑓∈𝐶(𝑋,ℝ) if and only if it has a unique tangent functional at 𝑓.

In Section 4, we discuss the Fréchet differentiability of the pre-image pressure. We obtain some equivalent conditions for 𝑃_pre(𝑇,•) to be Fréchet differentiable at 𝑓. Also, we show that the pre-image function 𝑃_pre(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 ℎ_pre(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 𝒞_𝑋, 𝒞^𝑜_𝑋, and 𝒫_𝑋, 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 by 𝐶(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 ∑^𝑛−1_𝑖=0𝑓(𝑇^𝑖(𝑥)) by (𝑆_𝑛𝑓)(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 ℎ_pre(𝑇) [19]. For a given TDS (𝑋,𝑇), the pre-image pressure of 𝑇 is a map 𝑃_pre(𝑇,•)∶𝐶(𝑋,ℝ)→ℝ which is convex, Lipschitz continuous, increasing, with 𝑃_pre(𝑇,0)=ℎ_pre(𝑇). More precisely, let 𝒰∈𝒞^𝑜_𝑋. For 𝑥∈𝑋 and 𝑘∈ℕ, we put𝑃_𝑛𝑇,𝑓,𝒰,𝑇^−𝑘(𝑥)∶=inf_{𝒱𝐵∈𝒱}sup_𝑦∈𝐵e^{(𝑆𝑛𝑓)(𝑦)}, (2.1) where the infimum is taken over all finite subcovers 𝒱 of ⋁^𝑛−1_𝑖=0𝑇^−𝑖𝒰 with respect to 𝑇^−𝑘(𝑥). We define the pre-image pressure of 𝑇 related to 𝒰 at 𝑓 as𝑃_pre𝑇,𝑓,𝒰∶=lim_𝑛→∞1𝑛log𝑃_pre,𝑛𝑇,𝑓,𝒰, (2.2) where 𝑃_pre,𝑛(𝑇,𝑓,𝒰)∶=sup_{𝑥∈𝑋,𝑘⩾𝑛}𝑃_𝑛(𝑇,𝑓,𝒰,𝑇^−𝑘(𝑥)). The pre-image pressure of 𝑇 at 𝑓 is defined by𝑃_pre𝑇,𝑓∶=sup_{𝒰∈𝒞𝑜𝑋}𝑃_pre𝑇,𝑓,𝒰. (2.3) It is clear that 𝑃_pre(𝑇,𝑓)⩽𝑃(𝑇,𝑓) (topological pressure, see [12]) and 𝑃_pre(𝑇,0)=ℎ_pre(𝑇). 𝑃_pre(𝑇,𝑓)⩽‖𝑓‖ if 𝑇 is a homeomorphism.

2.2. Measure-Theoretic Pre-Image Entropy

Denote by ℳ(𝑋), ℳ(𝑋,𝑇), and ℳ^𝑒(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)⊇ℳ^𝑒(𝑋,𝑇)≠∅, and both ℳ(𝑋) and ℳ(𝑋,𝑇) are convex compact metric spaces when endowed with the weak*-topology; ℳ^𝑒(𝑋,𝑇) is a 𝐺_𝛿 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:𝜇−𝜈∶=sup𝑔d𝜇−𝑔d𝜈∶𝑔∈𝐶(𝑋,ℝ),𝑔⩽1. (2.4) Note that ‖𝜇−𝜈‖=2 if 𝜇≠𝜈∈ℳ^𝑒(𝑋,𝑇).

Given 𝛼∈𝒫_𝑋, 𝜇∈ℳ(𝑋) and a sub-𝜎-algebra 𝒜⊆ℬ(𝑋), define𝐻_𝜇(𝛼∣𝒜)∶=_{𝐴∈𝛼𝑋}−𝔼1_𝐴∣𝒜log𝔼1_𝐴∣𝒜d𝜇, (2.5) where 𝔼(1_𝐴∣𝒜) is the conditional expectation of 1_𝐴 with respect to 𝒜. It is a standard fact that 𝐻_𝜇(𝛼∣𝒜) increases with respect to 𝛼 and decreases with respect to 𝒜. Note that 𝛽∈𝒫_𝑋 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∈𝒫_𝑋, that 𝐻_𝜇(𝛼∣𝛽)=𝐻_𝜇(𝛼∨𝛽)−𝐻_𝜇(𝛽). More generally, for a sub-𝜎-algebra 𝒜⊆𝐵(𝑋), we have𝐻_𝜇𝛼∨𝛽∣𝒜=𝐻_𝜇𝛽∣𝒜+𝐻_𝜇𝛼∣𝛽∨𝒜. (2.6)

When 𝜇∈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 𝑎_𝑛=𝐻_𝜇(⋁^𝑛−1_𝑖=0𝑇^−𝑖𝛼∣𝒜) is a nonnegative subadditive sequence for a given 𝛼∈𝒫_𝑋, that is, 𝑎_𝑛+𝑚≤𝑎_𝑛+𝑎_𝑚 for all positive integers 𝑛 and 𝑚. It is well known that lim_𝑛→∞𝑎_𝑛𝑛=inf_𝑛≥1𝑎_𝑛𝑛. (2.7) The conditional entropy of 𝛼 with respect to 𝒜 is then defined byℎ_𝜇(𝑇,𝛼∣𝒜)∶=lim_𝑛→∞1𝑛𝐻_𝜇^𝑛−1_𝑖=0𝑇^−𝑖𝛼∣𝒜=inf_𝑛≥11𝑛𝐻_𝜇^𝑛−1_𝑖=0𝑇^−𝑖𝛼∣𝒜. (2.8) Moreover, the metric conditional entropy of (𝑋,𝑇) with respect to 𝒜 is defined byℎ_𝜇(𝑇,𝑋∣𝒜)=sup_{𝛼∈𝒫𝑋}ℎ_𝜇(𝑇,𝛼∣𝒜). (2.9) Note that if 𝒩={∅,𝑋} is a trivial sub-𝜎-algebra, we recover the metric entropy, and we write ℎ_𝜇(𝑇,𝛼∣𝒩) and ℎ_𝜇(𝑇,𝑋∣𝒩) simple by ℎ_𝜇(𝑇,𝛼) and ℎ_𝜇(𝑇).

Particularly, set ℬ⁻=⋂^∞_𝑛=0𝑇^−𝑛ℬ(𝑋), then ℬ⁻ is a 𝑇-invariant sub-𝜎 algebra. We call ℬ⁻ the infinite past 𝜎-algebra related to ℬ(𝑋). We define the measure-theoretic (or metric) pre-image entropy of 𝛼 with respect to (𝑋,𝑇) byℎ_pre,𝜇(𝑇,𝛼)∶=ℎ_𝜇(𝑇,𝛼∣ℬ⁻)=lim_𝑛→∞1𝑛𝐻_𝜇^𝑛−1_𝑖=0𝑇^−𝑖𝛼∣ℬ⁻. (2.10) Moreover, we define the metric pre-image entropy of (𝑋,𝑇) byℎ_pre,𝜇(𝑇)∶=sup_{𝛼∈𝒫𝑋}ℎ_pre,𝜇(𝑇,𝛼). (2.11)

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, 𝑃_pre𝑇,𝑓=sup_{()𝜇∈ℳ𝑋,𝑇}ℎ_pre,𝜇(𝑇)+_𝑋𝑓d𝜇. (2.12)

We also have (see, e.g., [26]) the following proposition.

Proposition 2.2. Let (𝑋,𝑇) be a TDS, 𝛼∈𝒫_𝑋 and 𝜇∈ℳ(𝑋,𝑇). Then, 𝜇↦ℎ_pre,𝜇(𝑇,𝛼) and 𝜇↦ℎ_pre,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 ℎ_pre,𝜇(𝑇,𝛼)=_{ℳ𝑒(𝑋,𝑇)}ℎ_pre,𝜃(𝑇,𝛼)d𝜆(𝜃),ℎ_pre,𝜇(𝑇)=_{ℳ𝑒(𝑋,𝑇)}ℎ_pre,𝜃(𝑇)d𝜆(𝜃). (2.13)

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

Given 𝑓∈𝐶(𝑋,ℝ). A finite signed Borel measure 𝜇 on (𝑋,ℬ(𝑋)) is called a tangent functional to 𝑃_pre(𝑇,•) at 𝑓 if 𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓⩾𝑔d𝜇,∀𝑔∈𝐶(𝑋,ℝ). (2.14) Let 𝒯_𝑓(𝑋,𝑇) denote the collection of all tangent functionals to 𝑃_pre(𝑇,•) at 𝑓. An application of the Hahn-Banach theorem gives 𝒯_𝑓(𝑋,𝑇)≠∅. It is easy to see that 𝜇∈𝒯_𝑓(𝑋,𝑇) if and only if 𝑃_pre𝑇,𝑓−𝑓d𝜇=inf𝑃_pre𝑇,𝑔−𝑔d𝜇∶𝑔∈𝐶(𝑋,ℝ). (2.15) Also, we have 𝒯_𝑓(𝑋,𝑇)⊆ℳ(𝑋,𝑇) (see [25] for details).

Theorem 2.3. The following holds.(1)For 𝑓∈𝐶(𝑋,ℝ), 𝒯_𝑓(𝑋,𝑇)=^∞_𝑛=1𝜇∈ℳ(𝑋,𝑇)∶ℎ_pre,𝜇(𝑇)+𝑓d𝜇>𝑃_pre−1𝑇,𝑓𝑛. (2.16)(2)If 𝑓₁,𝑓₂∈𝐶(𝑋,ℝ) and 𝜇∈𝒯_𝑓1(𝑋,𝑇)∩𝒯_𝑓2(𝑋,𝑇), then 𝑃_pre𝑇,𝑝𝑓₁+1−𝑝𝑓₂=𝑃_pre𝑇,𝑓₁+1−𝑝𝑓₁−𝑓₁d𝜇,∀𝑝∈[0,1], (2.17) and 𝒯_{𝑝𝑓1+(1−𝑝)𝑓2}(𝑋,𝑇)⊆𝒯_𝑓1(𝑋,𝑇)∩𝒯_𝑓1(𝑋,𝑇).

Proof. (1) Let 𝜇∈𝐼_𝑓≡⋂^∞_𝑛=1{𝜇∈ℳ(𝑋,𝑇)∶ℎ_pre,𝜇(𝑇)+∫𝑓d𝜇>𝑃_pre(𝑇,𝑓)−1/𝑛}. By Theorem 2.1, there is 𝜇_𝑛∈ℳ(𝑋,𝑇) with 𝜇_𝑛→𝜇 and ℎ_pre,𝜇𝑛(𝑇)+∫𝑓d𝜇_𝑛→𝑃_pre(𝑇,𝑓). Hence, for each 𝑔∈𝐶(𝑋,ℝ), 𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓⩾lim_𝑛→∞ℎ_pre,𝜇𝑛(𝑇)+𝑓+𝑔d𝜇_𝑛−𝑃_pre𝑇,𝑓=𝑔d𝜇, (2.18) which follows that 𝜇∈𝒯_𝑓(𝑋,𝑇). Now suppose there is 𝜇₀∈𝒯_𝑓(𝑋,𝑇)⧵𝐼_𝑓. Since 𝐼_𝑓 is convex, the standard separation theorem [27, page 417] follows that there exists 𝑔∈𝐶(𝑋,ℝ) with 𝑔d𝜇₀>sup𝑔d𝜇∶𝜇∈𝐼_𝑓. (2.19) By Theorem 2.1, we can choose 𝜇_𝑛∈ℳ(𝑋,𝑇) such that ℎ_pre,𝜇𝑛(𝑇)+𝑔𝑓+𝑛d𝜇_𝑛>𝑃_pre𝑔𝑇,𝑓+𝑛−1𝑛². (2.20) Without loss of generality, we can assume 𝜇_𝑛→𝜇^∗. Then, 𝑔d𝜇₀𝑔=𝑛𝑛d𝜇₀⩽𝑛𝑃_pre𝑔𝑇,𝑓+𝑛−𝑃_preℎ𝑇,𝑓<𝑛_pre,𝜇𝑛(𝑇)+𝑔𝑓+𝑛d𝜇_𝑛+1𝑛²−ℎ_pre,𝜇𝑛(𝑇)−𝑓d𝜇_𝑛=𝑔d𝜇_𝑛+1𝑛→𝑔d𝜇^∗. (2.21) However, 𝜇^∗∈𝐼_𝑓 follows from the fact: ℎ_pre,𝜇𝑛(𝑇)+𝑓d𝜇_𝑛>𝑃_pre𝑔𝑇,𝑓+𝑛−𝑔𝑛d𝜇_𝑛−1𝑛²by(2.20)>𝑃_pre−𝑔𝑇,𝑓2⋅𝑛−1𝑛²,by[25,Lemma4.1(3)], (2.22) which is a contradiction.
(2) If 0<𝑝<1, then 𝑝𝑃_pre𝑇,𝑓₁+1−𝑝𝑃_pre𝑇,𝑓₂⩾𝑃_pre𝑇,𝑝𝑓₁+1−𝑝𝑓₂by[25,Lemma4.1(3)]=𝑃_pre𝑇,𝑓₁+1−𝑝𝑓₂−𝑓₁⩾𝑃_pre𝑇,𝑓₁+1−𝑝𝑓₂−𝑓₁d𝜇since𝜇∈𝒯_𝑓1(𝑋,𝑇). (2.23) Hence, 𝑃_pre𝑇,𝑓₂−𝑓₂d𝜇⩾𝑃_pre𝑇,𝑓₁−𝑓₁d𝜇. (2.24) By symmetry, 𝑃_pre(𝑇,𝑓₂)−∫𝑓₂d𝜇=𝑃_pre(𝑇,𝑓₁)−∫𝑓₁d𝜇, which implies 𝑃_pre𝑇,𝑓₁+1−𝑝𝑓₂−𝑓₁d𝜇=𝑝𝑃_pre𝑇,𝑓₁+1−𝑝𝑃_pre𝑇,𝑓₂⩾𝑃_pre𝑇,𝑝𝑓₁+1−𝑝𝑓₂. (2.25)

A member 𝜇∈ℳ(𝑋,𝑇) is called an equilibrium state for 𝑃_pre(𝑇,•) at 𝑓 if𝑃_pre𝑇,𝑓=ℎ_pre,𝜇(𝑇)+𝑓d𝜇. (2.26) Let ℳ_𝑓(𝑋,𝑇) denote the collection of all equilibrium states for 𝑃_pre(𝑇,•) at 𝑓.

The set 𝒯_𝑓(𝑋,𝑇) is convex and compact in the weak*-topology. The set ℳ_𝑓(𝑋,𝑇) is convex but it may be not closed in the weak*-topology. Note that ℳ_𝑓(𝑋,𝑇)⊆𝒯_𝑓(𝑋,𝑇)⊆ℳ(𝑋,𝑇) and ℳ_𝑓(𝑋,𝑇) could be empty (see Example 5.1, [25]). We also have ℳ_𝑓(𝑋,𝑇)=𝒯_𝑓(𝑋,𝑇) if and only if the metric pre-image entropy map ℎ_pre,•(𝑇) is upper semicontinuous at every element of 𝒯_𝑓(𝑋,𝑇), Theorem  5.2 [25]. The extreme points of ℳ_𝑓(𝑋,𝑇) are precisely the ergodic members of ℳ_𝑓(𝑋,𝑇) and if 𝜇∈ℳ_𝑓(𝑋,𝑇), then almost every measure in the ergodic decomposition of 𝜇 is a member of ℳ_𝑓(𝑋,𝑇) (see Proposition 2.1, [26]). When the metric pre-image entropy map ℎ_pre,•(𝑇) is upper semicontinuous on ℳ(𝑋,𝑇), then ⋃_{𝑓∈𝐶(𝑋,ℝ)}ℳ_𝑓(𝑋,𝑇) is dense in ℳ(𝑋,𝑇) in the norm topology, and given any finite collection of ergodic measures {𝜇₁,…,𝜇_𝑛}, there is some 𝑓∈𝐶(𝑋,ℝ) such that {𝜇₁,…,𝜇_𝑛}⊆ℳ_𝑓(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∈𝒯_𝑓(𝑋,𝑇)⧵ℳ_𝑓(𝑋,𝑇); (2)ℎ_pre,𝜇(𝑇)+∫𝑓d𝜇<𝑃_pre(𝑇,𝑓) and there exist {𝜇_𝑛}^∞_𝑛=1⊆ℳ(𝑋,𝑇) with 𝜇_𝑛→𝜇 and ℎ_pre,𝜇𝑛(𝑇)+∫𝑓d𝜇_𝑛→𝑃_pre(𝑇,𝑓);(3)𝜇∈𝒯_𝑓(𝑋,𝑇) and ℎ_pre,•(𝑇) is not upper semicontinuous at 𝜇.

Proof. (1)⇒(2)Follows from the variational principle and Theorem 2.3(1).(2)⇒(3)By Theorem 2.3(1), 𝜇∈𝒯_𝑓(𝑋,𝑇). If ℎ_pre,•(𝑇) is upper semicontinuous at 𝜇, then𝑃_pre𝑇,𝑓=lim_𝑛→∞ℎ_pre,𝜇𝑛(𝑇)+𝑓d𝜇_𝑛⩽ℎ_pre,𝜇(𝑇)+𝑓d𝜇. (2.27) Hence, 𝜇∈ℳ_𝑓(𝑋,𝑇) by the variational principle.(3)⇒(1)If (3) holds, then there are 𝜇_𝑛∈ℳ(𝑋,𝑇) with 𝜇_𝑛→𝜇 and lim_𝑛→∞ℎ_pre,𝜇𝑛(𝑇)>ℎ_pre,𝜇(𝑇). Hence, 𝑃_pre𝑇,𝑓⩾lim_𝑛→∞ℎ_pre,𝜇𝑛(𝑇)+𝑓d𝜇_𝑛>ℎ_pre,𝜇(𝑇)+𝑓d𝜇. (2.28) Therefore, 𝜇∈𝒯_𝑓(𝑋,𝑇)⧵ℳ_𝑓(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 ℎ_pre,•(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.

Given 𝑓,𝑔∈𝐶(𝑋,ℝ). Since 𝑃_pre(𝑇,•) is convex, the map 𝑡↦(𝑃_pre(𝑇,𝑓+𝑡𝑔)−𝑃_pre(𝑇,𝑓))/𝑡 is increasing and hence𝑑⁺𝑃_pre𝑇,𝑓𝑔=lim_𝑡→0+𝑃_pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓𝑡,𝑑⁻𝑃_pre𝑇,𝑓𝑔=lim_𝑡→0−𝑃_pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓𝑡 (3.1) exist. Note that 𝑑⁺𝑃_pre(𝑇,𝑓)(𝑔)=−𝑑⁻𝑃_pre(𝑇,𝑓)(−𝑔). The pre-image pressure function 𝑃_pre(𝑇,•) is said to be Gateaux differentiable at 𝑓 if, for all 𝑔∈𝐶(𝑋,ℝ),lim_𝑡→0𝑃_pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓𝑡 (3.2) exist. It is easy to check that 𝑃_pre(𝑇,•) is Gateaux differentiable at 𝑓 if and only if 𝑔↦𝑑⁺𝑃_pre(𝑇,𝑓)(𝑔) is linear.

Lemma 3.1. Let (𝑋,𝑇) be a TDS and 𝑓,𝑔∈𝐶(𝑋,ℝ). Then, 𝑑⁺𝑃_pre𝑇,𝑓𝑔=sup𝑔d𝜇∶𝜇∈𝒯_𝑓(𝑋,𝑇). (3.3)

Proof. If 𝜇∈𝒯_𝑓(𝑋,𝑇), then, for 𝑔∈𝐶(𝑋,ℝ), 𝑃𝑔d𝜇⩽^pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓𝑡,∀𝑡>0. (3.4) Hence, sup𝑔d𝜇∶𝜇∈𝒯_𝑓(𝑋,𝑇)⩽𝑑⁺𝑃_pre𝑇,𝑓𝑔. (3.5) Next, we prove the converse inequality. Set 𝑎=𝑑⁺𝑃_pre(𝑇,𝑓)(𝑔). Define a continuous linear functional 𝛾∶{𝑡𝑔∶𝑡∈ℝ}→ℝ by 𝛾𝑡𝑔=𝑡𝑎,𝑡∈ℝ. (3.6) The convexity of 𝑃_pre(𝑇,•) implies 𝛾𝑡𝑔=𝑡⋅𝑑⁺𝑃_pre𝑇,𝑓𝑔⩽𝑃_pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓,∀𝑡∈ℝ. (3.7) By the Hahn-Banach theorem, 𝛾 can be extended to a continuous linear functional on 𝐶(𝑋,ℝ) such that 𝛾(ℎ)⩽𝑃_pre𝑇,𝑓+ℎ−𝑃_pre𝑇,𝑓,∀ℎ∈𝐶(𝑋,ℝ). (3.8) By the Riesz representation theorem, there is 𝜇∈ℳ(𝑋) with 𝛾(ℎ)=ℎd𝜇,∀𝑔∈𝐶(𝑋,ℝ). (3.9) Combining (3.6), (3.8), and (3.9), we have 𝜇∈𝒯_𝑓(𝑋,𝑇), and 𝑔d𝜇=𝛾𝑔=𝑎=𝑑⁺𝑃_pre𝑇,𝑓𝑔. (3.10) 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 function 𝑃_pre(𝑇,•) is Gateaux differentiable at 𝑓∈𝐶(𝑋,ℝ);(2)the unique tangent functional to 𝑃_pre(𝑇,•) at 𝑓 is 𝜇;(3)for each 𝑔∈𝐶(𝑋,ℝ),lim_𝑡→0𝑃_pre𝑇,𝑓+𝑡𝑔−𝑃_pre𝑇,𝑓𝑡=𝑔d𝜇. (3.11)

Proof. (1)⇒(2)If the pre-image pressure function 𝑃_pre(𝑇,•) is Gateaux differentiable at 𝑓, then the function 𝑔↦𝑑⁺𝑃_pre(𝑇,𝑓)(𝑔) is linear. By Lemma 3.1,sup𝑔d𝜇∶𝜇∈𝒯_𝑓(𝑋,𝑇)=𝑑⁺𝑃_pre𝑇,𝑓g=−𝑑⁺𝑃_pre𝑇,𝑓−𝑔=−sup−𝑔d𝜇∶𝜇∈𝒯_𝑓(𝑋,𝑇)=inf𝑔d𝜇∶𝜇∈𝒯_𝑓(𝑋,𝑇) (3.12) for each 𝑔∈𝐶(𝑋,ℝ). This implies there is a unique tangent functional to 𝑃_pre(𝑇,•) at 𝑓.
(2)⇒(3)It directly follows from Lemma 3.1.(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 function 𝑃_pre(𝑇,•) is said to be Fréchet differentiable at 𝑓 if there is 𝛾∈𝐶(𝑋,ℝ)^∗ such thatlim_𝑔→0𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓−𝛾𝑔𝑔=0. (4.1) The pre-image pressure function 𝑃_pre(𝑇,•) is said to be Fréchet differentiable if it is Fréchet differentiable at each 𝑓∈𝐶(𝑋,ℝ).

Note that if 𝑃_pre(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 𝑃_pre(𝑇,•) at 𝑓.

Theorem 4.1. The following conditions are mutually equivalent:(1)𝑃_pre(𝑇,•) if Fréchet differentiable at 𝑓;(2)𝒯_𝑓(𝑋,𝑇)={𝜇_𝑓} and ‖𝜇_𝑛−𝜇_𝑓‖→0 for each {𝜇_𝑛}⊆ℳ(𝑋,𝑇) with ℎ_pre,𝜇𝑛(𝑇)+∫𝑓d𝜇_𝑛→𝑃_pre(𝑇,𝑓);(3)𝑃_pre(𝑇,•) is locally affine at 𝑓;(4)𝒯_𝑓(𝑋,𝑇)={𝜇_𝑓} and lim_𝑔→0sup𝜇−𝜇_𝑓∶𝜇∈𝒯_𝑓+𝑔(𝑋,𝑇)=0. (4.2)

Proof. (1)⇒(2)Suppose 𝑃_pre(𝑇,•) is Fréchet differentiable at 𝑓. Then, 𝑓 has a unique tangent functional 𝜇_𝑓 to 𝑃_pre(𝑇,•) at 𝑓. Let {𝜇_𝑛}^∞_𝑛=1⊆ℳ(𝑋,𝑇) with ℎ_pre,𝜇𝑛(𝑇)+∫𝑓d𝜇_𝑛→𝑃_pre(𝑇,𝑓). Given 𝜖>0, there are 𝑁∈ℕ and 𝛿>0 such that𝑃_pre𝑇,𝑓⩽ℎ_pre,𝜇𝑛(𝑇)+𝑓d𝜇_𝑛+𝜖𝛿,∀𝑛⩾𝑁,0⩽𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓−𝑔d𝜇_𝑓⩽𝜖⋅𝑔whenever𝑔<𝛿. (4.3)(4.4) Hence, if 𝑛⩾𝑁 and ‖𝑔‖<𝛿, then 𝑔d𝜇_𝑛−𝑔d𝜇=𝑃_pre𝑇,𝑓+𝑔d𝜇_𝑛−𝑃_pre𝑇,𝑓−𝑔d𝜇_𝑓⩽ℎ_pre,𝜇𝑛(𝑇)+𝑓+𝑔d𝜇_𝑛+𝜖𝛿−𝑃_pre𝑇,𝑓−𝑔d𝜇_𝑓by4.4⩽𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓−𝑔d𝜇_𝑓𝑔.+𝜖𝛿byTheorem2.1⩽𝜖⋅+𝛿⩽2𝜖𝛿by4.4 (4.5) Note that (4.5) is also true when −𝑔 instead of 𝑔. So, 𝑔d𝜇_𝑛−𝑔d𝜇⩽2𝜖𝛿whenever𝑔<𝛿,𝑛⩾𝑁. (4.6) Therefore, 𝜇_𝑛−𝜇_𝑓=sup𝑔d𝜇_𝑛−𝑔d𝜇_𝑓∶𝑔=1<1𝛿sup𝑔d𝜇_𝑛−𝑔d𝜇_𝑓∶𝑔.<𝛿⩽2𝜖by4.6 (4.7) By arbitrary of 𝜖, ‖𝜇_𝑛−𝜇_𝑓‖→0.(2)⇒(3)By the variational principle of pre-image pressure, we can pick ergodic measures 𝜇_𝑛 such that ℎ_pre,𝜇𝑛(𝑇)+∫𝑓d𝜇_𝑛→𝑃_pre(𝑇,𝑓), Then ‖𝜇_𝑛−𝜇_𝑓‖→0. Note that two distinct ergodic measures have norm-distance 2. So there is 𝑁 such that 𝜇_𝑛=𝜇_𝑓 for all 𝑛⩾𝑁. Hence, 𝜇_𝑓∈ℳ^𝑒(𝑋,𝑇) and𝜖∶=𝑃_pre𝑇,𝑓−supℎ_pre,𝜇(𝑇)+𝑓d𝜇∶𝜇∈ℳ^𝑒(𝑋,𝑇),𝜇≠𝜇_𝑓>0. (4.8) This implies for each 𝑔∈𝐶(𝑋,ℝ), with ‖𝑓−𝑔‖<𝜖/2, that supℎ_pre,𝜇(𝑇)+𝑔d𝜇∶𝜇∈ℳ^𝑒(𝑋,𝑇),𝜇≠𝜇_𝑓⩽𝑃_pre𝑇,𝑓−𝜖+𝑓−𝑔⩽𝑃_pre𝑇,𝑔−𝜖+2𝑓−𝑔<𝑃_pre𝑇,𝑔. (4.9) By the variational principle of pre-image pressure again, we have 𝑃_pre𝑇,𝑔=ℎ_pre,𝜇𝑓(𝑇)+𝑔d𝜇_𝑓<𝜖whenever𝑓−𝑔2. (4.10) Hence, 𝑃_pre(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)Let 𝑔∈𝐶(𝑋,ℝ) and 𝜇∈𝒯_𝑓+𝑔(𝑋,𝑇). By definition, we have𝑃_pre𝑇,𝑓−𝑃_pre𝑇,𝑓+𝑔⩾−𝑔d𝜇. (4.11) Hence, 0⩽𝑃_pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓−𝑔d𝜇_𝑓⩽𝑔d𝜇−𝑔d𝜇_𝑓()⩽𝑔⋅by4.11𝜇−𝜇_𝑓. (4.12) Therefore, 𝑃0⩽^pre𝑇,𝑓+𝑔−𝑃_pre𝑇,𝑓−∫𝑔d𝜇_𝑓𝑔⩽sup𝜇−𝜇_𝑓∶𝜇∈𝒯_𝑓+𝑔(𝑋,𝑇)⟶0 (4.13) as 𝑔→0. That is 𝑃_pre(𝑇,•) is Fréchet differentiable at 𝑓.

Corollary 4.2. Let (𝑋,𝑇) be a TDS with finite pre-image entropy. Then, 𝑃_pre(𝑇,•) is Fréchet differentiable if and only if 𝑇 is uniquely ergodic.

Proof. Using Theorem 4.1, 𝑃_pre(𝑇,•) is locally affine whenever 𝑃_pre(𝑇,•) is Fréchet differentiable. Hence, the map 𝑓∈𝐶(𝑋,ℝ)↦𝜇_𝑓∈ℳ(𝑋,𝑇) is locally constant, where 𝒯_𝑓(𝑋,𝑇)={𝜇_𝑓} for each 𝑓∈𝐶(𝑋,ℝ). Since 𝐶(𝑋,ℝ) is connected, the map is constant. So 𝜇_𝑓=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 𝑃_pre𝑇,𝑘𝑓⩾ℎ_pre,𝜇(𝑇)+𝑘𝑓d𝜇>ℎ_pre,𝜇0(𝑇)+𝑘𝑓d𝜇₀=𝑃_pre𝑇,𝑘𝑓, (4.14) which is a contradiction. Therefore, ℳ(𝑋,𝑇)={𝜇₀}.

Remark 4.3. In the situation of Corollary 4.2, there is only one invariant measure, and the pre-image pressure is the expectation with respect to this measure, hence, it is linear.

Acknowledgments

The authors would like to thank the referee for many valuable and constructive comments and suggestions for improving this paper. The first and second authors are supported by the National Natural Science Foundation of China (no. 11161029) and the Department of Education Research Foundation of Guangxi Province (nos. 200911MS298, 200103YB157); the third author is supported by the National Natural Science Foundation of China (no. 11126342) and the Natural Science Foundation of Guangxi Province (nos. 0897012, 2010GXNSFA013109).