Relative pressure functions and their equilibrium states

2021/09/05 by Yayama, Yuki
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.02180

Abstract

For a subshift (X, σ_X) and a subadditive sequence F=\log f_n\_n=1^∞ on X, we study equivalent conditions for the existence of h∈ C(X) such that lim_n→∞(1/n)∫ log f_n d μ=∫ h d μ for every invariant measure μ on X. For this purpose, we first we study necessary and sufficient conditions for F to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map π: X→ Y, where (X, σ_X) is an irreducible shift of finite type and (Y, σ_Y) is a subshift, applying our results and the results obtained by Cuneo [7] on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application, we study we study the projection πμ of an invariant weak Gibbs measure μ for a continuous function on an irreducible shift of finite type.

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