TY - JOUR
T1 - Uniqueness of Minimizer for Countable Markov Shifts and Equidistribution of Periodic Points
AU - Takahasi, Hiroki
N1 - Funding Information:
I thank anonymous referees for very useful comments, and Johannes Jaerisch for fruitful discussions. This research was supported by the JSPS KAKENHI 19K21835, 20H01811.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/12
Y1 - 2020/12
N2 - 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.
AB - 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.
KW - Countable Markov shift
KW - Gibbs-equilibrium state
KW - Large deviation principle
KW - Minimizer
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U2 - 10.1007/s10955-020-02670-5
DO - 10.1007/s10955-020-02670-5
M3 - Article
AN - SCOPUS:85095782564
SN - 0022-4715
VL - 181
SP - 2415
EP - 2431
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -