Almost-additive thermodynamic formalism for countable Markov shifts

Iommi, Godofredo; Yayama, Yuki

doi:10.1088/0951-7715/25/1/165

Cited by 16 publications

(45 citation statements)

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“…Remark 5.2. Theorem 5.1 generalizes [IY1,Theorem 4.1] in which almost-additive Bowen sequences on finitely primitive countable Markov shifts are considered. If G = {log g n } ∞ n=1 is an almost-additive Bowen sequence, then…”

Section: Invariant Gibbs Measures and Uniqueness Of Gibbs Equilibriummentioning

confidence: 84%

“…In all these cases, it was shown that the definition does not depend on the symbol a chosen. The Gurevich pressure was defined and studied for almostadditive sequences on topologically mixing countable Markov shifts by Iommi and Yayama [IY1]. We stress that the definition given here extends both the class of sequences of potentials and the class of shifts (satisfying the weak specification) previously considered in the literature.…”

Section: 2mentioning

confidence: 88%

“…Existence of Gibbs measure was studied in [IY1,IY2] for an almost-additive sequence on a topologically mixing countable Markov shift with BIP property and in [KR,Theorem 3.7] for a class of sub-additive Bowen sequences on the full shift on a countable alphabet satisfying (C2), (C3) and (C4). Here we will generalize these results by considering a finitely irreducible countable sofic shift.…”

Section: Invariant Gibbs Measures and Uniqueness Of Gibbs Equilibriummentioning

confidence: 99%

“…Differences with the work in [IY1,IY2] are discussed in Section 3.1. In particular, not every almost-additive sequence studied in [IY1] is in the class of sequences we study here (see Example 3.2). This phenomenon is different from what is observed in the compact case, in which every almost-additive sequence satisfies the assumptions we consider.…”

Section: Introductionmentioning

confidence: 99%

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Hidden Gibbs measures on shift spaces over countable alphabets

2019

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We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets.∞ n=0 B n (X), i.e., the union of all allowable words of X and the empty word ε.We now define several notions of specification that generalize the one first introduced by Bowen [Bo] with the purpose of proving that there exits a unique measure of maximal entropy for a large class of compact subshifts. Our definitions are given in terms of the language of X.Definition 2.1. We say that a subshift (X, σ) on a countable alphabet is irreducible if for any allowable words u, v ∈ B(X), there exists an allowable word w ∈ B(X) such that uwv ∈ B(X).Definition 2.2. We say that a subshift (X, σ) on a countable alphabet has the weak specification property if there exists p ∈ N such that for any allowable words u, v ∈ B(X), there exist 0 ≤ k ≤ p and w ∈ B k (X) such that uwv ∈ B(X). If in addition, k = p for any u and v, then X has the strong specification property. We call such p a weak (strong, respectively) specification number. Definition 2.3. A subshift (X, σ) is finitely irreducible if there exist p ∈ N and a finite subset W 1 ⊂ p n=0 B n (X) such that for every u, v ∈ B(X), there exists w ∈ W 1 such that uwv ∈ B(X). GODOFREDO IOMMI, CAMILO LACALLE, AND YUKI YAYAMADefinition 2.4. A subshift (X, σ) is finitely primitive if there exist p ∈ N and a finite subset W 1 ⊂ B p (X) such that for every u, v ∈ B(X), there exists w ∈ W 1 such that uwv ∈ B(X).Remark 2.1. Note that the weak specification property does not imply topologically mixing. However, if (Σ, σ) is a topologically mixing subshift of finite type defined on a finite alphabet with the weak specification property, then it has the strong specification property (see [J1, Lemma 3.2]). The class of general shifts on finite alphabets with the weak specification property include irreducible sofic shifts (see [J1] and Definition 2.11).As it is clear from the definition, the notion of finitely primitive (see Definition 2.4) is essentially the same as that of specification introduced by Bowen [Bo] in a non-compact symbolic setting. There is a closely related class of countable Markov shifts studied by Sarig [S3].Definition 2.5. A countable Markov shift (Σ, σ) is said to satisfy the big images and preimages property (BIP property) if there exists {b 1 , b 2 , . . . , b n } in the alphabet S such that for every a ∈ S there exist i, j ∈ {1, . . . , n} such that t bia t abj = 1.Remark 2.2. If the countable Markov shift (Σ, σ) satisfies the BIP property, then for every symbol in the a...

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Section: Invariant Gibbs Measures and Uniqueness Of Gibbs Equilibriummentioning

confidence: 84%

Section: 2mentioning

confidence: 88%

Section: Invariant Gibbs Measures and Uniqueness Of Gibbs Equilibriummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hidden Gibbs measures on shift spaces over countable alphabets

2019

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“…Our approach uses ideas from the non-additive thermodynamical formalism and we estimate the measure of deviation sets in the case that the reference measure satisfies a weak Gibbs property. Some recent results considering the thermodynamical formalism of almost additive or sub-additive sequences of potentials include [4,19,20,22,29]: in all cases the authors proved that there exists a unique equilibrium state µ and it is absolutely continuous with respect to a Gibbs measure ν with density bounded away from zero and infinity. Building on [3], Méson and Vericat [28] obtained bounds for large deviations processes for a family of non-additive potentials = {ϕ n }, namely those such that ϕ n − ϕ n−1 • f converge uniformly.…”

Section: Introductionmentioning

confidence: 99%

Weak specification properties and large deviations for non-additive potentials

Varandas

Yun

2013

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385713000667How to cite this article: PAULO VARANDAS and YUN ZHAO (2015). Weak specication properties and large deviations for non-additive potentials. Ergodic Theory and Dynamical Systems, 35, pp 968-993Abstract. We obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given. IntroductionThe purpose of the theory of large deviations is to study the rates of convergence of sequences of random variables to some limit distribution. Some applications of these ideas into the realm of dynamical systems have been particularly useful to estimate the velocity at which time averages of typical points of ergodic invariant measures converge to the space average as guaranteed by Birkhoff's ergodic theorem. More precisely, given a continuous transformation f on a compact metric space M and a reference measure ν, one interesting question is to obtain sharp estimates for the ν-measure of the deviation sets {x ∈ M : (1/n) n−1 j=0 g( f j (x)) > c} for all continuous functions g : M → R and real numbers c. We refer the reader to [1, 11, 16-18, 24-27, 31, 33, 35, 42, 43, 45] and the references therein for an account of recent large deviations results.Since many relevant quantities in dynamical systems arise from non-additive sequences, e.g. the largest Lyapunov exponent for higher-dimensional dynamical systems, Kingman's ergodic theorem becomes in many situations crucial in the study of deviation sets {x ∈ M : ϕ n (x) > cn} with respect to some not necessarily additive sequence = {ϕ n } n of continuous functions. Inspired by the pioneering work of Young [43] our purpose in this direction is to provide sharp large deviations estimates for a wide class of nonadditive sequences of continuous potentials. Our approach uses ideas from the non-additive

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On Entropy Production of Repeated Quantum Measurements I. General Theory

Benoist

Jakšić

Pautrat

et al. 2017

Commun. Math. Phys.

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Abstract. We study entropy production (EP) in processes involving repeated quantum measurements of finite quantum systems. Adopting a dynamical system approach, we develop a thermodynamic formalism for the EP and study fine aspects of irreversibility related to the hypothesis testing of the arrow of time. Under a suitable chaoticity assumption, we establish a Large Deviation Principle and a Fluctuation Theorem for the EP.

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Almost-additive thermodynamic formalism for countable Markov shifts

Cited by 16 publications

References 53 publications

Hidden Gibbs measures on shift spaces over countable alphabets

Hidden Gibbs measures on shift spaces over countable alphabets

Weak specification properties and large deviations for non-additive potentials

On Entropy Production of Repeated Quantum Measurements I. General Theory

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