A convex analysis approach to entropy functions, variational principles and equilibrium states
aa r X i v : . [ m a t h . D S ] N ov A CONVEX ANALYSIS APPROACH TO ENTROPY FUNCTIONS,VARIATIONAL PRINCIPLES AND EQUILIBRIUM STATES
ANDRZEJ BI´S, MARIA CARVALHO, MIGUEL MENDES, AND PAULO VARANDAS*
Abstract.
The first aims of this work are to endorse the advent of finitely additive set func-tions as equilibrium states and to explore the possibility to replace the metric entropy by anupper semi-continuous map associated to a general variational principle. More precisely, usingmethods from Convex Analysis, we construct for each generalized convex pressure function anupper semi-continuous entropy-like map (which, in the context of continuous transformationsacting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy), then establish a new abstractvariational principle and prove that equilibrium states, possibly finitely additive, always exist.This conceptual approach provides a new insight on dynamical systems without a measure withmaximal entropy, prompts the study of finitely additive ground states for non-uniformly hyper-bolic maps and grants the existence of finitely additive Lyapunov equilibrium states for singularvalue potentials generated by linear cocycles over continuous maps. We further investigateseveral applications, including a new thermodynamic formalism for systems driven by finitelygenerated semigroup or countable sofic group actions. On the final pages of the manuscript weprovide a list of open problems in a wide range of topics suggested by our main results.
Contents
Part 1. INTRODUCTION AND STATEMENTS
21. Introduction 22. Main results 62.1. An abstract variational principle 72.2. Finitely additive equilibrium states 82.3. Differentiability of the pressure 92.4. Applications 10
Part 2. ADDITIVE MEASURES AND CONVEX ANALYSIS
Part 3. THERMODYNAMIC FORMALISM FOR INDIVIDUAL DYNAMICS
Date : November 11, 2020.2010
Mathematics Subject Classification.
Primary: 37D20, 37A25; Secondary: 37C10.
Key words and phrases.
Finitely additive set function; Pressure; Variational principle; Equilibrium state;Transfer operator; Semigroup action; Amenable group; Sofic group; Upper Carath´eodory structure; Sub-additivesequence; Linear cocycle; Lyapunov exponent; Singular value potential; Phase transition; Nonlinear thermody-namic formalism; Sequential dynamical systems.A. Bi´s, Faculty of Mathematics, L´od´z University, Poland, [email protected]. M. Carvalho, CMUP& Departamento de Matem´atica, Faculdade de Ciˆencias, Universidade do Porto, rua do Campo Alegre s/n, 4169–007 Porto, Portugal [email protected]. M. Mendes, CMUP & Faculdade de Engenharia da Universidade doPorto, Portugal. P. Varandas, CMUP-Portugal & Departamento de Matem´atica, Universidade Federal da Bahia,Salvador, Brazil.* Corresponding author: [email protected], [email protected]. h and h ∗ : Proof of Corollary 6 247.4. Example without a measure with maximal entropy 267.5. Pressure derived from Ruelle-Perron-Frobenius transfer operators 287.6. Finitely additive equilibrium states and second order phase transition 29 Part 4. THERMODYNAMICS FOR NON-ADDITIVE SEQUENCES
Part 5. THERMODYNAMIC FORMALISM FOR SEMIGROUP ACTIONS P is a pressure function 4310.3. An alternative variational principle for finitely generated group actions 4511. Countable sofic group actions 4611.1. Pressure function for countable sofic group actions 4611.2. A variational principle for countable sofic group actions 4711.3. Sofic equilibrium states 4711.4. Countable amenable group actions 4812. Uncountable groups with a reference probability measure 4812.1. Pressure function 4912.2. A variational principle 49 Part 6. FINAL REMARKS
Part INTRODUCTION AND STATEMENTS Introduction
Classical thermodynamic formalism . The modern theory of Dynamical Systems has its ori-gins in the end of the nineteenth century with the pioneering work of Poincar´e who attempted tocompletely describe the solutions of the differential equations modeling the three body problemin Celestial Mechanics. It is in the course of this investigation that Poincar´e encounters theintriguing phenomenon of what was later named homoclinic tangency. Several decades afterthis first appearance of such complex dynamical behavior a common opinion had grown amongresearchers that geometric methods were insufficient to fully describe the asymptotic behaviorof dynamical systems in general. In the late 1950’s, the groundbreaking inspiration, put forwardby Kolmogorov, of bringing both Probability and Entropy Theories into the realm of DynamicalSystems Theory, faced intrinsic difficulties mainly due to the fact that the latter theory wasnot yet established as an independent domain of research. Notice for instance that, although
NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 3 some of the ideas from Statistical Mechanics (e.g., Boltzman ergodic hypothesis) dated fromthe previous century, the main theoretical results of the time, both von Neumann and Birkhoffergodic theorems, had just been published in the early 1930’s.The theory of uniformly hyperbolic dynamical systems, essentially born in the sixties, buildingover the existence of invariant foliations with exponential contracting and expanding behavior,brought forward both the models and the axiomatic framework which granted some under-standing of the homoclinic behavior and the realization of Kolmogorov’s insight. Meanwhile,the paradigmatic example of Smale’s horseshoe made clear that subshifts of finite type couldcodify the hyperbolic dynamical systems, and that such coding was possible due to the existenceof finite partitions with a Markov property. Actually, the finite Markov partitions for hyperbolicdynamical systems (both diffeomorphisms and flows) had a further benefit: they brought thethermodynamic formalism from Statistical Mechanics to Dynamical Systems, through keystonecontributions by Sinai, Ruelle, Bowen, Ratner and Walters [22, 25, 103, 109, 114, 118], amongothers. We refer the reader to Keller’s monograph [74] for a comprehensive survey on the relationbetween some dynamical systems and physical models from Statistical Physics.The search for a thermodynamic formalism for dynamical systems aims to prove the existenceof invariant probability measures which maximize the topological pressure, besides reporting ontheir statistical properties. We recall that the pressure generalizes the notion of topologicalentropy by way of weighting the orbits with a fixed potential map. Such measures, calledequilibrium states, are often Gibbs measures, and include as specific examples both probabilitymeasures absolutely continuous with respect to Lebesgue and measures of maximal entropy. Foruniformly hyperbolic diffeomorphisms and flows, when restricted to a basic piece of the non-wandering set, equilibrium states exist and are unique for every H¨older continuous potential (cf.[23, 108, 114]). In the case of diffeomorphisms, the basic strategy to prove this remarkable factwas to (semi)conjugate the dynamics to a subshift of finite type, via a finite Markov partition.Finer properties, including the analiticity of the pressure map and the relation between pressure,periodic orbits and dynamical zeta functions, were later considered by Parry and Pollicott [91].
Non-uniform hyperbolicity and phase transitions . The study of dynamical systems withweaker forms of hyperbolicity, such as partial hyperbolicity and dominated splittings, is still un-der development (cf. [21]). In particular, an extension of the thermodynamic formalism beyondthe scope of uniform hyperbolicity has been facing several difficulties. A very fruitful strategyto overcome some of them is the construction of induced and tower dynamics with hyperbolicbehavior (see e.g. [97, 111]). These induced maps can be well described by countable shifts,for which the thermodynamic formalism is more or less comprehended [112], but introducestwo new hindrances. On the one hand, not all invariant probability measures (even with non-zero Lyapunov exponents) may be lifted to the induced tower dynamics. On the other hand,equilibrium states for the tower dynamics may induce probability measures that are σ -finiteon the phase space, but this transfer process depends on the integrability of the return timefunction. While in the case of C ∞ -surface diffeomorphisms with positive entropy these problemshave been overcome by Buzzi, Crovisier and Sarig [34], who proved that there are finitely manymeasures of maximal entropy, the previous two issues are among the reasons why the theory ofthermodynamic formalism for non uniformly hyperbolic dynamical systems remains incompleteeven for partially hyperbolic systems.It is in this context that phase transitions (characterized either by the non-analiticity ofthe pressure map or by a discontinuity on the number of its equilibrium states) have been A. BI´S, M. CARVALHO, M. MENDES, AND P. VARANDAS thoroughly studied. Phase transitions are reasonably well understood in the one-dimensionalcontext (see [43, 44, 45, 68, 100, 99] and references therein) due to the recent tools to detect andcharacterize the sources of the non-hyperbolic behavior. Observe that, beyond hyperbolicity,even the existence of equilibrium states is far from being established, though the non-uniformhyperbolicity may be enough in special contexts, as happens in the case of Sinai billiard maps (cf.[8]). Moreover, Newhouse described in [88] sufficient conditions for the entropy map to be uppersemi-continuous; in particular, C ∞ maps always have equilibrium states for every continuouspotential. In addition, a codification mecanism by symbolic extensions, and its existence forsmooth dynamics, have also been explored by Downarowicz and Newhouse in [48]. Nonetheless,one can ask under what general conditions do equilibrium states exist, seeing that there are C r -surface diffeomorphisms, 1 r < + ∞ , having no measure with maximal entropy (cf. [33]). Oneof the main goals in this paper is to prove that they always exist if one drops the requirementthat they are countably additive. The intervention of Convex Analysis . One of the fundamental mathematical tools usedin Thermodynamics, Statistical Physics and Stochastic Finance is convexity. For instance,regarding the theory of lattice gases, Israel observes in [65] that the pressure exists in thethermodynamic limit as a convex function in the space of interactions, and uses the very fruitfulpioneer work of Bishop and Phelps on Convex Analysis [18] to construct equilibrium states, toprove that these satisfy Dobrushin-Lanford-Ruelle condition and to describe phase transitions.The aforesaid paper [18] is also a cornerstone for several recent applications in Economics, namelyon equilibrium theory, risk measures and stochastic finance (cf. [2, 56, 57] and references therein).More precisely, the seminal work [5] by Artzner et al. establishes an axiomatic framework withinwhich it is possible to prove a robust representation for what these authors call coherent riskmeasures. Subsequently, F¨ollmer and Schied [56] extended this theorem to the case when riskmeasures are convex instead of subadditive. It turns out that this result may be interpreted underthe light of thermodynamic formalism as an abstract variational principle, that is, independentof the underlying dynamics, provided we replace the roles of convex risk measures and penaltyfunctions by those of pressure and entropy maps, respectively.An important application of the Convex Analysis methods in the thermodynamic formalismis due to Israel and Phelps [66] which, inspired by mathematical models in the classical theoryof lattice gases [109], investigated the differentiability, the tangent functionals and the varia-tional entropy (defined by means of the Legendre-Frenchel transform) in the case of generalizedpressure functions acting on the space of affine real-valued continuous functions whose domainis a compact convex set. Apart from the natural convexity and continuity assumptions, and thefact that the pressure function acts on a set of affine maps, such pressure maps were assumedto satisfy also a strong positivity condition (cf. [66, p.136]), a hypothesis which seldom occursin the thermodynamic formalism context.Inspired by [66, 83], recent formulations of variational principles using Convex Analysis haveappeared in [39, 42, 61], extending [66] to the dynamical context of shifts on Borel standardspaces and dynamical systems whose transfer operators have a spectral gap. Since the usualKolmogorov-Sinai metric entropy is not adequate to the context of shifts on Borel standardspaces, these authors define a notion of entropy (which they also call variational entropy) formeasures that are eigenvectors of the dual of a normalized transfer operator, and prove, as avariational principle, that it coincides with the Legendre transform of the spectral radius ofa suitable transfer operator. Similar variational principles have been obtained by Antonevich,
NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 5
Bakhtin and Lebedev [4, 9, 10] in the context of weighted shift spaces acting on continuousor L -functions. In both situations, it may happen that the pressure function, defined as thelogarithm of the spectral radius of the corresponding transfer operators, even if well adaptedto the transfer operator, is hardly related to the classical notion of topological pressure fordynamical systems.It is worth noticing that there is a common feature in the previously mentioned works, namelythe advent of finitely additive equilibrium states in the thermodynamic formalism beyond thehyperbolic setting. Such a class of set functions contains the classical equilibrium states and canbe used to detect phase transitions, as we will clarify later (cf. [38, 39] and references therein). Semigroup actions . The thermodynamic formalism for semigroup actions is linked to geomet-ric objects in quotient dynamics, fractal and iterated function systems or foliation theory, justto mention a few. We illustrate this connection with a particularly important example. Ghys,Langevin and Walczak introduced in [58] the notion of geometric entropy for a foliation andthe concept of topological entropy for any pseudogroup induced by the foliation. The geometricentropy of a codimension-one foliation F of a compact manifold M is related to geometry of M .For instance, if the geometric entropy vanishes then so does the Godbillon-Vey class of F . Onthe other hand, if the geometric entropy of F is positive then there exists a so called resilientleaf, a counterpart of a horseshoe for classical dynamical systems. Moreover, any foliation withvanishing geometric entropy admits a nontrivial transverse invariant measure. Regarding thetopological entropy of the pseudogroup, if it vanishes, a condition which holds for instance if thepseudogroup has polynomial growth, then there exists a probability measure which is invariantby all the pseudogroup elements. However, typical finitely generated semigroups are expectedto be free (see [59] for the case of circle homeomorphisms) and to carry no common invariantmeasures. In particular, a main difficulty in setting up a unified thermodynamic formalism forsemigroup actions is precisely that, though the notion of pressure should match the kind ofgroup under study, a general concept of measure theoretical entropy ought to make no referenceto common invariant probability measures.Motivated by multifractal analysis, in order to describe the complexity of non-compact subsetsof finitely generated semigroup actions one can define topological pressure by means of theCarath´eodory structures developed in [94, Chapter 4]. These structures, which appear alsoin the definition of Hausdorff or box-counting dimensions, have been used in several contextsof semigroup actions (see [17, 121, 123]). While the topological objects are well understood,their measure theoretical counterpart and variational principle have not yet been successfullyaccomplished.Another relevant concept is the notion of sofic group introduced by Gromov in [62], includingthe countable amenable groups as specific examples. In [26], Bowen was the first to propose theconcept of entropy for the action of a countable sofic group on a standard probability space.In rough terms it measures the exponential growth of the complexity seen through a measure,and does not require invariance whatsoever. Later, a topological entropy for this kind of actionswas defined by Kerr and Li in [75], where the authors also established a variational principleunder quite general assumptions. More recently, a notion of sofic pressure and a correspondingvariational principle have been introduced in [40]. These are quite general concepts, whichgeneralize the entropy and pressure for countable amenable group actions. In future sections,we will brief the reader on these concepts, referring to [28] for a complete survey on sofic entropy. A. BI´S, M. CARVALHO, M. MENDES, AND P. VARANDAS
As far as we know, there are few classes of sofic group actions for which equilibrium states areknown to exist.Finally, we also mention continuous actions on a compact metric space by a group endowedwith a probability measure. In this case, groups may not be finitely generated, and someexamples include Lie group endowed with the Haar measure or random walks on groups. Wenote that the known notion of pressure in this setting complies with the assumptions of our firstmain result.
Our main contributions . The main source of inspiration for this paper stems from recentapplications of Convex Analysis to risk measures in Stochastic Finance [56], where variationalprinciples have been established under great generality. More precisely, we address the follow-ing problem: to find an abstract variational principle, valid for real-valued convex, monotoneand translation invariant operators defined on a suitable Banach space of potentials, which ispowerful enough to be applied to either the classical topological pressure map associated to asingle dynamics or to the more recently defined notions of pressure for semigroup actions. Sucha principle might require the intervention of an upper semi-continuous entropy-like map actingon the space of finitely additive normalized set functions, instead of the standard metric en-tropy for σ -additive invariant probability measures. But it should not only recover the classicalvariational principle for the topological pressure of continuous potentials but also improve thethermodynamic formalism of dynamics which do not admit equilibrium states (e.g. [33, 85]) orelse allow us to address more general families of potentials.We start by proving the existence of such a general variational principle and how it providesfinitely additive equilibrium states. Afterwards we pay particular attention to the consequencesof this unifying approach to dynamical systems determined by either a single dynamics acting ona compact metric space or a countable sofic group action. These are settings where there existprobability measures which are invariant by all generators. We do not confine to these contexts,though. The thermodynamic formalism we develop applies to finitely generated semigroup ac-tions of continuous maps (comprising groups with exponential growth), or to actions generatedby hyperbolic or Lie groups endowed with a random walk or the Haar measure, respectively, aswell as to non-additive sequences of continuous potentials and the search for Lyapunov equilib-rium states with respect to singular value potentials associated to linear cocycles over continuousmaps. 2. Main results
Let (
X, d ) be a metric space and let B stand for the σ -algebra of Borel subsets of X . Denoteby B a Banach space over the field R equal to either B d ( X ) = (cid:8) ϕ : X → R | ϕ is measurable and bounded (cid:9) (2.1)or C b ( X ) = (cid:8) ϕ ∈ B d ( X ) | ϕ is continuous (cid:9) or else C c ( X ) = (cid:8) ϕ ∈ C b ( X ) | ϕ has compact support (cid:9) endowed with the norm k ϕ k ∞ = sup x ∈ X | ϕ ( x ) | . In what follows, P a ( X ) will stand for the setof normalized finitely additive set functions on B , which we will simply call finitely additiveprobability measures , with the total variation norm (cf. [49, IV.2.15]). For future use, denote by P ( X ) ⊂ P a ( X ) the set of Borel σ -additive probability measures on X endowed with the weak ∗ topology, and by C ( X ) the space of real valued continuous maps whose domain is X . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 7
Definition 2.2.
A function Γ : B → R is called a pressure function if it satisfies the followingconditions:(C ) Monotonicity : ϕ ψ ⇒ Γ( ϕ ) Γ( ψ ) ∀ ϕ, ψ ∈ B .(C ) Translation invariance : Γ( ϕ + c ) = Γ( ϕ ) + c ∀ ϕ ∈ B ∀ c ∈ R .(C ) Convexity : Γ( t ϕ + (1 − t ) ψ ) t Γ( ϕ ) + (1 − t ) Γ( ψ ) ∀ ϕ, ψ ∈ B ∀ t ∈ [0 , ) and (C ) imply that any pressure function is Lipschitz continu-ous. Indeed, for every ϕ, ψ ∈ B ,Γ( ψ ) − k ϕ − ψ k ∞ = Γ( ψ − k ϕ − ψ k ∞ ) Γ( ϕ ) Γ( ψ + k ϕ − ψ k ∞ ) = Γ( ψ ) + k ϕ − ψ k ∞ and therefore, | Γ( ϕ ) − Γ( ψ ) | k ϕ − ψ k ∞ . An abstract variational principle.
Our first main result, inspired by [57], establishesa general variational principle and the existence of finitely additive equilibrium states for anypressure function.
Theorem 1.
Let
Γ : B → R be a pressure function. Then Γ( ϕ ) = max µ ∈ P a ( X ) (cid:26) h ( µ ) + Z ϕ dµ (cid:27) ∀ ϕ ∈ B (2.3) where h ( µ ) = inf ϕ ∈ A Γ (cid:26)Z ϕ dµ (cid:27) and A Γ = (cid:8) ϕ ∈ B : Γ( − ϕ ) (cid:9) . (2.4) Moreover, h ( µ ) is affine and upper semi-continuous; and if α : P a ( X ) → [0 , + ∞ ] is anotherfunction for which (2.3) holds, then α h . In addition, one has h ( µ ) = inf ϕ ∈ B (cid:26) Γ( ϕ ) − Z ϕ dµ (cid:27) ∀ µ ∈ P a ( X ) . If X is locally compact and B = C c ( X ) then the maximum in (2.3) is attained in P ( X ) . Some comments are in order. Firstly, we stress that we do not require compactness of thespace X just local compactness when considering B = C c ( X ) due to use of the Riesz-MarkovTheorem, and that the supremum in (2.3) can be computed taking only extremal measures(cf. (5.3)). Secondly, expression (2.4) asserts that the function h is computed by averagingpotentials whose additive inverses have non-positive pressure. Yet, for any pressure functionΓ satisfying the normalization condition Γ(0) = 0 (otherwise, Γ := Γ − Γ(0) is a normalizedpressure function due to translation invariance) one has − Γ( ψ ) Γ( − ψ ) for every ψ ∈ B as adirect consequence of convexity. In particular n ϕ : Γ( − ϕ ) o ⊆ n ϕ : Γ( ϕ ) > o and so inf { ϕ : Γ( ϕ ) > } (cid:26)Z ϕ dµ (cid:27) h ( µ ) . Note also that h ( µ ) > ⇔ Γ( ϕ ) > Z ϕ dµ ∀ µ ∈ P a ( X ) A. BI´S, M. CARVALHO, M. MENDES, AND P. VARANDAS a condition that in some dynamical contexts characterizes the invariance of probability measures(cf. [118, Theorem 9.11]). Observe, moreover, that any pressure function Γ : B → R not onlydetermines the convex set A Γ as it can be reconstructed from it through the following equalityΓ( ϕ ) = inf n a ∈ R : a − ϕ ∈ A Γ o (see [56]). Finally, we remark that the function h depends on both the pressure function Γ andits domain B , i.e., h = h Γ , B . Let us detail this issue a bit further in the case of a compact metricspace X in which case C b ( X ) = C c ( X ) = C ( X ). If Γ : B d ( X ) → R is a pressure function thenΓ | C : C ( X ) → R is a pressure function as well, though with a different domain, and accordingto (2.4) the corresponding functions h are computed respectively by h B d ( X ) ( µ ) = inf (cid:26)Z ϕ dµ : ϕ ∈ B d ( X ) and Γ( − ϕ ) (cid:27) ∀ µ ∈ P a ( X )and h C ( X ) ( µ ) = inf (cid:26)Z ϕ dµ : ϕ ∈ C ( X ) and Γ( − ϕ ) (cid:27) ∀ µ ∈ P ( X ) . In particular, h B d ( X ) ( µ ) h C ( X ) ( µ ) for every µ ∈ P ( X ).2.2. Finitely additive equilibrium states.
The variational principle stated in Theorem 1ensures that there always exist normalized finitely additive measures for which the right-handside of (2.3) attains the supremum; that is, the set E ϕ (Γ) = n µ ∈ P a ( X ) : Γ( ϕ ) = h ( µ ) + Z ϕ dµ o (2.5)is non-empty. This raises the subtle question of whether they are unique. To address thisissue we consider functionals tangent to the pressure in our abstract framework, inspired by thesimilar notion introduced in [119] in the context of topological pressure and metric entropy.Recall that if X is locally compact and B = C ( X ), it follows from the Riesz-Markov repre-sentation Theorem that the dual of B can be identified with the collection of all finite signedmeasures on ( X, B ) equipped with the weak ∗ topology, and that the subset of its positive nor-malized elements corresponds to the space P ( X ) (cf. [78, pp. 253]). Whenever B = C b ( X ) or B = B d ( X ), an extension of the Riesz-Markov Theorem informs that the dual of B is representedby the space of Borel finitely additive measures with the topology induced by the total variationnorm (see [55, 64]), whose subset of positive normalized elements corresponds to P a ( X ). Definition 2.6.
Consider a pressure function Γ : B → R and a potential ϕ ∈ B . We say that µ ∈ P a ( X ) is a tangent functional to Γ at ϕ (also known as sub-differential ) ifΓ( ϕ + ψ ) − Γ( ϕ ) > Z ψ dµ ∀ ψ ∈ B . (2.7)We denote by T ϕ (Γ) the set of tangent functionals to Γ at ϕ .The continuity of Γ and the Hahn-Banach Theorem guarantee that T ϕ (Γ) = ∅ for every ϕ ;and it is easily seen to be a convex and weak ∗ compact set. The next result states that the spaceof tangent functionals to Γ at ϕ ∈ B coincides with the space of finitely additive probabilitymeasures attaining the maximum on (2.3). NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 9
Theorem 2.
Let
Γ : B → R be a pressure function. Then E ϕ (Γ) = T ϕ (Γ) ∀ ϕ ∈ B . Moreover, if B = C b ( X ) or B = C c ( X ) , then there exists a residual subset R ⊂ B such that E ϕ (Γ) = 1 for every ϕ ∈ R . Differentiability of the pressure.
It is known that, in the setting of the topologicalpressure function, the uniqueness of equilibrium states is tied in with the differentiability of thepressure (cf. [119]). Within the more abstract setting we are addressing, we have the followinggeneralization.
Definition 2.8.
A pressure function Γ : B → R is locally affine at ϕ ∈ B if there exist aneighborhood V of 0 in B and (a unique) µ ϕ ∈ P a ( X ) such thatΓ( ϕ + ψ ) − Γ( ϕ ) = Z ψ dµ ϕ ∀ ψ ∈ V . (2.9)In particular, T ϕ (Γ) = { µ ϕ } . If Γ is locally affine at all elements of B then µ ϕ does not dependon ϕ , and Γ is said to be affine .The local affine property seems to be stronger than the Fr´echet differentiability of the pressurefunction at ϕ ∈ B since Γ : B → R is Fr´echet differentiable at ϕ ∈ B if there exists µ ϕ ∈ P a ( X )(again unique) such that lim ψ → k ψ k ∞ (cid:12)(cid:12)(cid:12) Γ( ϕ + ψ ) − Γ( ϕ ) − Z ψ dµ ϕ (cid:12)(cid:12)(cid:12) = 0 . (2.10)However, [119, Theorem 6] reveals that the local affine property is equivalent to Fr´echet dif-ferentiability when one considers the classical topological pressure function for an individualdynamics. We will show that the analogous statement for general pressure functions is also true.Recall that the total variation norm in P a ( X ) is given by k µ − ν k = sup (cid:26)(cid:12)(cid:12)(cid:12) Z ψ dµ − Z ψ dν (cid:12)(cid:12)(cid:12) : ψ ∈ B and k ψ k ∞ (cid:27) . Theorem 3.
Let
Γ : B → R be a pressure function. The following assertions are equivalent: (a) Γ is locally affine at ϕ . (b) There exists a unique tangent functional in T ϕ (Γ) and lim ψ → sup n k µ − µ ϕ k : µ ∈ T ϕ + ψ (Γ) o = 0 . (c) Γ is Fr´echet differentiable at ϕ .Therefore, the following statements are mutually equivalent as well: (¯ a ) Γ is affine. (¯ b ) S ϕ ∈ B T ϕ (Γ) is a singleton. (¯ c ) Γ is everywhere Fr´echet differentiable. As being affine is a rigid condition, the previous theorem also conveys the information that apressure function Γ is rarely everywhere Fr´echet differentiable. Consequently, typical pressurefunctions either exhibit more than one tangent functional at some element of B , or these do notvary continuously in the operator norm (see [47] for examples in the context of a single dynamics).We refer the reader to [60, pp. 147-148] for more information regarding the differentiation ofconvex functions.The previous discussion prompts us to consider the weaker notion of Gateaux differentiability.A pressure function Γ : B → R is said to be Gateaux differentiable at ϕ ∈ B if, for every ψ ∈ B the directional pressure map t ∈ R Γ( ϕ + tψ ) is differentiable. In other words, it is requiredthat the limit d Γ( ϕ )( ψ ) := lim t → t [ Γ( ϕ + tψ ) − Γ( ϕ ) ]exists and is finite for every ψ ∈ B . Concerning real valued convex functions on Banach spaces,Walters proved in [119, Corollary 2] a criterium for Gateaux differentiability. In our setting, thecorresponding statement reads as follows: Corollary 4.
Let
Γ : B → R be a pressure function. Then Γ is Gateaux differentiable at ϕ ifand only if there exists a unique tangent functional in T ϕ (Γ) . Applications.
In the remainder of this section we will list a few consequences of theprevious results on the thermodynamic formalism of semigroup actions. We start with thespecial case of a single continuous dynamical system, for which Theorem 1 provides a newvariational principle for the topological pressure.2.4.1.
Continuous maps . Given a continuous map f : X → X on a compact metric space X , denote by P f ( X ) the space of f -invariant Borel probability measures on X with the weak ∗ topology. Corollary 5.
Let f : X → X be a continuous transformation of a compact metric space X with h top ( f ) < + ∞ . Then there exists a map h f : P ( X ) → R satisfying h f ( µ ) = inf ϕ ∈ C ( X ) (cid:26) P top ( f, ϕ ) − Z ϕ dµ (cid:27) ∀ µ ∈ P ( X ) (2.11) and such that P top ( f, ϕ ) = max µ ∈ P ( X ) n h f ( µ ) + Z ϕ dµ o = max µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) ∀ ϕ ∈ C ( X ) . Moreover, every measure µ ∈ P a ( X ) which attains the maximum is f -invariant and h f ( µ ) > . When h top ( f ) < + ∞ and the entropy map µ ∈ P f ( X ) h µ ( f ) is upper semi-continuous (ashappens, for instance, when f is expansive), from both the equality (2.11) and [118, Theorem9.12] one deduces that h f ( µ ) = h µ ( f ) ∀ µ ∈ P f ( X ) . Thus, under these assumptions, the map h f : P ( X ) → R is an extension of the Kolmogorov-Sinaimetric entropy and max µ ∈ P f ( X ) h µ ( f ) = h top ( f ) = max µ ∈ P f ( X ) h f ( µ ) . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 11
Star-entropy . The lack of upper semi-continuity of the entropy map has led some au-thors to regularize the notion of metric entropy. For instance, given µ ∈ P f ( X ), the concept of star-entropy was introduced in [88] and later explored by Viana and Yang in [117], being definedby h ∗ µ ( f ) = sup n lim sup n → + ∞ h µ n ( f ) | ( µ n ) n ∈ N is a sequence in P f ( X ) with lim n → + ∞ µ n = µ o . It is known that, for every µ ∈ P f ( X ), one has h µ ( f ) h ∗ µ ( f ) h µ ( f ) + h loc ( f ) (2.12)where h loc ( f ) stands for the local entropy of f , defined by h loc ( f ) = lim ε → + lim δ → + lim sup n → + ∞ n sup x ∈ X log s n ( f, δ, B fn ( x, ε ))and s n ( f, δ, B fn ( x, ε )) denotes the maximal cardinality of an ( n, δ )-separated subset of B fn ( x, ε ).The first inequality of (2.12) is a straightforward consequence of the definition of h ∗ , while thesecond was proved in [88] (see also [32, Appendix B]), from which we conclude that h loc ( f )bounds the defect in upper semi-continuity of the map µ ∈ P f ( X ) h µ ( f ). The star-entropyfunction h ∗ : P f ( X ) → R is related to the entropy structures of Boyle and Downarowicz (cf. [29])since h ∗ is precisely the upper semi-continuous envelope of the metric entropy. Therefore (see[48, p. 466–467]) h ∗ µ ( f ) = inf n T ( µ ) : T : P f ( X ) → R is continuous and T ( µ ) > h µ ( f ) ∀ µ ∈ P f ( X ) o (2.13)and so h ∗ µ ( f ) is bounded from above by T = topological entropy. The advantage of consideringthe star-entropy is that the function µ ∈ P f ( X ) h ∗ µ ( f ) is upper semi-continuous when P f ( X )is endowed with the weak ∗ -topology. In particular, defining the star-pressure by P ∗ top ( f, ϕ ) = sup µ ∈ P f ( X ) (cid:26) h ∗ µ ( f ) + Z ϕ dµ (cid:27) one guarantees that there always exists an f -invariant probability measure which attains thesupremum. Meanwhile, this equality could be a drawback since the star-entropy and the star-pressure functions may differ substantially from the usual metric entropy and topological pres-sure (we refer the reader to [110] for examples of smooth maps without an invariant probabilitymeasure with maximal entropy), and so the corresponding maximum values and maximal mea-sures might fail to describe standard physical quantities. In general, however, we have thefollowing properties. Corollary 6.
Let f : X → X be a continuous transformation acting on a compact metric space X with h top ( f ) < + ∞ . Then: (a) P top ( f, ϕ ) = max µ ∈ P f ( X ) n h ∗ f ( µ ) + R ϕ dµ o ∀ ϕ ∈ C ( X ) . (b) h µ ( f ) h ∗ µ ( f ) = h f ( µ ) ∀ µ ∈ P f ( X ) . (c) h f ( µ ) = inf n T ( µ ) : T : P f ( X ) → R is continuous and T ( µ ) > h µ ( f ) ∀ µ ∈ P f ( X ) o . Ergodic optimization . The previous results also pave the way to the description ofmultifractal analysis for Birkhoff averages, large deviations or ergodic optimization in both hy-perbolic and non-hyperbolic contexts. Actually, while upper semi-continuity of entropy anduniqueness of equilibrium states are useful ingredients to provide a full description of the en-tropy map, the dimension of the level sets associated to Birkhoff averages and the maximizingprobability measures, these properties may fail beyond the realm of uniform hyperbolicity. Inthe sequel, we give a simple illustration through an application in ergodic optimization beyonduniform hyperbolicity.Given ϕ ∈ C ( X ), the map µ ∈ P f ( X ) R ϕ dµ is continuous and defined on a compactmetric space. Hence it has a maximum, which is realized by f -invariant probability measures.These are referred to as ϕ -maximizing probability measures , and may be not unique. It istherefore useful to be able to describe those which are most chaotic, meaning those which carrylarger metric entropy. It is known that the ϕ -maximizing probability measures obtained throughzero-temperature limits have this property (cf. [69, Theorem 4.1]). More precisely, on the onehand, if h top ( f ) < + ∞ , then1 t P top ( f, tϕ ) = sup ν ∈ P f ( X ) n t h ν ( f ) + Z ϕ dν o t → + ∞ −→ sup ν ∈ P f ( X ) Z ϕ dν = max ν ∈ P f ( X ) Z ϕ dν. On the other hand, if for each t > µ t for f with respect to tϕ , then any weak ∗ accumulation probability measure µ ∈ P f ( X ) of ( µ t ) t > as t goes to + ∞ is a ϕ -maximizing probability measure. Notice that there are non-uniformlyexpanding maps for which the entropy function is not upper semi-continuous, and for which onecan only ensure that tϕ has an equilibrium state for small values of the parameter t (see e.g.[115]). Yet, if h top ( f ) < + ∞ , then one can use Corollaries 5 and 6 to obtain1 t P top ( f, tϕ ) = max ν ∈ P f ( X ) n t h f ( ν ) + Z ϕ dν o t → + ∞ −→ max ν ∈ P f ( X ) Z ϕ dν and also conclude that, for every t >
0, there exists an f -invariant probability measure ν t ∈P f ( X ) such that P top ( f, tϕ ) = h f ( ν t ) + Z t ϕ dν t . (2.14)Altogether, this shows that, as t goes to + ∞ , any weak ∗ accumulation point ν ∞ ∈ P f ( X ) of( ν t ) t > is an f -invariant probability measure satisfying Z ϕ dν ∞ = max µ ∈ P f ( X ) Z ϕ dµ. In particular, this proves that f -invariant ϕ -maximizing probability measures obtained throughzero-temperature limits always exist; denote by M f ( X, ϕ ) its union set.Taking into account that h µ ( f ) h f ( µ ) for every µ in P f ( X ), the previous discussion togetherwith [69, Theorem 4.1] (replacing the usual entropy function by the upper semi-continuous map µ h f ( µ )) yield the following result. Corollary 7.
Let f : X → X be a continuous map on a compact metric space X such that h top ( f ) < + ∞ , and consider ϕ ∈ C ( X ) . For each t > , let ν t ∈ P a ( X ) be an f -invariantprobability measure satisfying (2.14) . Then, as t goes to + ∞ , any weak ∗ accumulation point ν ∞ ∈ P f ( X ) of ( ν t ) t > satisfies: NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 13 (a) R ϕ dν ∞ = max µ ∈ P f ( X ) R ϕ dµ . (b) h f ( ν ∞ ) = max ν ∈ M f ( X, ϕ ) h f ( ν ) > max ν ∈ M f ( X, ϕ ) h ν ( f ) . (c) lim t → + ∞ h f ( ν t ) = h f ( ν ∞ ) . Sub-additive sequences of potentials . On Section 4, we will provide a new thermo-dynamic formalism for sub-additive sequences of continuous potentials, which yields the con-struction of finitely additive equilibrium states for linear cocycles over continuous maps withrespect to generalized singular value potentials, and in particular, of finitely additive groundstates for non-uniformly hyperbolic maps.2.4.5.
Free semigroup actions . Let us now analyze the case of more general finitely generatedsemigroup actions. Consider a compact metric space (
X, d ) and a semigroup ( G, ◦ ) of continuousendomorphisms of X , where the semigroup operation ◦ is the composition of maps. Assumethat G is finitely generated, that is, there exists a finite set G = { id X , g , · · · , g p } ⊂ G suchthat G = S + ∞ n =1 G n , where, for each n ∈ N , G n = n g j ◦ · · · ◦ g j n : g j i ∈ G ∀ i n o . Each element g of G n may be seen as a word which originates from the concatenation of n elements in G . Yet, different concatenations may generate the same element in G . Whenconsidering free semigroup actions, we will regard the different concatenations instead of theelements in G they create. One way to interpret this statement is to define the itinerary map ι : F p → G n ⊂ Gj . . . j n g j n ◦ · · · ◦ g j where F p is the free semigroup with p generators, thus addressing concatenations on G as imagesby ι of finite words on F p . Thereby, each x ∈ X is endowed by the pair ( G, G ) with infinitelymany path-orbits, whose union is precisely the action of ( G, G ) on X .Among the several known definitions of topological entropy proposed for this context weshall focus on the notion considered by Ghys, Langevin and Walczak in [58], which we denoteby h top ( G, G ). We will define a corresponding pressure P top ( G, G , · ) and prove that it is apressure function (cf. Section 9.1, where more information will be disclosed). As far as we know,the next result establishes the first variational principle for this notion of topological entropyfor a semigroup action, whose existence has been discussed in [15]. Corollary 8.
Let ( G, G ) be a finitely generated free semigroup with h top ( G, G ) < + ∞ . Thenthere exists an upper semi-continuous function h G : P a ( X ) → R such that P top ( G, G , ϕ ) = max µ ∈ P a ( X ) (cid:26) h G ( µ ) + Z ϕ dµ (cid:27) . In particular, h top ( G, G ) = max µ ∈ P a ( X ) h G ( µ ) , so there is a finitely additive probability thatattains the maximum. Other motivating settings . On the last sections of this work we present a new ther-modynamic formalism for countable sofic group actions and uncountable group actions with areference measure. To ease the readability of this final part of the text, and cope with so manydifferent settings, we postpone this information till we reach Section 9.
Part ADDITIVE MEASURES AND CONVEX ANALYSIS A variational principle: Proof of Theorem 1
For the sake of completeness and rigor, we include a proof of the first part of Theorem 1 alongthe lines of that provided by F¨ollmer and Schied (see [56] and [57]). Let (
X, d ) be a metricspace, B = B d ( X ) and Γ : B → R be a pressure function. Define A Γ = (cid:8) ϕ ∈ B : Γ( − ϕ ) (cid:9) and h ( µ ) := inf ϕ ∈ A Γ (cid:26)Z ϕ dµ (cid:27) . We start showing that, for every ϕ ∈ B , one hasΓ( ϕ ) > sup µ ∈ P a ( X ) (cid:26) h ( µ ) + Z ϕ dµ (cid:27) . (3.1)Due to translation invariance it is clear that Γ( ϕ − Γ( ϕ )) = 0. Thus e ϕ := Γ( ϕ ) − ϕ belongs to A Γ . Therefore, for every µ ∈ P a ( X ) h ( µ ) Z e ϕ dµ = Γ( ϕ ) − Z ϕ dµ which implies (3.1). Conversely, given ϕ ∈ B we need to find µ ϕ ∈ P a ( X ) such thatΓ( ϕ ) h ( µ ϕ ) + Z ϕ dµ ϕ . (3.2)Yet, it is enough to do it for ϕ such that Γ( ϕ ) = 0 since the general case follows from taking ϕ − Γ( ϕ ) and the translation invariance property of Γ. So, consider ϕ ∈ B such that Γ( ϕ ) = 0.Thus the observable − ϕ does not belong to the set B Γ = { ψ ∈ B : Γ( − ψ ) < } which is convex and open due to the convexity and continuity of Γ. Therefore, by the geometricalversion of the Hahn-Banach Theorem there is a continuous, not identically zero, linear functional L : B → R which separates the sets {− ϕ } and B Γ in the sense that L ( − ϕ ) inf ψ ∈ B Γ L ( ψ ) . (3.3)By linearity of L , this is equivalent to saying that L ( ϕ ) + inf ψ ∈ B Γ L ( ψ ) > . (3.4) Lemma 3.5. L is positive and L (1) > .Proof. Consider ψ ∈ B such that ψ >
0. Firstly, let us see that for every λ > λψ + c ∈ B Γ , where c = Γ(0) + 1. Indeed, by translation invariance and monotonicity of Γ onehas Γ( − λψ − c ) = Γ( − λψ ) − c Γ(0) − c < . Due to (3.3), this in turn implies that L ( − ϕ ) L ( λψ + c ) = λL ( ψ ) + cL (1) ∀ λ > . Thus, if L ( ψ ) < L ( − ϕ ) = −∞ , leading to a contradiction with the fact that L is abounded functional. This proves that L is a positive functional. In particular, L (1) > NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 15
Let us now prove that L (1) = 0. As L is linear and not identically zero, we may take ψ ∈ B such that L ( ψ ) > k ψ k ∞ < . Write ψ = ψ +0 − ψ − , where ψ +0 = max { ψ , } and ψ − = max {− ψ , } . Then, as L is positive, we have L ( ψ +0 ) = L ( ψ ) + L ( ψ − ) > L ( ψ ) > L (1 − ψ +0 ) > − ψ +0 >
0. Using again both the linearity and the monotonicity of L we finally concludethat L (1) = L (1 − ψ +0 ) + L ( ψ +0 ) > L ( ψ +0 ) > . (cid:3) The previous lemma indicates that the continuous linear operator LL (1) is positive and nor-malized. Therefore, according to an extension of the Riesz-Markov representation theorem [49],there is a finitely additive probability measure µ ϕ ∈ P a ( X ) (which belongs to P ( X ) if B = C c ( X )and X is locally compact) such that Z ψ dµ ϕ = L ( ψ ) L (1) ∀ ψ ∈ B . We are left to show that µ ϕ satisfies (3.2). Observe that for every ψ ∈ A Γ and every ε > ψ + ε ∈ B Γ . In other words, A Γ , ε := { ψ + ε : ψ ∈ A Γ } ⊂ B Γ . Hence h ( µ ϕ ) = inf ψ ∈ A Γ Z ψ dµ ϕ inf ψ ∈ B Γ Z ψ dµ ϕ inf ψ ∈ A Γ , ε Z ψ dµ ϕ = inf ψ ∈ A Γ Z ψ dµ ϕ + ε. Since ε > h ( µ ϕ ) = inf ψ ∈ B Γ R ψ dµ ϕ . Consequently, h ( µ ϕ ) + Z ϕ dµ ϕ = (cid:18) inf ψ ∈ B Γ Z ψ dµ ϕ (cid:19) + L ( ϕ ) L (1) = (cid:18) inf ψ ∈ B Γ L ( ψ ) L (1) (cid:19) + L ( ϕ ) L (1)= 1 L (1) (cid:18) L ( ϕ ) + inf ψ ∈ B Γ L ( ψ ) (cid:19) > ϕ )where we have used relation (3.4) in the last step. This completes the proof of (3.2).Endowing P a ( X ) with the total variation topology, the function h is upper semi-continuoussince it is defined as infimum of the family of continuous functions (cid:0) µ ∈ P a ( X ) R ϕ dµ (cid:1) ϕ ∈ A Γ (cf. [7, 22]).We proceed showing the maximality (hence uniqueness) of the function h among those whichsatisfy (2.3). Let α be such a map. Then,Γ( − ψ ) > α ( µ ) + Z − ψ dµ ∀ ψ ∈ B ∀ µ ∈ P a ( X )or, equivalently, α ( µ ) Γ( − ψ ) + Z ψ dµ ∀ ψ ∈ B ∀ µ ∈ P a ( X )which implies that α ( µ ) inf ψ ∈ B (cid:26) Γ( − ψ ) + Z ψ dµ (cid:27) ∀ µ ∈ P a ( X ) . (3.6) Moreover, as A Γ ⊂ B and Γ( − ψ ) ψ ∈ A Γ we conclude that α ( µ ) inf ψ ∈ B (cid:26) Γ( − ψ ) + Z ψ dµ (cid:27) inf ψ ∈ A Γ (cid:26) Γ( − ψ ) + Z ψ dµ (cid:27) inf ψ ∈ A Γ (cid:26)Z ψ dµ (cid:27) = h ( µ ) ∀ µ ∈ P a ( X ) . The previous reasoning using α = h allows us to conclude that h ( µ ) = inf ψ ∈ B (cid:26) Γ( − ψ ) + Z ψ dµ (cid:27) ∀ µ ∈ P a ( X ) . (3.7)As B is a vector space the equality (3.7) can be rewritten as h ( µ ) = inf ψ ∈ B (cid:26) Γ( − ψ ) − Z − ψ dµ (cid:27) = inf ψ ∈ B (cid:26) Γ( ψ ) − Z ψ dµ (cid:27) . We are left to prove that h is affine. The convex set P a ( X ) is compact with respect to theweak ∗ topology (cf. [49, Theorem 2, V.4.2]) and so, by the Krein-Milman Theorem, it is theclosed convex hull of its extreme points. Moreover, we can use the Choquet RepresentationTheorem (cf. [6, Theorem 6.6] or [118, p.153]) to express each member of P a ( X ) in terms ofthe extreme elements of P a ( X ). More precisely, if E a ( X ) denotes the set of extreme points of P a ( X ) and µ belongs to P a ( X ) then there is a unique measure P µ on the Borel subsets of thecompact metrizable space P a ( X ) such that P µ ( E a ( X )) = 1 and Z X ψ ( x ) dµ ( x ) = Z E a ( X ) (cid:18)Z X ψ ( x ) dm ( x ) (cid:19) d P µ ( m ) ∀ ψ ∈ B d ( X ) . (3.8)Hence every µ ∈ P a ( X ) is a generalized convex combination of extreme finitely additive prob-ability measures. We write µ = R E a ( X ) m d P µ ( m ) and call this equality the decomposition inextremes of the finitely additive probability measure µ . Lemma 3.9.
Given µ ∈ P a ( X ) whose decomposition in extremes is µ = R E a ( X ) m d P µ ( m ) , then h ( µ ) = Z h ( m ) d P µ ( m ) . In particular, the function h is affine.Proof. Recall that h ( µ ) = inf ψ ∈ A Γ R ψ dµ and the map ψ ∈ B d ( X ) R ψ dµ is continuous.By (3.8) there exists a probability measure P µ giving full weight to the space E a ( X ) of extrememeasures of P a ( X ) and satisfying R ψ dµ = R E a ( X ) (cid:0)R ψ dm (cid:1) d P µ ( m ). In particular, h ( µ ) = inf ψ ∈ A Γ Z E a ( X ) (cid:18)Z ψ ( x ) dm ( x ) (cid:19) d P µ ( m ) . Moreover, as P µ is countably additive, the Monotone Convergence Theorem (cf. [104, TheoremIV.15, Vol. I]) for nets of continuous maps when applied to the net ( R ψ dµ ) ψ ∈ A Γ implies that h ( µ ) = inf ψ ∈ A Γ Z E a ( X ) (cid:18)Z ψ ( x ) dm ( x ) (cid:19) d P µ ( m ) = Z E a ( X ) (cid:18) inf ψ ∈ A Γ Z ψ ( x ) dm ( x ) (cid:19) d P µ ( m )= Z E a ( X ) h ( m ) d P µ ( m ) . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 17
Finally, by Choquet theorem, given µ , µ ∈ P a ( X ) there exist unique probability measures P µ , P µ giving full weight to E a ( X ) and such that µ i = R E a ( X ) m d P µ i ( m ) for i = 1 ,
2. In particular,for each 0 < α <
1, one has αµ + (1 − α ) µ = Z E a ( X ) m d [ α d P µ + (1 − α ) d P µ ]( m )and, using the first part of the lemma, one gets h ( αµ + (1 − α ) µ ) = Z E a ( X ) h ( m ) d [ α P µ + (1 − α ) P µ ]( m ) = α h ( µ ) + (1 − α ) h ( µ ) . This ends the proof of Theorem 1. (cid:3) Tangent functionals: Proof of Theorem 2
The argument we will present follows closely the one in [118, Theorems 9.14 and 9.15]. Con-sider ϕ ∈ B and assume that µ ∈ E ϕ (Γ). Then, by Theorem 1,Γ( ϕ + ψ ) − Γ( ϕ ) > h ( µ ) + Z ( ϕ + ψ ) dµ − h ( µ ) − Z ϕ dµ = Z ψ dµ ∀ ψ ∈ B . This shows that E ϕ (Γ) ⊆ T ϕ (Γ) . To establish the converse inclusion, fix µ ∈ T ϕ (Γ) and note thatΓ( ϕ + ψ ) − Γ( ϕ ) > Z ψ dµ ∀ ψ ∈ B m Γ( ϕ + ψ ) − Z ( ϕ + ψ ) dµ > Γ( ϕ ) − Z ϕ dµ ∀ ψ ∈ B . This equivalence together with the variational principle in Theorem 1, the fact that B is a vectorspace and equation (3.7) imply that h ( µ ) = inf ψ ∈ B n Γ( ϕ + ψ ) − Z ( ϕ + ψ ) dµ o > Γ( ϕ ) − Z ϕ dµ. Since the reverse inequality h ( µ ) Γ( ϕ ) − Z ϕ dµ is an immediate consequence of (2.3), we conclude that T ϕ (Γ) ⊆ E ϕ (Γ). The second claim in thestatement of Theorem 2 is a consequence of [84] (see also [96, page 12]), which ensures that theconvex function Γ, acting on the separable Banach space B = C b ( X ) or B = C c ( X ), admits aunique tangent functional for every ϕ in a residual subset of B .5. Fr´echet differentiability: Proof of Theorem 3
In this section we will show the characterization of Fr´echet differentiability of the pressurefunctional in terms of tangent functionals.( a ) ⇒ ( b ). As Γ is locally affine at ϕ , there exist (a unique) µ ϕ ∈ P a ( X ) (respectively, a signedprobability measure if X is locally compact and B = C c ( X )) such that, for every ψ , ψ ∈ B whose norms are small enough, one hasΓ( ϕ + ψ ) − Γ( ϕ + ψ ) = Z ( ψ − ψ ) µ ϕ . This implies that T ϕ + ψ (Γ) = { µ ϕ } for every ψ ∈ B with small enough norm. Thus,lim ψ → sup n k µ − µ ϕ k : µ ∈ T ϕ + ψ (Γ) o = lim ψ → { } = 0 . ( b ) ⇒ ( c ). Assume that there exists a unique tangent functional µ ϕ ∈ T ϕ (Γ) and thatlim ψ → sup n k µ − µ ϕ k : µ ∈ T ϕ + ψ (Γ) o = 0 . (5.1)As µ ϕ ∈ T ϕ (Γ), one has Γ( ϕ + ψ ) − Γ( ϕ ) > Z ψ dµ ϕ ∀ ψ ∈ B . Moreover, the uniqueness of the tangent functional µ ϕ at ϕ and Theorem 2 imply thatΓ( ϕ ) = h ( µ ϕ ) + Z ϕ dµ ϕ and Γ( ϕ ) > h ( µ ) + Z ϕ dµ ∀ µ = µ ϕ . So, given ψ ∈ B and µ ∈ T ϕ + ψ (Γ) = E ϕ + ψ (Γ), one gets0 Γ( ϕ + ψ ) − Γ( ϕ ) − Z ψ dµ ϕ = h ( µ ) + Z ( ϕ + ψ ) dµ − Γ( ϕ ) − Z ψ dµ ϕ h ( µ ) + Z ( ϕ + ψ ) dµ − h ( µ ) − Z ϕ dµ − Z ψ dµ ϕ = Z ψ dµ − Z ψ dµ ϕ k ψ k ∞ k µ − µ ϕ k . Therefore, by assumption (5.1), one has lim ψ → k ψ k ∞ (cid:12)(cid:12) Γ( ϕ + ψ ) − Γ( ϕ ) − R ψ dµ ϕ (cid:12)(cid:12) = 0.( c ) ⇒ ( a ). The map h is affine, but the pressure function Γ is not associated to an underly-ing dynamics. Therefore, part of the argument to prove Theorem 6 of [119] (the analogue ofTheorem 3 for the topological pressure) has to be adapted to the general setting we are dealingwith. Assume that Γ is Fr´echet differentiable at ϕ ∈ B . Then, as Γ is convex, there is a uniquetangent functional to Γ at ϕ (cf. [105, Chapt. IV, § T ϕ (Γ) = { µ ϕ } , and µ ϕ satisfieslim ψ → k ψ k ∞ (cid:12)(cid:12)(cid:12) Γ( ϕ + ψ ) − Γ( ϕ ) − Z ψ dµ ϕ (cid:12)(cid:12)(cid:12) = 0 . By (2.7), one also has Γ( ϕ + ψ ) − Γ( ϕ ) > Z ψ dµ ϕ ∀ ψ ∈ B . We are left to prove the reverse inequality for ψ inside a neighborhood of 0 in B . Lemma 5.2.
Let ( µ n ) n ∈ N be a sequence in P a ( X ) such that lim n → + ∞ h ( µ n ) + R ϕ dµ n = Γ( ϕ ) .Then lim n → + ∞ k µ n − µ ϕ k = 0 . Proof.
Given ε >
0, take δ > ψ ∈ B , k ψ k ∞ δ ⇒ Γ( ϕ + ψ ) − Γ( ϕ ) − Z ψ dµ ϕ ε k ψ k ∞ . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 19
For each n ∈ N consider ε n = Γ( ϕ ) − h ( µ n ) − R ϕ dµ n , and let N ∈ N be such that for every n > N one has 0 ε n < εδ . Therefore, if k ψ k ∞ δ and n > N , Z ψ dµ n − Z ψ dµ ϕ = Γ( ϕ ) + Z ψ dµ n − Γ( ϕ ) − Z ψ dµ ϕ = h ( µ n ) + Z ϕ dµ n + ε n + Z ψ dµ n − Γ( ϕ ) − Z ψ dµ ϕ = h ( µ n ) + Z ( ϕ + ψ ) dµ n + ε n − Γ( ϕ ) − Z ψ dµ ϕ Γ( ϕ + ψ ) + ε n − Γ( ϕ ) − Z ψ dµ ϕ = Γ( ϕ + ψ ) − Γ( ϕ ) − Z ψ dµ ϕ + ε n < δε. Since these estimates are also valid for − ψ , one has (cid:12)(cid:12)(cid:12) R ψ dµ n − R ψ dµ ϕ (cid:12)(cid:12)(cid:12) < δε for every k ψ k ∞ δ and n > N . Thus, if n > N , k µ n − µ ϕ k = sup (cid:26)(cid:12)(cid:12)(cid:12) Z ψ dµ n − Z ψ dµ ϕ (cid:12)(cid:12)(cid:12) : k ψ k ∞ (cid:27) = 1 δ sup (cid:26)(cid:12)(cid:12)(cid:12) Z ψ dµ n − Z ψ dµ ϕ (cid:12)(cid:12)(cid:12) : k ψ k ∞ δ (cid:27) < δ δε = 2 ε. (cid:3) Recall that, by Lemma 3.9, one has h ( µ ) = R h ( m ) d P µ ( m ) for every µ ∈ P a ( X ) whosedecomposition in extreme points is µ = R E a ( X ) m d P µ ( m ). Thus,Γ( ϕ ) = sup n h ( µ ) + Z ϕ dµ | µ ∈ E a ( X ) o . (5.3) Lemma 5.4.
The tangent functional µ ϕ is an extreme point of P a ( X ) and Γ( ϕ ) > sup n h ( µ ) + Z ϕ dµ | µ ∈ E a ( X ) and µ = µ ϕ o . Proof.
Using (5.3), one can choose a sequence ( µ n ) n ∈ N of extreme finitely additive probabilitymeasures with lim n →∞ (cid:0) h ( µ n ) + Z ϕ dµ n (cid:1) = Γ( ϕ ) . Then, by Lemma 5.2, one has lim n → + ∞ k µ n − µ ϕ k = 0 . Since distinct extreme points in P a ( X )have norm distance equal to 2 (cf. [6]), there is N ∈ N such that µ n = µ N for every n > N .Therefore, µ N = µ ϕ , so µ ϕ is an extreme point of P a ( X ). In addition, observe that, since T ϕ (Γ) = E ϕ (Γ) = { µ ϕ } , the previous argument also shows that one cannot haveΓ( ϕ ) = sup n h ( µ ) + Z ϕ dµ | µ ∈ E a ( X ) and µ = µ ϕ o . Thus, Γ( ϕ ) > sup n h ( µ ) + R ϕ dµ | µ ∈ E a ( X ) and µ = µ ϕ o . (cid:3) Lemma 5.5.
There is a neighborhood U of µ ϕ in the total variation norm such that h ( µ ϕ ) > sup n h ( µ ) | µ ∈ U , µ ∈ E a ( X ) and µ = µ ϕ o . Proof.
Let a = Γ( ϕ ) − sup n h ( µ ) + R ϕ dµ | µ ∈ E a ( X ) and µ = µ ϕ o and U the neighborhoodof µ ϕ given by U = n µ ∈ P a ( X ) : (cid:12)(cid:12)(cid:12) Z ϕ dµ − Z ϕ dµ ϕ (cid:12)(cid:12)(cid:12) < a/ o . If µ ∈ U is an extreme of P a ( X ) and µ = µ ϕ , then h ( µ ) h ( µ ) + Z ϕ dµ − Z ϕ dµ ϕ + a Γ( ϕ ) − a − Z ϕ dµ ϕ + a
2= Γ( ϕ ) − Z ϕ dµ ϕ − a h ( µ ϕ ) − a T ϕ (Γ) = E ϕ (Γ) = { µ ϕ } . (cid:3) We are finally ready to show that Γ is locally affine at ϕ . Let a be as in the proof of Lemma 5.5.Then, for every ψ ∈ B d ( X ) satisfying k ψ − ϕ k ∞ < a/
2, one hassup n h ( µ ) + Z ψ dµ | µ ∈ E a ( X ) and µ = µ ϕ o Γ( ϕ ) − a + k ψ − ϕ k ∞ Γ( ψ ) − a + 2 k ψ − ϕ k ∞ < Γ( ψ ) . Thus, by (5.3), all such maps ψ which are a/ ϕ have µ ϕ as unique extreme Γ-equilibriumstate in P a ( X ). In particular, k ψ − ϕ k ∞ < a/ ⇒ Γ( ψ ) = h ( µ ϕ ) + Z ψ dµ ϕ . So Γ is locally affine at ϕ . This ends the proof of the first part of Theorem 3.Regarding the second list of equivalent assertions stated in Theorem 3, firstly assume that Γis everywhere Fr´echet differentiable. The previous equivalent conditions imply that Γ is locallyaffine at every ϕ ∈ B , so, by the connectedness of the vector space B , we conclude that Γ isaffine. Conversely, affine functions are clearly Fr´echet differentiable. Hence items (¯ a ) and (¯ b )are equivalent. Assume now that Γ is affine. Then there exists µ ∈ P a ( X ) such that for every ϕ ∈ B one has Γ( ϕ + ψ ) − Γ( ϕ ) = Z ψ dµ ∀ ψ ∈ B. Thus µ is a tangent functional to every ϕ ∈ B . As any element in P a ( X ) is determined by itsintegrals over B , the previous equality implies that µ is the unique tangent functional at every ϕ ∈ B . So, S ϕ ∈ B T ϕ (Γ) = { µ } , and (¯ a ) implies (¯ c ). Finally, condition (¯ c ) implies (¯ a ) due tothe corresponding local property ( c ) ⇒ ( a ). This completes the proof of Theorem 3. Part THERMODYNAMIC FORMALISM FOR INDIVIDUAL DYNAMICS Entropy and pressure
We start defining the topological and free energy for one dynamics, along with the recollectionof some of its properties.
NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 21
Topological entropy.
Let f : X → X be a continuous transformation of a compact metricspace ( X, d ). Given n ∈ N , define the dynamical distance d n : X × X → [0 , + ∞ ) by d n ( x, z ) = max n d ( x, z ) , d ( f ( x ) , f ( z )) , . . . , d ( f n ( x ) , f n ( z )) o which generates the same topology as d . For every x ∈ X , n ∈ N and ε >
0, denote by B fn ( x, ε )the ball centered at x with radius ε for the distance d n , that is, B fn ( x, ε ) = n y ∈ X : d ( f i ( x ) , f i ( y )) < ε ∀ i n o . Having fixed n ∈ N and ε >
0, we say that a set E ⊂ X is ( n, ε )–separated by f if d n ( x, z ) > ε ∀ x = z ∈ E. Denote by s n ( f, ε ) the maximal cardinality of all ( n, ε )–separated subsets of X by f . Due tothe compactness of X , the number s n ( f, ε ) is finite. The topological entropy of f is defined by h top ( f ) = lim ε → + lim sup n → + ∞ n log s n ( f, ε ) . Topological pressure.
More generally, given a continuous map ϕ : X → R (also called apotential), the topological pressure of f and ϕ is defined by P top ( f, ϕ ) = lim ε → + lim sup n → + ∞ n log P n ( f, ϕ, ε ) (6.1)where, for every n ∈ N , P n ( f, ϕ, ε ) = sup E n X x ∈ E e S fn ϕ ( x ) : E ⊂ X is ( n, ε )-separated o (6.2)and S fn ϕ ( x ) = ϕ ( x ) + ϕ ( f ( x )) + · · · + ϕ ( f n ( x )) . This way, one assigns to each point x ∈ X the weight e S fn ϕ ( x ) determined by the potential ϕ along the block of the first n iterates of f at x . In particular, P top ( f,
0) = h top ( f ).As X is compact, C ( X ) is a subspace of B d ( X ) where k . k ∞ is the norm of the uniformconvergence. The pressure map P top ( f, . ) : C ( X ) → R ∪ { + ∞} satisfies, for every ϕ, ψ ∈ C ( X ) and constant c ∈ R , the following properties [118]:(1) ϕ ψ ⇒ P top ( f, ϕ ) P top ( f, ψ ) . (2) h top ( f ) + min ϕ P top ( f, ϕ ) h top ( f ) + max ϕ. (3) P top ( f, . ) is either finite valued or constantly + ∞ .(4) If P top ( f, . ) < + ∞ , then P top ( f, . ) is convex.(5) P top ( f, ϕ + c ) = P top ( f, ϕ ) + c .(6) P top ( f, ϕ + ψ ◦ f − ψ ) = P top ( f, ϕ ).(7) If P top ( f, . ) < + ∞ , then | P top ( f, ϕ ) − P top ( f, ψ ) | k ϕ − ψ k ∞ . We remark that (6.1) and (6.2) can be used to define a pressure function on the space B d ( X )of bounded potentials which, by some abuse of notation, we still denote by P top ( f, . ) : B d ( X ) → R ∪ { + ∞} . To avoid any confusion, in order to distinguish the pressure functions we will alwaysmention their domains.6.3. Free energy.
Denote by P f ( X ) the space of f -invariant Borel probability measures on X endowed with the weak ∗ topology. Given µ ∈ P f ( X ) and a continuous potential ϕ : X → R ,the free energy of f , µ and ϕ is given by P µ ( f, ϕ ) = h µ ( f ) + Z ϕ dµ where h µ ( f ) is the metric entropy of f with respect to µ (definition and properties may be readin [118, Chapter 4]). A measure µ ∈ P f ( X ) is called an equilibrium state for f and the potential ϕ if P µ ( f, ϕ ) = sup ν ∈ P f ( X ) (cid:26) h ν ( f ) + Z ϕ dν (cid:27) . Variational principles
In the mid seventies the thermodynamic formalism was brought from statistical mechanics todynamical systems by the pioneering work of Sinai, Ruelle and Bowen [23], which established apowerful correspondence between one-dimensional lattices and uniformly hyperbolic dynamicsand conveyed several notions from one setting to the other. The success of this approach ulti-mately relies on a variational principle for the topological pressure, along with the constructionof equilibrium states as the class of pressure maximizing invariant probability measures. Inthis section we first show that Theorem 1 extends the classical thermodynamic formalism forcontinuous self maps on compact metric spaces, and then we complete the proof of Corollary 5.7.1.
Classical variational principle.
Given a continuous transformation f : X → X actingon a compact metric space ( X, d ), the variational principle (cf. [118, § ϕ : X → R , P top ( f, ϕ ) = sup µ ∈ P f ( X ) n h µ ( f ) + Z X ϕ dµ o . Moreover, the previous least upper bound coincides with the supremum evaluated on the set ofergodic probability measures. An equilibrium state for f and the potential ϕ attains the previoussupremum. For instance, an equilibrium state for f and the potential ϕ ≡ h top ( f ) < + ∞ and the entropy function µ ∈ P f ( X ) h µ ( f ) is uppersemi-continuous, the variational principle has a dual version (see [118, Theorem 9.12]), whoseorigin lies on Convex Analysis, telling that h µ ( f ) = inf ϕ ∈ C ( X ) n P top ( f, ϕ ) − Z ϕ dµ o = sup n lim sup n → + ∞ h µ n | ( µ n ) n ∈ N is a sequence in P f ( X ) with lim n → + ∞ µ n = µ o NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 23 where C ( X ) stands for the space of real valued continuous maps whose domain is X . In thiscase, an easy computation using [118, Theorem 9.12] also yields h µ ( f ) = inf ϕ ∈ Λ Z ϕ dµ where Λ = { P top ( f, ϕ ) − ϕ : ϕ ∈ C ( X ) } . New variational principle: Proof of Corollary 5.
Keeping the classical notion oftopological pressure and summoning Theorem 1, we replace the entropy map (metric or star)acting on the space P f ( X ) by a more general real valued function h f whose domain is the space P ( X ) of the Borel probability measures on X . More precisely, assume that h top ( f ) < + ∞ ; then P top ( f, . ) : C ( X ) → R is a pressure function (cf. Section 6) to which we may apply Theorem 1.This way, we conclude that the map h f : P ( X ) → R given by h f ( µ ) = inf ϕ ∈ A P top Z ϕ dµ where A P top = { ϕ ∈ C ( X ) : P top ( f, − ϕ ) } is upper semi-continuous, satisfies h f ( µ ) = inf ϕ ∈ C ( X ) (cid:26) P top ( f, ϕ ) − Z ϕ dµ (cid:27) ∀ µ ∈ P ( X ) (7.1)and P top ( f, ϕ ) = max µ ∈ P ( X ) n h f ( µ ) + Z ϕ dµ o ∀ ϕ ∈ C ( X ) . (7.2)It is immediate from (2.11) that h f ( µ ) h top ( f ) for every µ ∈ P ( X ). Moreover, using theaforementioned strategy, it is clear that, given ϕ ∈ C ( X ), there exists µ ϕ ∈ P ( X ) such that P top ( f, ϕ ) = h f ( µ ϕ ) + R ϕ dµ ϕ . In the special case of ϕ ≡ h top ( f ) = max µ ∈ P ( X ) h f ( µ )and µ ∈ P ( X ) where h f attains its maximum value h top ( f ). This ends the proof of Corollary 5. Corollary 7.3.
Given ϕ ∈ C ( X ) , every µ ϕ ∈ P ( X ) attaining the maximum at (7.2) is f -invariant.Proof. Recall that µ ∈ P ( X ) is said to be f -invariant if R ( ψ ◦ f ) dµ = R ψ dµ for every ψ in C ( X ). Fix µ ϕ ∈ P ( X ) such that P top ( f, ϕ ) = h f ( µ ϕ ) + R ϕ dµ ϕ , and consider ψ ∈ C ( X ). Bythe variational relation (7.2) applied to both ϕ + ψ ◦ f − ψ and ϕ + ψ − ψ ◦ f we may take µ , µ ∈ P ( X ) such that P top ( f, ϕ + ψ ◦ f − ψ ) = h f ( µ ) + Z ϕ dµ + Z ( ψ ◦ f ) dµ − Z ψ dµ and P top ( f, ϕ + ψ − ψ ◦ f ) = h f ( µ ) + Z ϕ dµ + Z ψ dµ − Z ( ψ ◦ f ) dµ . Using the equalities P top ( f, ϕ + ψ ◦ f − ψ ) = P top ( f, ϕ ) = P top ( f, ϕ + ψ − ψ ◦ f ) (cf. Subsection 6) together with (7.2), we conclude that h f ( µ ϕ ) + Z ϕ dµ ϕ = h f ( µ ) + Z ϕ dµ + Z ( ψ ◦ f ) dµ − Z ψ dµ > h f ( µ ϕ ) + Z ϕ dµ ϕ + Z ( ψ ◦ f ) dµ ϕ − Z ψ dµ ϕ so R ( ψ ◦ f ) dµ ϕ − R ψ dµ ϕ . In a similar way, we deduce that h f ( µ ϕ ) + Z ϕ dµ ϕ = h f ( µ ) + Z ϕ dµ + Z ψ dµ − Z ( ψ ◦ f ) dµ > h f ( µ ϕ ) + Z ϕ dµ ϕ + Z ψ dµ ϕ − Z ( ψ ◦ f ) dµ ϕ hence R ψ dµ ϕ − R ( ψ ◦ f ) dµ ϕ . (cid:3) One consequence we draw at once from the classical variational principle and the fact thatthe metric entropy is always non-negative is that the pressure operator determines P f ( X ), inthe sense that (cf. [118, Theorem 9.11]) µ ∈ P f ( X ) ⇔ Z ϕ dµ P top ( f, ϕ ) ∀ ϕ ∈ C ( X ) . By (2.11), this is equivalent to say that µ ∈ P f ( X ) if and only if h f ( µ ) > . Corollary 5 rendersthe following generalization.
Corollary 7.4.
Let f : X → X be a continuous transformation of a compact metric space X with h top ( f ) < + ∞ . If µ ∈ P ( X ) and h f ( µ ) > then µ is f -invariant.Proof. Assume that µ ∈ P ( X ) and h f ( µ ) >
0. Then P top ( f, − ϕ ) + R ϕ > ϕ ∈ C ( X ).Fix ψ ∈ C ( X ) and n ∈ Z \ { } . By the assumption and the property (7) of the topologicalpressure (cf. Subsection 6.2), one has n Z ( − ψ ◦ f + ψ ) dµ P top ( f, − nψ ◦ f + nψ ) = h top ( f ) . Dividing by n > n go to + ∞ , gives R ( − ψ ◦ f + ψ ) dµ . Similarly, when n − −∞ , we obtain R ( − ψ ◦ f + ψ ) dµ > . Therefore µ is f -invariant. (cid:3) Linking h and h ∗ : Proof of Corollary 6. We now relate h f ( µ ), h µ ( f ) and h ∗ µ ( f ) when µ belongs to P f ( X ). As previously mentioned, for every µ ∈ P f ( X ) one has h µ ( f ) h ∗ µ ( f ).Moreover, by the classical variational principle, for each µ ∈ P f ( X ) we get h µ ( f ) P top ( f, ϕ ) − Z ϕ dµ ∀ ϕ ∈ C ( X )and so h f ( µ ) = inf ϕ ∈ C ( X ) (cid:26) P top ( f, ϕ ) − Z ϕ dµ (cid:27) > h µ ( f ) . (7.5)If h ∗ µ ( f ) > h f ( µ ) for some µ ∈ P f ( X ), then there exists ν ∈ P f ( X ) satisfying h ν ( f ) > h f ( ν ),contradicting (7.5). This proves that h µ ( f ) h ∗ µ ( f ) h f ( µ ) for every µ ∈ P f ( X ). Theseinequalities together with Theorem 1 yieldsup µ ∈ P f ( X ) (cid:26) h µ ( f ) + Z ϕ dµ (cid:27) max µ ∈ P f ( X ) (cid:26) h ∗ µ ( f ) + Z ϕ dµ (cid:27) max µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 25
On the other hand, as P f ( X ) ⊂ P ( X ), one has for every ϕ ∈ C ( X )max µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) max µ ∈ P ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) . Therefore, from both the classical and the new variational principle (7.2) we deduce thatmax µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) max µ ∈ P ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) = P top ( f, ϕ ) = sup µ ∈ P f ( X ) (cid:26) h µ ( f ) + Z ϕ dµ (cid:27) max µ ∈ P f ( X ) (cid:26) h ∗ µ ( f ) + Z ϕ dµ (cid:27) max µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) . and, consequently, P top ( f, ϕ ) = max µ ∈ P f ( X ) (cid:26) h f ( µ ) + Z ϕ dµ (cid:27) = max µ ∈ P f ( X ) (cid:26) h ∗ µ ( f ) + Z ϕ dµ (cid:27) . (7.6)Notice that the supremum is attained since h f is upper semi-continuous on P f ( X ), so the map µ ∈ P f ( X ) h f ( µ ) + R ϕ dµ is upper semi-continuous as well; similar reasoning regarding h ∗ .This proves item (a) of Corollary 6. We are left to show that h = h ∗ . Lemma 7.7.
The map µ ∈ P f ( X ) h ∗ µ ( f ) is concave.Proof. Consider α, β ∈ P f ( X ) and 0 < p <
1. If ( α n ) n ∈ N and ( β n ) n ∈ N are sequences of f -invariant probability measures converging in the weak ∗ -topology to α and β , respectively, thenthe f -invariant probability measure (cid:0) p α n +(1 − p ) β n (cid:1) n ∈ N converges to p α +(1 − p ) β . Therefore,as the metric entropy map is affine, we obtain h ∗ p α + (1 − p ) β ( f ) > h p α n + (1 − p ) β n ( f ) = p h α n ( f ) + (1 − p ) h β n ( f ) ∀ n ∈ N . So, h ∗ p α + (1 − p ) β ( f ) > p h ∗ α ( f ) + (1 − p ) h ∗ β ( f )which proves the lemma. (cid:3) To end the proof of the equality h = h ∗ stated on item (b) we just need to take into accountthat the map h ∗ is upper semi-continuous, that it satisfies (7.6) and that, by Lemma 7.7, itis concave. These properties are enough to apply the argument in [118, Theorem 9.12] andconclude that, for every µ ∈ P f ( X ), h ∗ µ ( f ) = inf ϕ ∈ C ( X ) (cid:26) P top ( f, ϕ ) − Z ϕ dµ (cid:27) a formula which the function h also satisfies. Item (c) is now a direct consequence of the equalityin item (a) together with (2.13). The proof of the Corollary 6 is complete.We note that, whenever the metric entropy map is not upper semi-continuous, h f = h ∗ is astrict upper bound for the Kolmogorov-Sinai entropy; thus, Corollary 6 motivates the searchfor an optimal upper semi-continuous upper bound. Clearly, Theorem 1 applied to the pressure function P top ( f, . ) : B d ( X ) → R ∪ { + ∞} provides in general a better bound than h f = h ∗ , sinceit guarantees that P top ( f, ϕ ) = max µ ∈ P a ( X ) (cid:26) h ∞ ( µ ) + Z ϕ dµ (cid:27) ∀ ϕ ∈ B d ( X )where the upper semi-continuous map h ∞ is defined by h ∞ ( µ ) = inf ϕ ∈ B d ( X ) (cid:26) P top ( f, ϕ ) − Z ϕ dµ (cid:27) ∀ µ ∈ P a ( X )and so h µ ( f ) h ∞ ( µ ) h f ( µ ) ∀ µ ∈ P ( X ) . Example without a measure with maximal entropy.
Given ϕ ∈ C ( X ), denote by P ϕ ( f, X ) ⊂ P f ( X ) the space of (classical) equilibrium states for f and ϕ . Both P ϕ ( f, X )and T ϕ ( P top ) are convex sets, but whereas T ϕ ( P top ) is always non-empty and compact for theweak ∗ topology, this is sometimes not true for P ϕ ( f, X ), as we will now check. In general,one has P ϕ ( f, X ) ⊂ T ϕ ( P top ), with equality if and only if the metric entropy map is uppersemi-continuous at every element of T ϕ ( P top ) (cf. [118]). As stated in Theorem 2, it is the set E ϕ (Γ) defined by (2.5), of the f -invariant probability measures which maximize the operatorΓ( ϕ ) = max µ ∈ P f ( X ) (cid:8) h f ( µ ) + R ϕ dµ (cid:9) , that fills in the gap between P ϕ ( f, X ) and T ϕ ( P top ).Let us briefly recall the example given on [118, p. 193] of a homeomorphism without ameasure with maximal entropy. We start describing the β -shift. Let β > β − , that is, 1 = P + ∞ n = 1 a n β − n where a = [ β ] and a n = (cid:2) β n − n − X j = 1 a j β n − j (cid:3) ∀ n > . Then 0 a n k − n ∈ N , where k = [ β ] + 1 . So we can consider a = ( a n ) n ∈ N as a pointin the space Σ + k = Q + ∞ i = 1 { , , · · · , k − } , within which we define the lexicographical ordering,that is, ( x n ) n ∈ N < ( y n ) n ∈ N if x j < y j for the smallest j with x j = y j . Let σ + : Σ + k → Σ + k bethe one-sided shift transformation. Note that σ n + ( a ) a for every n ∈ N . Define Y β = n x = ( x n ) n ∈ N ∈ Σ + k : σ n + ( x ) a ∀ n ∈ N o . This is a closed subset of Σ + k , and one has σ + ( Y β ) = Y β and h top ( σ + | Y β ) = log β . Besides, ifΣ k = Q + ∞ i = −∞ { , , · · · , k − } and X β = n x = ( x n ) n ∈ Z ∈ Σ k : ( x i , x i +1 , · · · ) ∈ Y β ∀ i ∈ Z o then X β is closed in Σ k , invariant under the two-sided shift σ and h top ( σ | X β ) = log β as well.Now choose an increasing sequence ( β n ) n ∈ N such that 1 < β n < n → + ∞ β n = 2.Let f n : X β n → X β n denote the two-sided β n -shift and consider on Σ k a metric d n inducing theproduct topology and satisfying d n ( x, y ) x, y ∈ Σ k . Define a new space X as the NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 27 disjoint union of all the spaces X β n together with a compactification point x ∞ , and put on X the metric ρ ( x, y ) = n d n ( x, y ) , if x, y ∈ X β n P pj = n j , if x ∈ X β n , y ∈ X β p and n < p P + ∞ j = n j , if x = x ∞ and y ∈ X β n .Then ( X, ρ ) is a compact metric space and the sequence of subsets (cid:0) X β n (cid:1) n ∈ N converges to x ∞ ,that is, the sequence n ∈ N τ n = inf (cid:8) ρ ( z, x ∞ ) : z ∈ X β n (cid:9) converges to 0. Moreover, the map f : X → X defined as f | X βn = f n and f ( x ∞ ) = x ∞ is ahomeomorphism of ( X, ρ ); and the Borel f -invariant probability measures are given by + ∞ X n = 1 p n µ n + (cid:16) − + ∞ X n = 1 p n (cid:17) δ x ∞ where µ n ∈ P f n ( X β n ) for every n ∈ N , and the numbers p n are non-negative and satisfy P + ∞ n = 1 p n
1. Hence the ergodic elements of P f ( X ) are either ergodic measures in P f n ( X β n )for some n or δ x ∞ . Therefore, if E f ( X ) stands for the subset of ergodic measures in P f ( X ), then h top ( f ) = sup (cid:8) h µ ( f ) : µ ∈ E f ( X ) (cid:9) = sup n ∈ N sup (cid:8) h µ n ( f n ) : µ n ∈ E f n ( X β n ) (cid:9) = sup n ∈ N h top ( f n ) = lim n → + ∞ log β n = log 2 . Now, if f had a maximal entropy measure, then there should exist an ergodic maximal entropymeasure µ . Thus µ would belong to E f n ( X β n ) for some n , and so h µ ( f ) = log β n < log 2 . Let us look instead for a maximizing probability measure of h ∗ . Lemma 7.8.
Let ε > be given and, for each n ∈ N , consider µ n ∈ P f n ( X β n ) such that h top ( f n ) = h ∗ µ ( f n ) . Then any accumulation point of ( µ n ) n ∈ N in the weak ∗ -topology is δ x ∞ .Proof. Take ψ ∈ C ( X ). Our aim is to show that lim n → + ∞ R ψ dµ n = ψ ( x ∞ ). As ψ is continuouson the compact X , the subsets (cid:0) X β n (cid:1) n ∈ N are pairwise disjoint and converge to x ∞ with respectto the metric ρ , then the sequence of continuous maps (cid:0) ψ n = ψ | X βn (cid:1) n ∈ N converges uniformlyto ψ ( x ∞ ). Therefore, (cid:12)(cid:12)(cid:12)(cid:12)Z ψ dµ n − ψ ( x ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:0) ψ n − ψ ( x ∞ ) (cid:1) dµ n (cid:12)(cid:12)(cid:12)(cid:12) k ψ n − ψ ( x ∞ ) k ∞ n → + ∞ −→ . (cid:3) From Lemma 7.8 and the upper semi-continuity of h ∗ we also conclude that h ∗ δ x ∞ ( f ) > lim sup n → + ∞ h ∗ µ n ( f ) = lim sup n → + ∞ h ∗ µ n ( f n ) = log 2 . Since by definition h ∗ δ x ∞ ( f ) h top ( f ) = log 2, the measure δ x ∞ maximizes h ∗ . On the contrary, h δ x ∞ ( f ) = 0 . Pressure derived from Ruelle-Perron-Frobenius transfer operators.
Some of thestatistical properties of equilibrium states are often proved using the so-called transfer operators,and topological pressure arises as the logarithm of the spectral radius of such an operator (seee.g. [74]). However, the spectral theory of these operators is more powerful when the transferoperator preserves the spaces of H¨older continuous or bounded variation potentials. In whatfollows, we recall some of these concepts and show that Theorem 1 also imparts a new insightin the thermodynamic formalism of piecewise continuous maps.Let f : X → X be a piecewise continuous map on a metric space ( X, d ) and assume that κ := sup x ∈ X f − ( x ) < + ∞ . Then, given a potential ϕ ∈ B d ( X ), the Ruelle-Perron-Frobeniustransfer operator with weight ϕ is well defined by L ϕ : B d ( X ) → B d ( X ) ψ
7→ L ϕ ( ψ ) : x ∈ X X f ( y ) = x e ϕ ( y ) ψ ( y ) . Denote by r ( L ϕ ) the spectral radius of L ϕ which, according to Gelfand’s formula (cf. [50]), maybe computed by r ( L ϕ ) = lim n → + ∞ n q kL nϕ k . Lemma 7.9.
The function P : B d ( X ) → R given by P ( ϕ ) = log r ( L ϕ ) is a pressure function.Proof. Fix ϕ ∈ B d ( X ). Since the space B d ( X ) is endowed with the supremum norm and L ϕ isa positive operator, for every n ∈ N one has kL nϕ k = sup k ψ k ∞ = 1 kL nϕ ( ψ ) k ∞ = kL nϕ ( ) k ∞ . So r ( L ϕ ) = lim n → + ∞ n q kL nϕ ( ) k ∞ , which is bounded by κ e k ϕ k ∞ . Now, given a ∈ [0 , ϕ , ϕ ∈ B d ( X ) and n ∈ N , we write L na ϕ +(1 − a ) ϕ ( )( x ) = X f n ( y ) = x e S n ( a ϕ +(1 − a ) ϕ )( y ) = X f n ( y ) = x (cid:0) e S n ϕ ( y ) (cid:1) a (cid:0) e S n ϕ ( y ) (cid:1) − a and apply H¨older’s inequality to get kL na ϕ +(1 − a ) ϕ ( ) k ∞ kL nϕ ( ) k a ∞ kL nϕ ( ) k − a ∞ . Taking logarithm, dividing by n and letting n go to + ∞ , we obtainlog (cid:0) r ( L a ϕ +(1 − a ) ϕ ) (cid:1) a log (cid:0) r ( L ϕ ) (cid:1) + (1 − a ) log (cid:0) r ( L ϕ ) (cid:1) thereby showing the convexity of the function P . The monotonicity follows from the positivityof the operator L ϕ and the proof of the translation invariance is immediate. (cid:3) Consequently, Theorem 1 yields the following variational principle.
Corollary 7.10.
Let f : X → X be a piecewise continuous map on a metric space X such that κ := sup x ∈ X f − ( x ) < + ∞ . Given ϕ ∈ B = B d ( X ) , there exists an upper semi-continuousmap h B : P a ( X ) → R such that log r ( L ϕ ) = max µ ∈ P a ( X ) n h B ( µ ) + Z ϕ dµ o NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 29
In particular, there is µ ϕ ∈ P a ( X ) satisfying Rohlin’s-like formula (cf. [107] ) h B ( µ ϕ ) = Z log (cid:0) r ( L ϕ ) e − ϕ (cid:1) dµ ϕ . We illustrate this result with the following class of examples.
Example 7.11 (Piecewise expanding maps) . Consider a C -piecewise expanding map f : X → X whose domain is the union of a finite number of subintervals X i = [ a i , b i ) or X i = ( a i , b i ],where a i < b i , within which f is continuous. Let ϕ = − log | f ′ | , which we assume to be piecewisecontinuous and bounded, though it may not exhibit any further regularity. The correspondingtransfer operator is given by ψ ∈ B d ( X )
7→ L ϕ ( ψ )( x ) = X f ( y ) = x | f ′ ( y ) | ψ ( y )and Corollary 7.10 ensures that there exists µ ϕ ∈ P a ( X ) such that h B ( µ ϕ ) + Z log | f ′ | dµ ϕ = log r ( L ϕ ) . For instance, the Lorenz maps satisfy the previous assumptions with X = [ − , ∪ (0 , X and exhibiting a spectral gap, Giulietti et al [61, Theorem F] showed a variational principlesimilar to the one in Corollary 7.10 with an entropy-like function h X computed by h X ( µ ) = inf φ ∈ X n log λ X ( φ ) − Z φ dµ o for every f -invariant probability measure µ , where λ X ( φ ) denotes the spectral radius of thetransfer operator L φ : X → X . In general, since X ( B d ( X ) one has h µ ( f ) h B ( µ ) h X ( µ )for every µ ∈ P f ( X ). In the special case that f is a Ruelle expanding map on a compact metricspace X and X = C α ( X ), α >
0, the spectral radius of the operator L φ acting on both spaces C α ( X ) and C ( X ) coincide and the three notions of entropy (with B = C ( X )) are the same.7.6. Finitely additive equilibrium states and second order phase transition.
Considerthe Manneville-Pomeau family of maps f α : [0 , → [0 , α >
0, given by f α ( x ) = ( x (1 + 2 α x α ) if x ∈ [0 , [2 x − x ∈ [ , . (7.12)It is known that the entropy map for each f α is upper semi-continuous, and that this familyexhibits phase transitions with respect to the potentials ϕ α,t = − t log | f ′ α | , parameterized by t ∈ R . We refer the reader to [31, 82, 116] for an ample discussion on phase transitions of theManneville-Pomeau family. For instance:(MP ) If α > t ∈ ] − ∞ , µ α,t for f α and ϕ α,t .(MP ) If α >
0, the map t ∈ [1 , + ∞ [ P top ( − t log | f ′ α | ) is equal to zero and the Dirac measure δ is an equilibrium state with respect to ϕ α,t .(MP ) If 0 < α <
1, there exist two equilibrium states with respect to ϕ α, , namely the Dirac δ and an f α -invariant probability measure µ α, which is absolutely continuous with respectto the Lebesgue measure; moreover, the map t ∈ R P top ( − t log | f ′ α | ) is not C . (MP ) If α >
1, there exists a unique equilibrium state for ϕ α,t for any t ∈ R ; besides, the map t ∈ R P top ( − t log | f ′ α | ) is C , but not C , and there is an f α -invariant, σ -finite andinfinite measure which is absolutely continuous with respect to the Lebesgue measure.It is worth noticing that the C -smoothness of the pressure is compatible with the presenceof second-order phase transitions. Yet, in the special case of the Manneville-Pomeau maps and α >
1, the second order phase transitions for the potential ϕ α,t ∈ C α ([0 , ⊂ C ([0 , T -invariantprobabilities by the larger space of T -invariant finitely additive measures. More precisely, if P top : B d ([0 , → R is the pressure function defined by (6.1)-(6.2), then P top ( f α , − log | f ′ α | ) = 0 = max µ ∈ P a ([0 , n h ∞ ( µ ) − Z log | f ′ α | dµ o (7.13)where h ∞ ( µ ) = inf g ∈ B d ([0 , n P top ( f α , g ) − Z g dµ o for every f α -invariant finitely additive measure µ . Observe now that h ∞ ( δ ) P top ( f α , − log | f ′ α | ) − Z log | f ′ α | dδ = 0 . On the other hand, as { } is an ( n, ε )-separated set for every n ∈ N and any ε >
0, then forevery g ∈ B d ([0 , P top ( f α , g ) > lim n → ∞ n log e S n g (0) = g (0) . This proves that h ∞ ( δ ) = 0 and, consequently, δ attains the maximum in (7.13). A secondequilibrium state appears while looking for absolutely continuous finitely additive invariantmeasures, which satisfy an analogous of the Pesin formula. More precisely: Theorem 7.14. [38] If ( f α ) α > is the Manneville-Pomeau family defined by (7.12) then thereexists a finitely additive normalized measure µ α, ∈ P a ([0 , which is f α -invariant and absolutelycontinuous with respect to the Lebesgue measure. Moreover, h L ( µ α, ) = Z log | f ′ α | dµ α, (7.15) where h L ( µ α, ) := inf g ∈ L ∞ ( Leb ) n P top ( f α , g ) − R g dµ α, o . It follows from the definitions and equality (7.15) that Z log | f ′ α | dµ α, = h L ( µ α, ) h ∞ ( µ α, ) = inf g ∈ B d ([0 , n P top ( f α , g ) − Z g dµ α, o Z log | f ′ α | dµ α, . In particular, not only h L ( µ α, ) and h ∞ ( µ α, ) coincide as the finitely additive measure µ α, attains the maximum in (7.13). Thus, summoning the previous information on δ , Corollary 4and Theorem 7.14, one deduces that: Corollary 7.16. If ( f α ) α > is the Manneville-Pomeau family (7.12) then T ϕ α, ( P top ) > .In particular, for every α > , the map P top : C ([0 , → R is Gateaux differentiable, thoughits extension to B d ([0 , is not Gateaux differentiable at ϕ α, . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 31
Part THERMODYNAMICS FOR NON-ADDITIVE SEQUENCES Non-additive sequences of continuous potentials
In this section, we fix a continuous endomorphism of a compact metric space X and, insteadof the classic pressure operator with respect to a given potential ϕ : X → R and the sequence ofBirkhoff sums ( S n ϕ ) n ∈ N , we consider non-additive sequences of continuous potentials. Althoughthese objects arise naturally in the study of Lyapunov exponents and dimension theory, thenon-additive thermodynamic formalism is still barely understood, and only when one restrictsto sequences displaying some stronger additive property. We refer the reader to [12, 13, 54, 46]for a thorough discussion on these topics.Let f : X → X be a continuous map on a compact metric space ( X, d ). We say that asequence Φ = ( ϕ n ) n ∈ N ∈ C ( X ) N of continuous potentials is(1) sub-additive if ϕ m + n ϕ m + ϕ n ◦ f m ∀ m, n ∈ N ;(2) almost additive if there exists a uniform constant C > ϕ m + ϕ n ◦ f m − C ϕ m + n ϕ m + ϕ n ◦ f m + C ∀ m, n ∈ N ;(3) asymptotically additive if for any ε > ϕ ε ∈ C ( X ) such thatlim sup n → + ∞ n (cid:13)(cid:13)(cid:13) ϕ n − n − X j =0 ϕ ε ◦ f j (cid:13)(cid:13)(cid:13) ∞ < ε. It is known that every almost additive sequence is asymptotically additive, and that for everyasymptotically additive sequence Φ = ( ϕ n ) n ∈ N ∈ C ( X ) N there exists ϕ ∈ C ( X ) such thatlim sup n → + ∞ n (cid:13)(cid:13)(cid:13) ϕ n − n − X j =0 ϕ ◦ f j (cid:13)(cid:13)(cid:13) ∞ = 0(cf. [46, 53]). Therefore, both the variational principle and the existence of finitely additiveequilibrium states established in Corollary 5 admit an immediate generalization to this context(the modifications necessary to deal with sequences in other Banach spaces are left as an easyexercise to the interested reader.) Henceforth, we will aim at the more general context of sub-additive sequences of continuous potentials. Definition 8.1.
Given a sequence Φ = ( ϕ n ) n ∈ N ⊂ C ( X ) of continuous potentials, the non-additive topological pressure is defined by P ( f, Φ) = lim ε → + lim sup n → + ∞ n log (cid:16) sup E X x ∈ E e ϕ n ( x ) (cid:17) (8.2)where the supremum is taken over the ( n, ε )-separated subsets E of X .This definition coincides with the usual notion of topological pressure P top ( f, ϕ ) when thereis ϕ ∈ C ( X ) such that ϕ n = P n − j =0 ϕ ◦ f j for every n ∈ N . It is known (cf. [13, 87]) thatevery almost additive sequence of continuous potentials which have bounded distortion admitsa unique equilibrium state, which is a Gibbs measure. More recently, it was proved in [46]that any almost additive or asymptotically additive sequence of continuous potentials have thesame pressure of an additive sequence associated to a continuous potential. However, it is notknown whether this potential inherits the distortion properties of the original almost additive sequence. Besides, for sub-additive sequences of continuous potentials no general constructionof equilibrium states is known, though for these sequences it was established by Cao, Feng andHuang [36] the following general variational principle. Theorem 8.3. [36] If Φ = ( ϕ n ) n ∈ N is a sub-additive sequence of continuous potentials suchthat P ( f, Φ) > −∞ , then P ( f, Φ) = sup µ ∈ P f ( X ) n h µ ( f ) + F ∗ (Φ , µ ) o (8.4) where, for every f -invariant probability measure µ , F ∗ (Φ , µ ) := lim n → + ∞ n Z ϕ n dµ. We note that if Φ = ( ϕ n ) n ∈ N is a sub-additive sequence of continuous potentials then F ∗ (Φ , µ )is well defined for every µ ∈ P f ( X ). In fact, the sequence of real numbers ( a n ) n ∈ R given by a n = R ϕ n dµ is sub-additive, hence lim n → + ∞ n a n = inf n ∈ N n a n by Fekete’s Lemma.8.1. An alternative variational principle for sub-additive sequences.
One might expectto obtain a counterpart of Theorem 1 for a more general context of Banach spaces of sequencesof functions. This faces non-trivial difficulties, though. Contrary to what happens within thesimpler case of almost additive sequences, albeit providing a convex cone in the space of sequencesof potentials, sub-additivity is not preserved under multiplication by negative numbers. Thisis a major obstruction since entropy in Theorem 1 is defined using observable maps ϕ suchthat − ϕ has non-positive pressure, which makes the Banach space generated by sub-additivesequences not suitable to this approach. Therefore, our strategy will explore the general contextof Theorem 1 and address a bounded representative for natural classes of sub-additive sequences.As sub-additive sequences depend on Kingman’s Sub-additive Ergodic Theorem, we are ledto narrow our analysis to the set S b = n Φ = ( ϕ n ) n ∈ N ∈ C ( X ) N : Φ is sub-additive and inf x ∈ X h inf n ∈ N n ϕ n ( x ) i > −∞ o . The set S b comprises relevant families of sequences of continuous potentials arising within thetheory of linear cocycles, as we will detail on Subsection 8.2. We also observe that Φ ∈ S b ifand only it is sub-additive and F ∗ (Φ , µ ) > −∞ for every f -invariant probability measure µ (cf.[113, pp. 336–337]). Moreover: Lemma 8.5.
Given
Φ = ( ϕ n ) n ∈ N ∈ S b , the map ψ Φ ∈ B d ( X ) defined by x ∈ X ψ Φ ( x ) = inf n ∈ N n ϕ n ( x ) is measurable, upper semi-continuous and satisfies Z ψ Φ dµ = F ∗ (Φ , µ ) ∀ µ ∈ P f ( X ) . Proof.
For every Φ = ( ϕ n ) n ∈ N ∈ S b , the corresponding map ψ Φ is measurable and upper semi-continuous, hence upper bounded on the compact X . Since Φ belongs to S b , the map ψ Φ isalso lower bounded. Moreover, by the Kingman’s Sub-additive Ergodic Theorem [77], the mapsinf n ∈ N n ϕ n and lim inf n → + ∞ n ϕ n coincide in a set with full measure and, for every µ ∈ P f ( X )one has R lim inf n → + ∞ n ϕ n dµ = F ∗ (Φ , µ ) . So, R ψ Φ dµ = F ∗ (Φ , µ ) as well. (cid:3) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 33
In the remaining of this subsection we restrict to S b and consider P ( f, . ) in order to improve thevariational relation (8.4). More precisely, we will show the following counterpart of Corollary 5in this context, using B d ( X ) instead of C ( X ). Corollary 8.6.
Let f be a continuous map acting on a compact metric space ( X, d ) whosetopological entropy is finite. Then there exists an affine and upper semi-continuous entropy map h : P a ( X ) → R so that P ( f, Φ) = max µ ∈ P a ( X ) n h ( µ ) + Z ψ Φ dµ o ∀ Φ ∈ S b . In particular, every sub-additive sequence Φ ∈ S b has a finitely additive equilibrium state µ Φ .Proof. An effortless computation shows that the operator Γ : B d ( X ) → R defined by ψ ∈ B d ( X ) Γ ( ψ ) = sup µ ∈ P f ( X ) n h µ ( f ) + Z ψ dµ o is a pressure function. Therefore, we may apply Theorem 1 and conclude that there exists anaffine and upper semi-continuous map h : P a ( X ) → R such that, for every µ ∈ P a ( X ), h ( µ ) = inf ψ ∈ A Γ1 Z ψ dµ and, for each ψ ∈ B d ( X ), Γ ( ψ ) = max µ ∈ P a ( X ) n h ( µ ) + Z ψ dµ o . Besides, for every ( ϕ n ) n ∈ N ∈ S b one has (cf. [36]) P ( f, Φ) > inf µ ∈ P f ( X ) F ∗ (Φ , µ ) > −∞ . Inaddition, from Lemma 8.5 and Theorem 8.3 one deduces that, for every Φ = ( ϕ n ) n ∈ N ∈ S b , P ( f, Φ) = Γ ( ψ Φ ) . That is, P ( f, Φ) = sup µ ∈ P f ( X ) n h µ ( f ) + F ∗ (Φ , µ ) o = max µ ∈ P a ( X ) n h ( µ ) + Z ψ Φ dµ o . The last sentence in the corollary is a direct consequence of the previous variational principle. (cid:3)
Application to linear cocycles and Lyapunov equilibrium states.
Non-uniformhyperbolicity is defined in terms of
Lyapunov exponents : a diffeomorphism is non-uniformlyhyperbolic if it has no zero Lyapunov exponents. These numbers measure the exponentialasymptotic rates of contraction or expansion along fixed directions, and became a fundamentaltool to characterize chaotic dynamics. The linear cocycles turns to be a powerful mean to attestthe abundance of non-uniformly hyperbolic behavior, as it allows to detach the underlyingdynamics from the action it induces on a vector space. Here we are mainly interested in theexistence of Lyapunov equilibrium states for linear cocycles. Some recent contributions on thistopic comprise [11, 19, 53, 54].
Lyapunov exponents . We start recalling some preliminary notions. Let f be a contin-uous map on a compact metric space ( X, d ). Given an integer ℓ >
1, a field K = R or C and ameasurable matrix-valued map A : X → GL ( ℓ, K ), the linear cocycle generated by A and drivenby f is the map F A : X × K ℓ → X × K ℓ ( x, v ) ( f ( x ) , A ( x ) v ) . Its iterates are F nA ( x, v ) = ( f n ( x ) , A n ( x ) v ), where A n ( x ) = A ( f n − ( x )) · · · A ( f ( x )) A ( x ) forevery n ∈ N , A ( x ) = x and, if f is invertible, A n ( x ) = A ( f n ( x )) − · · · A ( f − ( x )) − when n < f, A ). A natural example of linear cocycle is given bythe derivative cocycle associated to a diffeomorphism f ∈ Diff ( X ) on a compact Riemannianmanifold X , in which case the cocycle is generated by A ( x ) = Df ( x ) for each x ∈ X .Oseledets’ Theorem asserts that, under mild conditions, the Lyapunov exponents of the co-cycle ( f, A ) are well defined. More precisely, given an f -invariant probability measure µ , iflog k A ± k ∈ L ( µ ) then for µ -almost every x ∈ X there exist an integer k ( x ) >
1, a splitting K ℓ = E ,Ax ⊕ · · · ⊕ E k ( x ) ,Ax and real numbers (called Lyapunov exponents ) λ ( A, µ, x ) > · · · > λ k ( x ) ( A, µ, x )such that, for every v ∈ E i,Ax \ { } and 1 i k ( x ), A ( x ) (cid:0) E i,Ax (cid:1) = E i,Af ( x ) and λ i ( A, µ, x ) = lim n → + ∞ n log k A n ( x ) v k . If, in addition, µ is ergodic, then k ( x ), the Lyapunov exponents λ i ( A, µ, x ) and the dimensions ofthe subspaces E i,Ax are µ -almost everywhere constant, in which case one simplifies the notationby writing λ i ( A, µ ).8.2.2.
Singular value sub-additive potentials . In what follows, ∧ k L stands for the k thexterior power of the linear map L . Assume that the linear cocycle A : X → GL ( ℓ, K ) is con-tinuous. Then the Lyapunov exponents can be computed using exterior powers and a family ofsub-additive sequences of potentials. More precisely, if µ is an f -invariant and ergodic proba-bility measure and one takes for each k ∈ N the sub-additive sequence Φ k = ( ϕ k,n ) n ∈ N of thecontinuous functions x ∈ X ϕ k,n ( x ) = log k ∧ k A n ( x ) k then lim n → + ∞ n ϕ k,n ( x ) = k X i =1 λ i ( A, µ ) at µ -almost every x ∈ X .Motivated by applications in dimension theory and aiming to apply their results to Falconer’ssingular value function and affine iterated function systems with invertible affinities, Bochi andMorris [19] studied the following continuous parameterized family of sub-additive sequences ofpotentials. Given ~α = ( α , α , . . . , α ℓ ) ∈ R ℓ with α > α > · · · > α ℓ , consider the sequenceΦ ~α = ( ϕ ~α,n ) n ∈ N defined by ϕ ~α,n ( x ) = log (cid:16) ℓ Y i =1 s i ( A n ( x )) α i (cid:17) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 35 where s i ( L ) denotes the i th singular value of the linear map L . Then it is shown in [19] that if µ is an f -invariant and ergodic probability one haslim n → + ∞ n ϕ ~α,n ( x ) = k X i =1 α i · λ i ( A, µ ) at µ -almost every x ∈ X .Now, the variational principle for the previous family of sub-additive sequences, establishedby Theorem 8.3, says that, if h top ( f ) < + ∞ and P ( f, A, Φ ~α ) is the pressure function definedby (8.2) when Φ = Φ ~α , then P ( f, A, Φ ~α ) = sup µ ∈ P f ( X ) n h µ ( f ) + Z k X i =1 α i · λ i ( A, µ, x ) dµ o . (8.7)Invariant measures attaining the previous equality, so called Lyapunov equilibrium states , arein general hard to find. In the special context of cocycles over a full shift, the metric entropyfunction is upper semi-continuous and so these equilibrium states always exist. Moreover, withintotally disconnected spaces, it has been shown under great generality that, for typical one-stepcocycles and H¨older continuous fiber-bunched cocycles, the previous families of sequences ofpotentials satisfy a quasi-additivity property, and so they have unique Lyapunov equilibriumstates (see e.g. [52, 54, 90]). More recently, Bochi and Morris (cf. [19, Theorem 1]) proved that,in this setting, there are finitely many Lyapunov equilibrium states, these have full support, andare unique for potentials at large temperatures.It is known that the existence of Lyapunov equilibrium states for the family ( β Φ ~α ) β > carriesinformation on Lyapunov optimizing measures. For example, given Lyapunov equilibrium states µ β with respect to β Φ ~α , β >
0, any weak ∗ accumulation point of ( µ β ) β > is an ergodic optimizingmeasure for the potential Φ ~α . Additionally, in the case of 2 × Finitely additive equilibrium states and ergodic optimization . We are unawareof any results guaranteeing the existence of equilibrium states or Lyapunov optimizing measuresin this non-additive context when the dynamical system f does not exhibit a certain amount ofexpansion. Yet, after the information imparted on the previous subsections, it is clear that thefollowing result on singular value potentials is a direct consequence of Corollary 8.6. We remarkthat, as far as we know, the next statement brings forth novelty even in the case when f is theone-sided full shift. Corollary 8.8.
Let f be a continuous map on a compact metric space ( X, d ) . There is an uppersemi-continuous map h f : P a ( X ) → R such that, for every cocycle A ∈ C ( X, GL ( ℓ, R )) , everyvector ~α = ( α , α , . . . , α ℓ ) ∈ R ℓ with α > α > · · · > α ℓ and the the corresponding non-additivesequence Φ ~α of singular value potentials, one has P ( f, A, Φ ~α ) = max µ ∈ P a ( X ) n h f ( µ ) + Z k X i =1 α i · λ i ( A, x ) dµ o . Moreover, the set of finitely additive equilibrium states is non-empty for every linear cocycle in C ( X, GL ( ℓ, R )) and the zero temperature limits of finitely additive equilibrium states have thelargest value of h f amongst the Lyapunov optimizing measures. Part THERMODYNAMIC FORMALISM FOR SEMIGROUP ACTIONS
We now apply the previous information to the context of both semigroup and group actions.9.
Finitely generated free semigroup actions
We start briefly recalling the main definitions and results concerning a classical notion ofentropy for this type of action.9.1.
Entropy and pressure.
There have been several proposals to generalize the previous no-tions of entropy and pressure for a single dynamics to the setting of finitely generated semigroupactions. For an account on some of them we refer the reader to [14], [37] and references therein.In this section we have opted for the following definition inspired by [58].9.1.1.
Topological entropy . Given n ∈ N and ε >
0, the ( n, ε )-Bowen ball generated by thesemigroup action and centered at x is defined by B Gn ( x, ε ) = n y ∈ X : d ( g ( x ) , g ( y )) < ε ∀ g ∈ G n o . One says that two points x, y ∈ X are ( n, ε )-separated by G if there exists g ∈ G n such that d ( g ( x ) , g ( y )) > ε , that is, y does not belong to B Gn ( x, ε ). A subset E of X is ( n, ε )-separated ifany two distinct points of E are ( n, ε )-separated by G . Having fixed n ∈ N and ε >
0, consider s n ( G, G , ε ) = max {| E | : E ⊂ X is ( n, ε )-separated } . Since X is compact, s n ( G, G , ε ) is finite for every n ∈ N and ε >
0. Moreover, the map ε > lim sup n → + ∞ n log s n ( G, G , ε )is monotone. Definition 9.1.
The topological entropy of the free semigroup G generated by G is given by h top ( G, G ) = lim ε → + lim sup n → + ∞ n log s n ( G, G , ε ) . Topological pressure . Our aim now is to generalize the previous notion to any potential ϕ ∈ C ( X ). Given n ∈ N , ε > g ∈ G n presented by the concatenation g = g j n · · · g j (which may be one of many such presentations), where g j i ∈ G for every i ∈ { , · · · , n } , define x ∈ X S gn ϕ ( x ) = ϕ ( x ) + ϕ ( g j ( x )) + ϕ ( g j g j ( x )) + · · · + ϕ ( g j n · · · g j ( x )) . Definition 9.2.
The topological pressure of the free semigroup G generated by G and thepotential ϕ is given by P top ( G, G , ϕ ) = lim ε → + lim sup n → + ∞ n log P n ( G, G , ϕ, ε )where, for every n ∈ N and ε > P n ( G, G , ϕ, ε ) = 1 | G n | X g ∈ G n sup E n X x ∈ E e S gn ϕ ( x ) : E ⊂ X is ( n, ε )-separated o . (9.3) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 37
Properties of the pressure.
We now study the behavior of the operator P top ( G, G , . )when we change either the potential or the semigroup. The following properties are similar tothe ones listed on [118, Theorem 9.7] (see Subsection 6.2) and [118, Theorem 9.8] for the pressureassociated to one dynamics. Their proofs are also identical to [118, § Variation of P top ( G, G , . ) with the potential . We start verifying that the maps P top ( G, G , . ) : C ( X ) → R satisfies the three axioms requested on the definition of a pressurefunction. Lemma 9.4. If P top ( G, G , . ) < + ∞ , then P top ( G, G , . ) is a pressure function.Proof. Consider ϕ, ψ ∈ C ( X ), c ∈ R and ε >
0. From the Definition 9.2 it is clear that(1) P top ( G, G ,
0) = h top ( G, G ) . (2) P top ( G, G , ϕ + c ) = P top ( G, G , ϕ ) + c. (3) ϕ ψ ⇒ P top ( G, G , ϕ ) P top ( G, G , ψ ) . In particular, h top ( G, G ) + min ϕ P top ( G, G , ϕ ) h top ( G, G ) + max ϕ from whose inequalities we also conclude that P top ( G, G , . ) is either finite valued or identically+ ∞ . Besides, the relation sup a j sup b j sup (cid:16) a j b j (cid:17) for any collection of positive real numbers ( a j ) j and ( b j ) j implies that P n ( G, G , ϕ, ε ) P n ( G, G , ψ, ε ) e n k ϕ − ψ k ∞ which, if P top ( G, G , ϕ ) < + ∞ , yields | P top ( G, G , ϕ ) − P top ( G, G , ψ ) | k ϕ − ψ k ∞ . Moreover, the pressure map P top ( G, G , . ) is convex. Indeed, given 0 < λ < E of X , by Holder’s inequality we have X x ∈ E e S gn (cid:0) λϕ +(1 − λ ) ψ (cid:1) ( x ) X x ∈ E e S gn ϕ ( x ) ! λ X x ∈ E e S gn ϕ ( x ) ! − λ . Therefore P n ( G, G , λϕ + (1 − λ ) ψ, ε ) P n ( G, G , ϕ, ε ) λ P n ( G, G , ψ, ε ) − λ , which impliesthat P top ( G, G , λϕ + (1 − λ ) ψ ) λP top ( G, G , ϕ ) + (1 − λ ) P top ( G, G , ψ ) . (cid:3) A variational principle: Proof of Corollary 8.
Firstly note that P top ( G, G , ϕ ) h top ( G, G ) + max x ∈ X ϕ < + ∞ ∀ ϕ ∈ C ( X ) . Consequently, applying Theorem 1 to the pressure function P top ( G, G , . ) we deduce that thereexists a map h G : P ( X ) → R given by h G ( µ ) = inf ϕ ∈ A P top (cid:26)Z ϕ dµ (cid:27) where A P top = { ϕ ∈ C ( X ) : P top ( G, G , − ϕ ) } , such that h G ( µ ) = inf ϕ ∈ C ( X ) (cid:26) P top ( G, G , ϕ ) − Z ϕ dµ (cid:27) and P top ( G, G , ϕ ) = max µ ∈ P ( X ) (cid:26) h G ( µ ) + Z ϕ dµ (cid:27) . In particular, h top ( G, G ) = max µ ∈ P ( X ) h G ( µ )and there is a probability measure µ where h G attains its maximum value. Remark . Due to the previous variational principle, several new concepts are now available forfinitely generated free semigroup actions with finite topological entropy. For instance, we maysay that µ ∈ P ( X ) is a ( G, G )-invariant probability measure if h G ( µ ) >
0, or, equivalently, if P top ( G, G , ϕ ) > R ϕ dµ for every ϕ ∈ C ( X )). Another hint is to take h G ( µ ) as a natural notionof measure theoretic entropy of a free semigroup action G , finitely generated by G and withfinite topological entropy, with respect to a ( G, G )-invariant probability measure µ ∈ P ( X ). Remark . The concept of topological pressure though as an extension of Ghys, Langevin,Walczak entropy in Definition 9.2 is not the only possibility. A second one would be to takealternatively P n ( G, G , ϕ, ε ) = sup E n X x ∈ E e max g S gn ϕ ( x ) : E ⊂ X is ( n, ε )-separated o instead of (9.3). It is not hard to check that this also defines a pressure function and that itleads to a variational principle which differs from the one in Corollary 8.9.4. Generalization.
The previous pressure operator P top ( G, G , . ) works fine for free semi-group actions. It may be reshaped to comply with more general finitely generated semigroupsas follows. Denote by Con( X ) the family of all continuous self-maps of the compact metricspace ( X, d ). Consider a semigroup G generated by a finite set G ⊂ Con( X ) which containsthe identity map. For each n ∈ N , recall that G n := (cid:8) g i n ◦ g i n − ◦ ... ◦ g g : g i j ∈ G (cid:9) . Thereforeany element g ∈ G n is represented by a concatenation g i n , g i n − , ..., g i , but not necessarily in aunique way.Given a continuous potential ϕ : X → R , n ∈ N , x ∈ X and g = g i n ◦ g i n − ◦ ... ◦ g i ∈ G n ,define S ( g in ,g in − ,...,g i ) ϕ ( x ) := ϕ ( x ) + ϕ ( g i ( x )) + ... + ϕ ( g i n ◦ g i n − ◦ ... ◦ g g ( x )) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 39 and, for each g ∈ G n ,Max gn ( ϕ ( x )) := max n S ( g in ,g in − ,...g i ) ϕ ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o . Definition 9.7.
The topological pressure of a finitely generated semigroup (
G, G ) and a po-tential ϕ , where G ⊂ Con( X ), is given by P top (( G, G ) , ϕ ) = lim ε → + lim sup n → + ∞ n log P n (( G, G ) , ϕ, ε )where P n (( G, G ) , ϕ, ε ) = 1 | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( ϕ ( x )) : E ⊂ X is ( n, ε )-separated o . In the following lemmas we verify that the previous expression defines a pressure function.
Lemma 9.8.
For any potentials φ, ψ with φ ψ one has P top (( G, G ) , ϕ ) P top (( G, G ) , ψ ) . Proof. If φ ψ then S ( g in ,g in − ,...,g i ) φ ( x ) = φ ( x ) + φ ( g i ( x )) + ... + φ ( g i n ◦ g i n − ◦ ... ◦ g g ( x )) ψ ( x ) + ψ ( g i ( x )) + ... + ψ ( g i n ◦ g i n − ◦ ... ◦ g g ( x ))= S ( g in ,g in − ,...,g i ) ψ ( x ) . Therefore, for any g ∈ G n ,Max gn ( φ ( x )) = max n S ( g in ,g in − ,...g i ) φ ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o max n S ( g in ,g in − ,...g i ) ψ ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o = Max gn ( ψ ( x ))and consequently P n (( G, G ) , φ, ǫ ) = 1 | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( φ ( x )) : E ⊂ X is ( n, ε )-separated o | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( ψ ( x )) : E ⊂ X is ( n, ε )-separated o = P n (( G, G ) , ψ, ε ) . Letting n → + ∞ and ε → + we get P top (( G, G ) , φ ) P top (( G, G ) , ψ ) . (cid:3) Lemma 9.9.
For any potential φ and constant c ∈ R one has P top (( G, G ) , φ + c ) = P top (( G, G ) , φ ) + c. Proof.
Firstly, notice that S ( g in ,g in − ,...,g i ) φ + c ( x ) = φ ( x ) + φ ( g i ( x )) + ... + φ ( g i n ◦ g i n − ◦ ... ◦ g g ( x )) + n c = S ( g in ,g in − ,...,g i ) φ ( x ) + n c. Therefore, for any g ∈ G n ,Max gn ( φ ( x ) + c ) = max n S ( g in ,g in − ,...g i ) φ + c ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o = max n S ( g in ,g in − ,...g i ) φ ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o + n c = Max gn ( φ ( x )) + n c and so P n (( G, G ) , φ + c, ε ) = 1 | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( φ ( x )+ c ) : E ⊂ X is ( n, ε )-separated o = e n c | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( φ ( x )) : E ⊂ X is ( n, ε )-separated o = e n c P n (( G, G ) , φ, ε ) . Taking logarithms and passing to the limit with n → + ∞ and ε → + we get P top (( G, G ) , φ + c ) = P top (( G, G ) , φ ) + c. (cid:3) Lemma 9.10.
For any potential φ, ψ and t ∈ [0 , one has P top (( G, G ) , t φ + (1 − t ) ψ ) t P top (( G, G ) , φ ) + (1 − t ) P top (( G, G ) , ψ ) . Proof.
Notice that S ( g in ,g in − ,...,g i ) t φ + (1 − t ) ψ ( x ) = ( t φ + (1 − t ) ψ )( x ) + ... + ( t φ + (1 − t ) ψ )( g i n ◦ g i n − ◦ ... ◦ g g ( x ))= t S ( g in ,g in − ,...,g i ) φ ( x ) + (1 − t ) S ( g in ,g in − ,...,g i ) ψ ( x ) . Therefore, for any g ∈ G n ,Max gn (( t φ + (1 − t ) ψ )( x )) = max n S ( g in ,g in − ,...g i ) t φ +(1 − t ) ψ ( x ) : g i n ◦ g i n − ◦ ... ◦ g g = g o t Max gn ( φ ( x )) + (1 − t ) Max gn ( ψ ( x ))thus, due to Lemma 9.8 and Jensen’s inequality, P n (( G, G ) , t φ + (1 − t ) ψ, ε )= 1 | G n | X g ∈ G n sup E n X x ∈ E e Max gn ( t φ +(1 − t ) ψ )( x ) : E ⊂ X is ( n, ε )-separated o | G n | X g ∈ G n sup E n X x ∈ E e t Max gn ( φ ( x ))+(1 − t ) Max gn ( ψ ( x )) : E ⊂ X is ( n, ε )-separated o | G n | X g ∈ G n sup E n t X x ∈ E e Max gn ( φ ( x )) + (1 − t ) X x ∈ E e Max gn ( ψ ( x )) : E ⊂ X is ( n, ε )-separated o = t P n (( G, G ) , ψ, ε ) + (1 − t ) P n (( G, G ) , φ, ε ) . Again, by taking logarithms and the limits n → + ∞ and ε → + we get P top (( G, G ) , t φ + (1 − t ) ψ ) t P top (( G, G ) , φ ) + (1 − t ) P top (( G, G ) , ψ ) . (cid:3) NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 41
The previous lemmas ensure that a result similar to Corollary 8 holds in this more generalsetting.9.5.
Homogeneous and Gibbs probability measures.
In [15, 24], to overcome the absenceof a unifying notion of metric entropy for finitely generated group actions, the authors proposea local entropy formula which is similar to the Brin-Katok formula in [30], and prove that, whenthe group G is amenable, the corresponding group action is finitely generated and G admits ahomogeneous probability measure µ , then µ is a measure with maximal entropy. Let us be moreprecise. We say that a probability µ on the Borel subsets of a compact metric space ( X, d ) is G -homogeneous if given ε > δ ε > C ε > µ ( B Gn ( y, δ ε )) C ε µ ( B Gn ( x, ε )) ∀ n ∈ N ∀ x, y ∈ X. The space of finitely generated groups which admit G -homogeneous measures includes bothfinitely generated groups of isometries on a Riemannian manifold and finitely generated groupsof homeomorphisms on a compact topological group (cf. [15, Section 4.2]). Proposition 9.11. [15, Lemma 4.10 and Corollary 4.13]
Let G be a group generated by a finitecollection G of homeomorphisms. If the corresponding group action admits a G -homogeneousprobability measure µ , then the limit h µ ( G, G , x ) := lim ε → + lim sup n → ∞ − n log µ (cid:0) B Gn ( x, ε ) (cid:1) (9.12) exists, does not depend on x ∈ X and is equal to h top ( G, G ) . This result may be generalized to the context of G -Gibbs probability measures. We say thata probability measure µ is a G - Gibbs measure with respect to a continuous potential ϕ and anincreasing sequence ( F n ) n of compact subsets of G which exhaust G if given ε > C ε > C − ε µ (cid:0) B Gn ( x, ε ) (cid:1) e − Γ( ϕ ) n + | Fn | P g ∈ Fn ϕ ( g ( x )) C ε ∀ n ∈ N ∀ x ∈ X. Notice that a G -Gibbs measure for ϕ ≡ G -homogeneous measure, in which case one hasΓ(0) = lim ε → + lim sup n → ∞ n log s n ( G, G , ε )where s n ( G, G , ε ) denotes the maximal cardinality of ( n, ε )-separated sets by ( G, G ). Recallalso that a locally compact group ( G ) is said to be amenable if for any compact set K ⊂ G and δ > F ⊂ G such that m L ( F ∆ KF ) < δ m L ( F ), where m L standsfor the left Haar measure on G (or the counting measure in the case of a discrete group G ).Such a set F is said to be ( K, δ )-invariant. A sequence ( F n ) n of compact subsets of G is saidto be Følner if for every compact K ⊂ G and δ > n that F n is( K, δ )-invariant. A Følner sequence ( F n ) n is called tempered if there exists C > m L (cid:16) [ k < n F − k F n (cid:17) C m L ( F n ) ∀ n ∈ N . It is known that every Følner sequence has a tempered subsequence and that every amenablegroup has a tempered Følner sequence (cf. [81, Proposition 1.4]). The Pointwise ErgodicTheorem for amenable group actions (cf. [81, Theorem 1.2]) says that if µ is a Borel probability measure preserved by all the elements of the group and ( F n ) n is a tempered Følner sequence,then for every ϕ ∈ L ( µ ) the limit¯ ϕ ( x ) := lim n → + ∞ | F n | Z F n ϕ ( g ( x )) dm L ( g )exists for µ almost every x ∈ X , and it is G -invariant. If, in addition, µ is ergodic then¯ ϕ ( x ) = R ϕ dµ at µ almost everywhere. The following is an immediate consequence of theprevious information. Corollary 9.13.
Let G be an amenable group generated by a finite collection G of homeomor-phisms, and take ϕ ∈ C ( X ) . If the group action admits an invariant ergodic G -Gibbs measure µ with respect to the potential ϕ and a tempered Følner sequence ( F n ) n , then Γ( ϕ ) = h µ ( G, G ) + Z ϕ dµ. Carath´eodory structures for finitely generated group actions
In [95], Pesin and Pitskel used an approach inspired by dimension theory to introduce a notionof pressure for invariant but not necessarily compact sets by a single continuous map. This is aparticular case of the so-called Carath´eodory structures (also known as Carath´eodory capacities)described in great generality later in [94, Chapter 4], which turned to have a wide range ofapplications in many different dynamical contexts. We refer the reader e.g. to [17, 115, 123]and references therein for some of these applications arising in the context of non-uniformhyperbolicity, free and amenable group actions.10.1.
Upper Carath´eodory capacities.
Let G be a finitely generated group acting on acompact metric space ( X, d ) and G be a generating set. For each n ∈ N , let G n ⊂ G denotethe ball of radius n in the group G for the distance D ( f, g ) = min n k ∈ N : f g − = g i k . . . g i and g i j ∈ G o . For each finite set F ⊂ G , consider the F -dynamical ball centered at x ∈ X defined by B F ( x, ε ) := n y ∈ X : d ( g ( x ) , g ( y )) < ε ∀ g ∈ F o . and, given ϕ ∈ C ( X ), set S F ϕ ( x ) := P h ∈ F ϕ ( h ( x )) . Take ϕ ∈ C ( X ) and fix a subset Z ⊂ X , areal number s >
0, a natural N ∈ N , and a strictly increasing sequence ( F n ) n ∈ N of finite subsetsof G . Define M ϕ ( Z, N, ε, s, ( F n ) n ) := inf C X B Fn ( x, ε ) ∈ C exp (cid:0) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) (cid:1) (10.1)where the infimum is taken over the collection C N ( Z, ε, ( F n ) n ) of all finite or countable covers C = (cid:8) B F ni ( x i , ε ) : n i > N and x i ∈ X } of Z . The quantity M ϕ ( Z, N, ε, s, ( F n ) n ) does notdecrease as N increases, therefore there exists a limit M ϕ ( Z, ε, s, ( F n ) n ) := lim N → ∞ M ϕ ( Z, N, ε, s, ( F n ) n ) . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 43
It is known that the function s M ϕ ( Z, ε, s, ( F n ) n ) has a critical point where it jumps frominfinity to zero. Thus one defines P ( Z, ε, ϕ, ( F n ) n ) := inf n s > M ϕ ( Z, ε, s, ( F n ) n ) = 0 o . (10.2)One can prove that the function ε P ( Z, ε, ϕ, ( F n ) n ) is monotone, therefore the following limitexists P ( Z, ϕ, ( F n ) n ) := lim ε → + P ( Z, ε, ϕ, ( F n ) n ) . The pressure map we will deal with on this section is precisely ϕ P ( X, ϕ, ( F n ) n ).10.2. P is a pressure function. In this section we show that, having fixed Z , ( F n ) n and ( a n ) n as before, the function ϕ P ( Z, ϕ, ( F n ) n ) is a pressure function. We remark that the firstterm appearing in the summands in (10.1) might be more generally written as e − s a n for somesequence ( a n ) n of real numbers, and regarding some dynamical contexts the scale a n = n hasappeared in the literature. Yet, as may be attested during the proof of Lemma 10.4, the map P is translation invariant only if the sequences ( a n ) n and ( | F n | ) n have the same growth rate. Lemma 10.3. P ( Z, ϕ, ( F n ) n ) P ( Z, ψ, ( F n ) n ) , for every ϕ, ψ ∈ C ( X ) satisfying ϕ ψ .Proof. Consider ϕ, ψ ∈ C ( X ) with ϕ ψ . Therefore S F n ϕ ( y ) S F n ψ ( y ) for every y ∈ Y and n ∈ N . Consequently, if C is an arbitrary finite or countable cover of Z , then X B Fn ( x, ε ) ∈ C exp (cid:0) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) (cid:1) X B Fn ( x, ε ) ∈ C exp (cid:0) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ψ ( y ) (cid:1) . Taking the infimum over covers
C ∈ C N ( Z, ε, ( F n ) n ) and letting N → + ∞ , we get M ϕ ( Z, N, ε, s, ( F n ) n ) M ψ ( Z, N, ε, s, ( F n ) n ) M ϕ ( Z, ε, s, ( F n ) n ) M ψ ( Z, ε, s, ( F n ) n )which imply that P ( Z, ε, ϕ, ( F n ) n ) = inf n s : M ϕ ( Z, ε, s, ( F n ) n ) = 0 o inf n s : M ψ ( Z, ε, s, ( F n ) n ) = 0 o = P ( Z, ε, ψ, ( F n ) n ) . Taking ε → + , we get P ( Z, ϕ, ( F n ) n ) P ( Z, ψ, ( F n ) n ) as claimed. (cid:3) Lemma 10.4. P ( Z, ϕ + c, ( F n ) n ) = P ( Z, ϕ, ( F n ) n ) + c for every ϕ ∈ C ( X ) and c ∈ R .Proof. It is immediate that for every ϕ ∈ C ( X ), c ∈ R and n ∈ N , one has S F n ( ϕ + c )( x ) = X h ∈ F n ( ϕ + c )( h ( x )) = c | F n | + X h ∈ F n ϕ ( h ( x )) . Thus, evaluating on dynamical balls and summing over each arbitrary finite or countable cover C , we conclude that, for every s ∈ R , X B Fn ( x, ε ) ∈ C exp( − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ( ϕ + c )( x ))= X B Fn ( x, ε ) ∈ C exp(( c − s ) | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( x )) . Taking the infimum over covers
C ∈ C N ( Z, ε, ( F n ) n ) and letting N → + ∞ , we deduce that M ( ϕ + c ) ( Z, ε, s, ( F n ) n ) = M ϕ ( Z, ε, s − c, ( F n ) n )and so, inf n s : M ϕ + c ( Z, ε, s, ( F n ) n ) = 0 o = inf n s − c : M ϕ ( Z, ε, s − c, ( F n ) n ) = 0 o . Finally, taking the limit with ε → + we obtain the desired equality. (cid:3) The laborious step to verify that the Carath´eodory structure defines a pressure function is toprove the convexity condition.
Lemma 10.5.
For every finite set I , a ∈ [0 , and arbitrary choices ( x i ) i ∈ I , ( y i ) i ∈ I and ( z i ) i ∈ I , the following inequality holds X i ∈ I e z i e a x i +(1 − a ) y i (cid:16) X i ∈ I e z i + x i (cid:17) a (cid:16) X i ∈ I e z i + y i (cid:17) − a . Proof.
Write P i ∈ I e z i e a x i +(1 − a ) y i = P i ∈ I e a ( z i + x i )+(1 − a ) ( z i + y i ) and apply Holder’s inequality. (cid:3) Lemma 10.6.
For every ϕ ∈ C ( X ) and arbitrary a > one has P ( Z, a ϕ, ( F n ) n ) = a P ( Z, ϕ, ( F n ) n ) . Proof.
It is clear that S F n ( a ϕ )( x ) = X h ∈ F n ( a ϕ )( h ( x )) = a X h ∈ F n ϕ ( h ( x ))Thus, for any dynamical ball B F n ( x, ε ) and arbitrary finite or countable cover C , we can write X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) a S F n ( ϕ )( x ) (cid:17) = e a X B Fn ( x, ε ) ∈ C exp (cid:16) ( − sa ) | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( x ) (cid:17) . Taking the infimum over covers
C ∈ C N ( Z, ε, ( F n ) n ) and letting N → + ∞ , we get M ( a ϕ ) ( Z, ε, s, ( F n ) n ) = e a M ϕ ( Z, ε, sa , ( F n ) n )which yields P ( Z, ε, a ϕ, ( F n ) n ) = a P ( Z, ε, ϕ, ( F n ) n ) for any ε >
0. The lemma follows takingthe limit as ε → + . (cid:3) We are now ready to compare the values of P ( · ) on convex combinations. Lemma 10.7.
For any ϕ, ψ ∈ C ( X ) and arbitrary a ∈ [0 , one has P ( Z, a ϕ + (1 − a ) ψ, ( F n ) n ) a P ( Z, ϕ, ( F n ) n ) + (1 − a ) P ( Z, ψ, ( F n ) n ) . Proof.
As the generalized Birkhoff sums used on the definition of P are affine, we have S F n ( a ϕ + (1 − a ) ψ )( y ) = a S F n ϕ ( y ) + (1 − a ) S F n ψ ( y ) ∀ y ∈ X. Thus, for any dynamical ball B F n ( x, ε ) and arbitrary finite or countable cover C , we can writesup y ∈ B Fn ( x, ε ) S F n ( a ϕ + (1 − a ) ψ )( y ) a sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) + (1 − a ) sup y ∈ B Fn ( x, ε ) S F n ψ ( y ) . NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 45
Therefore, X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ( a ϕ + (1 − a ) ψ )( y ) (cid:17) X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + a sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) + (1 − a ) sup y ∈ B Fn ( x, ε ) S F n ψ ( y ) (cid:17) . Lemma 10.5 now implies that the right-hand side is bounded above by the product of (cid:16) X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) (cid:17)(cid:17) a and (cid:16) X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ψ ( y ) (cid:17)(cid:17) − a . In order to complete the proof it is enough to show that if s > a P ( Z, ϕ, ( F n ) n ) + (1 − a ) P ( Z, ψ, ( F n ) n )then P ( Z, a ϕ + (1 − a ) ψ, ( F n ) n ) s. Given such an s , either s > P ( Z, ϕ, ( F n ) n ) or s > P ( Z, ψ, ( F n ) n ). Assume that the first inequal-ity holds and consider a family of covers ˜ C ∈ C N ( Z, ε, ( F n ) n ) such that M ϕ ( Z, s, ( F n ) n ) = inf ˜ C X B Fn ( x, ε ) ∈ ˜ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ϕ ( y ) (cid:17) . Then M a ϕ +(1 − a ) ψ ( Z, s, ( F n ) n )= inf C ∈ C N ( Z, ε, ( F n ) n ) X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ( a ϕ + (1 − a ) ψ )( y ) (cid:17) inf ˜ C X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ( a ϕ + (1 − a ) ψ )( y ) (cid:17) M ϕ ( Z, s, ( F n ) n ) a · inf ˜ C (cid:16) X B Fn ( x, ε ) ∈ C exp (cid:16) − s | F n | + sup y ∈ B Fn ( x, ε ) S F n ψ ( y ) (cid:17)(cid:17) − a = 0which guarantees that P ( Z, a ϕ + (1 − a ) ψ, ( F n ) n ) s as claimed. (cid:3) An alternative variational principle for finitely generated group actions.
Wecan now apply Theorem 1 and deduce the following consequence.
Corollary 9.
Let ( G, G ) be a finitely generated group and choose a sequence ( F n ) n ∈ N . Thenthere exists an upper semi-continuous function h G : P ( X ) → R such that P ( X, ϕ, ( F n ) n ) = max µ ∈ P ( X ) (cid:26) h G ( µ ) + Z ϕ dµ (cid:27) . Countable sofic group actions
Pressure function for countable sofic group actions.
For any integer ℓ >
1, letSym( ℓ ) denote the group of permutations of the set { , , . . . , ℓ } . A countable group G is called sofic if there exist a sequence of positive integers ( ℓ i ) i ∈ N with limit + ∞ and a sequence ofpermutations, called sofic approximation sequence of G , we denote byΣ = { σ i : G → Sym( ℓ i ) | i ∈ N } satisfying(a) lim i → + ∞ ℓ i (cid:8) k ℓ i : σ i ( h ) ◦ σ i ( g )( k ) = σ i ( hg )( k ) (cid:9) = 1 for all g, h ∈ G ;(b) lim i → + ∞ ℓ i (cid:8) k ℓ i : σ i ( h )( k ) = σ i ( g )( k ) (cid:9) = 1 for all distinct g, h ∈ G .When no confusion arises, to simplify the notation we will write σ g ( · ) instead of σ ( g )( · ) for everymap σ : G → Sym( ℓ ). If S is a continuous action of a countable sofic group G on a compactmetric space ( X, d ), then it induces an action G × C ( X ) → C ( X ) of the group G on C ( X ) givenby ( g, ϕ ) ϕ g ∈ C ( X ) where ϕ g ( x ) := ϕ (cid:16) S ( g ) − ( x ) (cid:17) . Consider a pseudo-metric ρ on the space F ( { , , . . . , ℓ } , X ) of functions from { , , . . . , ℓ } to X defined by ρ ( ψ , ψ ) = 1 ℓ (cid:16) X j ℓ d ( ψ ( j ) , ψ ( j )) (cid:17) . Having fixed a finite subset F ⊂ G and σ : X → Sym( ℓ ), defineMap( F, σ, δ ) = n ψ : { , , . . . , ℓ } → X | max g ∈ F ρ (cid:0) S ( g ) − ◦ ψ, ψ ◦ σ g (cid:1) < δ o . Given a probability measure µ on the Borel subsets of X , non-empty finite subsets F ⊂ G and L ⊂ C ( X ), a map σ : X → Sym( ℓ ) and δ >
0, consider the setMap µ ( F, σ, L, δ ) = n ψ ∈ Map(
F, σ, δ ) : (cid:12)(cid:12)(cid:12) ℓ ℓ − X j =0 ϕ ( ψ ( j )) − Z ϕ dµ (cid:12)(cid:12)(cid:12) < δ ∀ ϕ ∈ L o . Definition 11.1.
Given a probability measure µ on the Borel subsets of X and a countablesofic group G with a sofic approximation sequence Σ, the sofic metric entropy of the continuousaction determined by G with respect to µ is defined by h Σ , µ ( G ) = sup ε > inf F inf L inf δ > h ε Σ , µ ( G, ϕ, F, L, δ ) , (11.2)where h ε Σ , µ ( G, ϕ, F, L, δ ) = lim sup i → + ∞ ℓ i log s ε Σ , µ ( F, σ i , L, δ )and s ε Σ , µ ( F, σ i , L, δ ) denotes the maximal cardinality of the ( ρ, ε )-separated subsets of mapswhich belong to the family Map µ ( F, σ i , L, δ ). We have omitted the dependence of h Σ , µ ( G ) on ρ since this notion turns out to be independent of the pseudo-metric as far as we keep it compatiblewith the topology induced by ρ (cf. [75]). NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 47
The concept of sofic pressure of a continuous action, which we will now recall, was introducedin [40] as an extension of the sofic entropy. To simplify the notation we shall omit the dependenceof this notion on the space X and the pseudo-metric ρ . Let S be a continuous action of acountable sofic group G on the metric space ( X, d ), and let Σ be a sofic approximation sequenceof G . Given a non-empty finite subset F ⊂ G , ϕ ∈ C ( X ), σ : G → Sym( ℓ ), δ > ε > M ε Σ ( ϕ, F, δ, σ ) = sup E n X ψ ∈ E e P ℓj =1 ϕ ( ψ ( j )) o where the supremum is taken over all ( ρ, ε )-separated subsets E of Map( F, σ, δ ). Moreover, set P ε Σ ( G, ϕ, F, δ ) := lim sup i → + ∞ ℓ i log M ε Σ ( ϕ, F, δ, σ i ) . Definition 11.3.
The sofic topological pressure of ϕ under the action of G is defined by P Σ ( G, ϕ ) = sup ε > inf F inf δ > P ε Σ ( G, ϕ, F, δ ) (11.4)where the sets F ⊂ G are chosen non-empty and finite.It is known that the sofic entropy of an action may depend on the choice of the sofic approxi-mation, and may have different positive values even for mixing subshifts of finite type (see [1]).11.2. A variational principle for countable sofic group actions.
Denote by P G ( X ) theset of probability measures on the Borel subsets of X which are preserved by all elements of thegroup G . The next result establishes a variational principle and shows that the finiteness of thesofic pressure is a sufficient condition for P G ( X ) = ∅ . Theorem 11.5. [40, Theorem 1.2]
Given a countable sofic group G with a sofic approximationsequence Σ , let S be a continuous action of G on a metric space ( X, d ) and ϕ : X → R be acontinuous potential. Then P Σ ( G, ϕ ) = sup µ ∈ P G ( X ) n h Σ , µ ( G ) + Z ϕ dµ o . In particular, if P Σ ( G, ϕ ) = −∞ then P G ( X ) = ∅ . The existence of equilibrium states for countable sofic group actions is not known under greatgenerality. If G is a countable sofic group and X = { , , . . . , d } G , then every local potential hasan equilibrium state which is a Gibbs measure (cf. [40, Theorem 5.3] and [3]). More generally,if the group action is expansive then the Σ-entropy function varies upper semi-continuously andequilibrium states do exist for continuous potentials [41]. In the next subsection we discuss theexistence of finitely additive equilibrium states for countable sofic group actions.11.3. Sofic equilibrium states.
Firstly, let us register that the sofic pressure satisfies theaxioms of a pressure function listed in Definition 2.2.
Lemma 11.6. [40, Proposition 6.1]
The sofic pressure function ϕ ∈ C ( X ) P Σ ( G, ϕ ) ismonotone, translation invariant and convex, provided that P Σ ( G, · ) = ±∞ . Therefore we may apply Theorems 1 and 2.
Corollary 11.7.
Given a countable sofic group G with a sofic approximation sequence Σ , let S be a continuous action of G on a metric space ( X, d ) . Assume that P Σ ( G, · ) = ±∞ . Then thereexists a map h Σ : P ( X ) → R satisfying h Σ ( µ ) = inf ϕ ∈ C ( X ) (cid:26) P Σ ( G, ϕ ) − Z ϕ dµ (cid:27) ∀ µ ∈ P ( X ) (11.8) such that, for every ϕ ∈ C ( X ) , one has P Σ ( G, ϕ ) = max µ ∈ P ( X ) n h Σ ( µ ) + Z ϕ dµ o = max µ ∈ P G ( X ) (cid:26) h Σ ( µ ) + Z ϕ dµ (cid:27) . Moreover, every measure µ ∈ P a ( X ) that attains the maximum is G -invariant and h Σ ( µ ) > .If, in addition, the function µ ∈ P f ( X ) h Σ , µ ( G ) is upper semi-continuous, then h Σ ( µ ) = h Σ , µ ( G ) for every µ ∈ P G ( X ) . Countable amenable group actions.
Following [81], a locally compact group G issaid to be amenable if for any compact set K ⊂ G and δ > F ⊂ G such that m L ( F ∆ KF ) < δ m L ( F ), where m L stands for the left Haar measure on G (or the counting measure in the case of a discrete group G ). Amenable groups admit invariantprobability measures which are preserved by all elements of the group. Regarding a variationalprinciple for general amenable group actions we refer the reader to [89] and references therein.In the special case of expansive Z d -actions with the specification property, Ruelle constructedequilibrium states and proved that they are Gibbs measures [108].Let G be a countable amenable group (hence sofic). Given ϕ ∈ C ( X ) denote by P ( G, ϕ )the classical pressure function defined by Ollagnier and Pinchon in [89] (see also [124]). Boththis classical and the sofic metric entropies (respectively, the classical and sofic topological pres-sures) coincide for this class of group actions, as proved in [40, 27, 76]. Therefore, combiningCorollary 11.7, Theorem 2 and [40, Theorem 1.1], one obtains the following variational prin-ciple for countable amenable group actions, which bridges between the classical and the soficthermodynamic objects.
Corollary 10.
Let S be a continuous action of a countable amenable group G on a metric space ( X, d ) . Then, given a continuous potential ϕ , one has P ( G, ϕ ) = P Σ ( G, ϕ ) and there exists anupper semi-continuous map h Σ : P ( X ) → R satisfying P ( G, ϕ ) = sup µ ∈ P G ( X ) n h Σ ,µ ( G ) + Z ϕ dµ o = max µ ∈ P G ( X ) n h Σ ( µ ) + Z ϕ dµ o . Additionally, there is a Baire residual subset R ⊂ C ( X ) such that every ϕ ∈ R has a unique G -invariant maximizing probability measure. Uncountable groups with a reference probability measure
In this section we consider continuous actions on a compact metric space (
X, d ) determinedby an uncountable group. In order to measure the complexity of these group actions we requirethe group G to have a metric structure. So, assume that G is endowed with a distance d G ,and that there exists a probability measure η G on the σ -algebra B of the Borel subsets of G ,though that there is no invariance requirement on η G . A number of examples satisfy the previousassumptions. For instance, finitely generated group actions have a natural distance computedin terms of the number of generators of a group element; and if, in addition, G is a free group NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 49 endowed with a random walk, then this probability measure is invariant by translation. When G is a Lie group, it has a natural metric structure and a natural probability measure, namelythe Haar measure. Regarding other examples of random walks on groups, we refer the readerto [70, 120] and references therein.12.1. Pressure function.
Under the previous assumptions, given ϕ ∈ C ( X ) we define the topological pressure of the group action G × X → X with respect to ϕ by P top ( G, η G , ϕ ) := lim ε → + lim sup n → + ∞ n log Z { g ∈ G : d G ( g, id ) < n } (cid:16) sup E g,n,ε X x ∈ E g,n,ε e ϕ ( g ( x )) (cid:17) d η G ( g )(12.1)where E g,n,ε denotes a ( g, n, ε )-separated set. If the d G -diameter of G is finite, the condition d G ( g, id ) < n must be replaced by d G ( g, id ) < diam( G ) − n . Observe that, when ϕ ≡
0, weobtain P top ( G, η G ,
0) = lim ε → + lim sup n → + ∞ n log Z { g ∈ G : d G ( g, id ) < n } (cid:16) sup E g,n,ε | E g,n,ε | (cid:17) d η G ( g )which, were it applied to finitely generated group actions, provides a value upper bounded bythe estimate in Definition 9.1. Yet, the map P top ( G, η G , . ) extends the concept of pressure usedin [37, 106], where the authors considered the special case of finitely generated free semigroupactions endowed with a random walk η G .We proceed proving that P top ( G, η G , . ) defines a pressure function. One easily checks that,given ϕ, ψ ∈ C ( X ) such that ϕ ψ and c ∈ R , then P top ( G, η G , ϕ ) P top ( G, η G , ψ ) P top ( G, η G , ϕ + c ) = P top ( G, η G , ϕ ) + c. Finally, H¨older inequality ensures that, for an arbitrary 0 < a <
1, one hassup E X x ∈ E e ( a ϕ +(1 − a ) ψ )( g ( x )) (cid:16) sup E X x ∈ E e ϕ ( g ( x )) (cid:17) a (cid:16) sup E X x ∈ E e ψ ( g ( x )) (cid:17) − a hence the convexity: for every ϕ, ψ ∈ C ( X ) and 0 < a < P top ( G, η G , a ϕ + (1 − a ) ψ ) a P top ( G, η G , ϕ ) + (1 − a ) P top ( G, η G , ψ ) . A variational principle.
One can now apply Theorem 1, which implies that:
Corollary 11.
Assume that η G is a Borel probability measure on the Borel sets of a metricgroup ( G, d G ) , and let G × X → X be a continuous group action of G on a compact metricspace ( X, d ) . Then there exists an upper semi-continuous function h G,η G : P ( X ) → R such that, P top ( G, η G , ϕ ) = max µ ∈ P ( X ) n h G,η G ( µ ) + Z ϕ dµ o ∀ ϕ ∈ C ( X ) . Part FINAL REMARKS
Phase transitions and finer ergodic properties
A system is said to undergo a phase transition if, after a small perturbation in some controlvariable, an abrupt change in one or more large-scale physical properties of the system happens.In general, phase transitions are grouped by the lowest derivative of the free energy which undergoes a discontinuity, though frequently further subdivided according to symmetries ofthe phases at each side of the transition. Within the statistical physics perspective, phasetransitions are described by changes on the thermodynamic properties, and are often detectedthrough either the non-smoothness of the pressure function, or the absence / non-uniquenessof equilibrium states. In the context of quadratic maps and rational maps on the Riemannsphere, phase transitions and a thermodynamic formalism are nowadays well understood (see[43, 44, 45, 100, 99] and references therein).In the special case of an expansive continuous map f : X → X on a compact metric space X (or an asymptotically entropy-expansive map [85], or else a C ∞ map [88]), the function µ ∈ P f ( X ) h µ ( f ) is upper semi-continuous, and so it coincides with h f on the space P f ( X )of f -invariant (countably additive) probability measures. Moreover, there are equilibrium statesfor every continuous potential. When these equilibrium states are unique, one expects thatphase transitions, if there are any, may be detected through subtler thermodynamic objects.Our results suggest the following question. Problem 1.
Assume that f is a continuous transformation on a compact metric space X suchthat the entropy map is upper semi-continuous. If each potential (on some selected family) hasa unique equilibrium state in P f ( X ) , can phase transitions be uncovered by the appearance ofpurely finitely additive equilibrium states? In the ongoing work [38], a first attempt to describe the phase transitions for the Manneville-Pomeau maps suggests that Problem 1 may have a positive solution.Under the extra assumptions of expansiveness and specification, Haydn and Ruelle [63] provedthat regular potentials have unique equilibrium states and that these are Gibbs measures. Hencethe following question:
Problem 2.
Assume that f : X → X is a continuous expansive map on a compact metric space X satisfying both expansiveness and the specification property. Do generic H¨older continuouspotentials have unique finitely additive equilibrium states? If so, are these weak Gibbs measures? The previous problem also makes sense if one requires C -generic instead of both the ex-pansiveness and specification properties. Indeed, it is known that C -generic continuous mapssatisfy the shadowing property (cf. [80]).A third main problem concerns the link between the exponential decay of correlations (withrespect to some suitable Banach space of observable maps) with the existence of phase transi-tions. Recall that one of the main advantages in proving the exponential decay of correlations isthat the exponential memory loss allows one to approximate the stationary processes, describedby Birkhoff sums, by martingales, for which a wide range of limit theorems is known. Never-theless, the central limit theorem, the law of iterated logarithm and other similar results applyto dynamics exhibiting weaker decay rates. Furthermore, there exist general principles whichenable one to recover limit theorems for sequences of independent random variables in a finitelyadditive setting from their analogues in a probability space (see e.g. [71, 101]); and there arealso counterparts of the Birkhoff’s ergodic theorem in this context [102]. Further investigationof this topic seems of interest, including the following question. Problem 3.
Assume that f : X → X is a continuous map on a compact metric space and let µ be an f -invariant finitely additive equilibrium state associated to a H¨older continuous potential.Describe general conditions under which µ satisfies the central limit theorem. NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 51
Partially hyperbolic maps and flows A C -diffeomorphism f ∈ Diff ( X ) on a compact Riemannian manifold X is called (strongly) partially hyperbolic if there exists a Df -invariant splitting T X = E s ⊕ E c ⊕ E u and N ∈ N suchthat k Df N | E s k < , k Df − N | E u k < and the following domination properties hold k Df N ( x ) v s kk Df N ( x ) v c k
12 and k Df N ( x ) v c kk Df N ( x ) v u k . for every unit vectors v s ∈ E sx , v c ∈ E cx and v u ∈ E ux . A particularly important class of suchdiffeomorphisms are the time-1 maps of hyperbolic flows, such as the geodesic flows on negativelycurved surfaces. While the general construction of equilibrium states for partially hyperbolicdiffeomorphisms remains out of reach, Theorem 1 guarantees that finitely additive equilibriumstates always exist. It is a challenge to find sufficient conditions under which these measuresare actually countably additive, hence classical equilibrium states. A similar question can beaddressed for flows. Here is a class of examples to start with: Problem 4.
Assume that dim X = 3 , that the one-dimensional central subbundle E c is uniquelyintegrable and that the central leaves are compact. Provide conditions under which a finitelyadditive invariant measure is countably additive. Multifractal analysis beyond hyperbolicity
It is known that, for each one-dimensional differentiable map f , the Hausdorff dimension oflevel sets of Birkhoff averages can be described in terms of equilibrium states of the family ofpotentials − t log | f ′ | (see [51, Theorem 3]). Now, one of the key features of the h map providedby Theorem 1 is certainly its upper semi-continuity. As finitely additive equilibrium states existand estimates of Hausdorff dimension (and measure) concern finitely many sets, for which finiteadditivity is enough, one can look at multifractal analysis beyond hyperbolicity. Here is anexample. Problem 5.
Let f : [0 , → [0 , be a C map and X α ⊂ [0 , denote the set of points havingLyapunov exponent equal to α . May one find effective bounds on the Hausdorff dimension dim H ( X α ) using the family ( µ β ) β ∈ R of finitely additive equilibrium states for f with respect tothe potentials ( − β log | f ′ | ) β > ? Nonlinear thermodynamic formalism
A nonlinear thermodynamical formalism is obtained by replacing the usual linear term (av-erage by integration of the potential) by a nonlinear function of the observable map. For in-stance, requiring both expansiveness and upper semi-continuity of the entropy map and entropy-denseness of the ergodic measures, Buzzi and Leplaideur proved in [35] a variational principlefor the nonlinear pressureΠ F ( f, ϕ ) = sup µ ∈ P f ( X ) n h µ ( f ) + F (cid:0) Z ϕ dµ (cid:1)o besides characterizing the nonlinear equilibrium measures and relating them to specific classicalequilibrium measures. In view of the results in Part 2, it is likely that the following problem hasa positive answer. Problem 6.
Assuming that f is expansive and F : R → R is a continuous nonlinear func-tion, does h f coincide with the classical entropy function? Are nonlinear and finitely additiveequilibrium states precisely the same? Sequential dynamical systems
Kolyada and Snoha introduced in [79] the concept of topological entropy of non-autonomousdynamical systems. Afterwards, Kawan proposed in [73] the concept of measure theoreticalentropy for a sequence of probability measures, and used it to prove that the topological entropybounds the measure theoretical entropies of invariant sequences of probability measures ( µ n ) n satisfying ( f n ) ∗ µ n = µ n +1 for all n ∈ N . This way, Kawan obtaining a partial variationalprinciple. As far as we know, a variational principle using this approach is still an open question. Problem 7.
Consider a sequence ( f n ) n ∈ N of continuous maps on a compact metric space X and take the map F : N × X → N × X given by F ( n, x ) = ( n + 1 , f n ( x )) . Under what conditionson F does there exist a version of Theorem 1 for F , thus establishing a new variational principleusing sequences of invariant measures? Recall from Subsection 9.5 that an alternative approach is due to Bi´s [16] and Xu and Zhou[122], who use instead Charat´eodory structures to define the topological entropy, apply an ana-logue of Brin-Katok formula to define the measure-theoretic entropy for every Borel probabilitymeasure, and then deduce a variational principle.18.
Geodesic flows in non-compact phase space
The classical variational principles for topological pressure consider continuous maps actingon compact metric spaces. Even assuming continuity, the non-compactness of the ambient spacebrings forward several difficulties, the first of which is to define a general notion of topologicalpressure. Under these assumptions, some authors define topological pressure as the supremumof the measure theoretical free energies (cf. [98]), hence it satisfies a variational principle.However, while in the compact context the topological entropy of geodesic flows turns out to bedifferentiable or analytic (cf. [72]), the non-compact setting is much less understood.In [42], building over [83], Cioletti, Silva and Stadlbauer considered shift spaces E N on stan-dard Borel sets E , and then gave a variational definition for the entropy and took the logarithmof the spectral radius of a suitable Ruelle-Perron-Frobenius operator, defined using an a priori measure on E , as the definition of topological pressure. While this approach is fitted to transferoperators and is likely to be ready to provide decay rates for these finitely additive equilibriumstates (compare with Problem 3), this topological pressure might fail the paramount goal ofdescribing weighted distinct orbits.A natural dynamical context, arising from geometry, where the ambient space fails to becompact is the context of geodesic flows. For instance, in the context of compact manifoldstopological entropy can be computed as the exponential growth rate of the number of geodesicsconnecting two given points in the manifold [92]. In [67] the authors address the thermodynamicformalism for a class of geodesic flows defined on non-compact manifolds (due to the presence ofcusps) benefiting from the fact that these flows can be modeled by suspension flows defined overcountable Markov shifts. A different strategy to get a thermodynamic formalism of geodesicflows appeared in [93], though not making use of suspension flows. The topological pressure P top,g of the geodesic flow generated by the metric g is a convex function (see Definition 6.2, NTROPY FUNCTIONS AND VARIATIONAL PRINCIPLES 53
Remark 6.5 and Proposition 6.9 in [67]) and satisfies the other two axioms that characterizea pressure function. Thus, Theorem 1 implies that finitely additive equilibrium states alwaysexist. Yet, [67, Theorem 1.3] presents level-one phase transitions for certain parameterizedfamilies of potentials ( tϕ ) t , where ϕ ∈ F , for which classical equilibrium states do not exist aftera transition parameter. This information suggests the following question: Problem 8.
In the context of [67, Theorem 1.3] , does there exist a unique finitely additiveequilibrium state for every potential? If so, can it be used to compute topological pressure usinggeodesic arcs between two points in the manifold as did in [92] for the compact setting?
Actually, the existence of a unique finitely additive equilibrium state is equivalent to theGateaux differentiability of the topological pressure function (cf. [119, Corollary 2]). In particu-lar, a positive answer to the previous question allows one to compare different pressure functions.Let us be more precise. Observe that, under the assumption of upper semi-continuity of theclassical entropy function, the Gateaux differentiability of t ∈ R Γ( tϕ ), where Γ : C ( X ) → R is a pressure function, is equivalent to the uniqueness of the equilibrium state for the potential tϕ , for every t ∈ R (recall Corollary 4). Moreover, the results in Subsection 7.6 assure that thereexists a pressure function Γ : C ([0 , → R which is Gateaux differentiable at some potential ϕ ∈ C ([0 , B d ([0 , ϕ . So,the following question concerns a finer structure of the space of equilibrium states: Problem 9.
Do there exist a continuous map f : X → X on a metric space X whose entropyfunction is upper semi-continuous and for which one may find a pressure function Γ : B d ( X ) → R and a potential ϕ ∈ C ( X ) such that Γ | C ( X ) : C ( X ) → R is Fr´echet differentiable at ϕ while Γ isGateaux but not Fr´echet differentiable at ϕ ? Acknowledgments.
The authors are grateful to G. Iommi, B. Kloeckner, A. Lopes and M.Todd for useful comments. MC, MM and PV were partially supported by CMUP, which is fi-nanced by national funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., under theproject with reference UIDB/00144/2020. PV also benefits from the grant CEECIND/03721/2017of the Stimulus of Scientific Employment, Individual Support 2017 Call, awarded by FCT, andby the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017).
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Faculty of Mathematics, L´od´z University, Poland
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