Existence of a measurable saturated compensation function between subshifts and its applications

Yayama, Yuki

doi:10.1017/s0143385710000404

Cited by 18 publications

(42 citation statements)

References 21 publications

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“…y n−1 ]. This function agrees with the usual relative topological entropy almost everywhere with respect to every invariant measure [11], and −g • π serves as a measurable, although likely not continuous, compensation function [17]. Some of the notation used in this paper deviates from that which is used in its predecessors.…”

Section: Introductionsupporting

confidence: 52%

“…Saturated compensation functions are those which can be written as g • π such as the relative topological entropy above. They have been studied in [13] and [17] for their relation to weighed entropy and Hausdorff measure. In [16] it was shown that factors with a compensation function in the Walters class of continuous functions have nice lifting properties for equilibrium states.…”

Section: Introductionmentioning

confidence: 99%

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Compensation functions for factors of shifts of finite type

Antonioli¹

2014

Ergod. Th. Dynam. Sys.

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We consider an infinite-to-one factor map from an irreducible shift of finite type X to a sofic shift Y. A compensation function relates equilibrium states on X to equilibrium states on Y. The p-Dini condition is given as a way of measuring the smoothness of a continuous function, with 1-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. We show that the relative equilibrium states of a 1-Dini function f over a fully supported invariant measure on Y are themselves fully supported, and have positive relative entropy. We then show that there exists a compensation function which is p-Dini for all p > 1 which has relative equilibrium states supported on a finite-to-one subfactor.Comment: 20 pages, 1 figure submitted to Ergodic Theory Dyn. Sys

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Section: Introductionsupporting

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Compensation functions for factors of shifts of finite type

Antonioli¹

2014

Ergod. Th. Dynam. Sys.

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“…Using arguments similar to those in the final part of the proof of Lemma 4.8 of [34], we obtainC 1 ,C 2 > 0 such that…”

Section: Proof Of Corollary 32mentioning

confidence: 91%

“…In [12,34,35], this problem was studied by using equilibrium states for sequences of continuous functions. The factor of equilibrium states for sequences of continuous functions and their preimages were studied (see Section 4 and Example 5.2 [34]).…”

Section: Introductionmentioning

confidence: 99%

On factors of Gibbs measures for almost additive potentials

Yayama¹

2014

Ergod. Th. Dynam. Sys.

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Let $(X, \sigma_X), (Y, \sigma_Y)$ be one-sided subshifts with the specification property and $\pi:X\rightarrow Y$ a factor map. Let $\mu$ be a unique invariant Gibbs measure for a sequence of continuous functions $\F=\{\log f_n\}_{n=1}^{\infty}$ on $X$, which is an almost additive potential with bounded variation. We show that $\pi\mu$ is also a unique invariant Gibbs measure for a sequence of continuous functions $\G=\{\log g_n\}_{n=1}^{\infty}$ on $Y$. When $(X, \sigma_X)$ is a full shift, we characterize $\G$ and $\mu$ by using relative pressure. This almost additive potential $\G$ is a generalization of a continuous function found by Pollicott and Kempton in their work on the images of Gibbs measures for continuous functions under factor maps. We also consider the following question: Given a unique invariant Gibbs measure $\nu$ for a sequence of continuous functions $\F_2$ on $Y$, can we find an invariant Gibbs measure $\mu$ for a sequence of continuous functions $\F_1$ on $X$ such that $\pi\mu=\nu$? We show that such a measure exists under a certain condition. If $(X, \sigma_X)$ is a full shift and $\nu$ is a unique invariant Gibbs measure for a function in the Bowen class, then we can find a preimage $\mu$ of $\nu$ which is a unique invariant Gibbs measure for a function in the Bowen class.Comment: 33 pages, To appear in Ergodic Theory and Dynamical System

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“…This notion of pressure was also extended for sequences of regular functions defined on subshifts of finite type by Barreira [B1,B2,B3], Falconer [F1], Feng [Fe1,Fe2,Fe3] and Cao, Feng and Huang [CFH] among others. Actually, assumption (C2) was introduced by Feng [Fe3] while studying thermodynamic formalism for potentials related to product of matrices and appeared also in the study of dimension of non-conformal repellers [Fe4,Y1]. Moreover, when (X, σ) is a subshift on a finite alphabet, Feng [Fe4] studied thermodynamic formalism for the class of sequences which satisfies (C1) and (C2) (see Theorem 5.2).…”

Section: 2mentioning

confidence: 99%

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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Existence of a measurable saturated compensation function between subshifts and its applications

Cited by 18 publications

References 21 publications

Compensation functions for factors of shifts of finite type

Compensation functions for factors of shifts of finite type

On factors of Gibbs measures for almost additive potentials

Hidden Gibbs measures on shift spaces over countable alphabets

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