Uniqueness of Minimizer for Countable Markov Shifts and Equidistribution of Periodic Points
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Uniqueness of Minimizer for Countable Markov Shifts and Equidistribution of Periodic Points Hiroki Takahasi1 Received: 14 May 2020 / Accepted: 2 November 2020 / Published online: 10 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen’s Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that the set of periodic points weighted with the potential equidistributes with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group with parabolic elements. Keywords Large deviation principle · Countable Markov shift · Gibbs-equilibrium state · Minimizer Mathematics Subject Classification 37A45 · 37A50 · 37A60 · 60F10
1 Introduction The theory of large deviations aims to characterize limit behaviors of measures in terms of rate functions. A sequence {μn }∞ n=1 of Borel probability measures on a topological space X is said to satisfy the Large Deviation Principle (LDP) if there exists a lower semi-continuous function I : X → [0, ∞] such that for every Borel subset B of X the following holds: 1 1 log μn (Bo ) ≤ lim log μn (B) ≤ − inf I , n→∞ n n→∞ n B
− info I ≤ lim B
where log 0 = −∞, inf ∅ = ∞, Bo and B denote the interior and the closure of B respectively. The function I is called a rate function, and is called a good rate function if the level set {x ∈ X : I (x) ≤ α} is compact for every α ∈ (0, ∞).
Communicated by Aernout van Enter.
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Hiroki Takahasi [email protected] Department of Mathematics, Keio Institute of Pure and Applied Sciences (KiPAS), Keio University, Yokohama 223-8522, Japan
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If X is a metric space, then the rate function is unique. We call x ∈ X a minimizer if I (x) = 0. For a closed set B of X which is disjoint from the set of minimizers, the LDP ensures that μn (B) decays exponentially as n → ∞. If moreover I is a good rate function, the support of any accumulation point of {μn }∞ n=1 is contained in the set of minimizers. Hence, it is important to determine the set of minimizers. The non-uniqueness of minimizer is referred to as a phase transition. The uniqueness of minimizer implies several strong conclusions. The setting in our mind is that X is a topological space on which a dynamical system σ acts, and X is the space of Borel probability measures on X . The sequence {μn }∞ n=1 is a sequence of Borel probability measures on X such that μn is determined by the time evolution σ, σ 2 , . . . , σ n−1 . The LDP in this case is called the level-2 LDP. This paper is concerned with the level-2 LDP in the thermodynamic formalism. For finite topological Markov shifts and Hölder continuous potentials, the variational principle and
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