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arXiv · 2109.02180

Relative pressure functions and their equilibrium states

Yuki Yayama
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Abstract

For a subshift $(X, σ_X)$ and a subadditive sequence $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on $X$, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim_{n\rightarrow\infty}(1/{n})\int \log f_n d μ=\int h d μ$ for every invariant measure $μ$ on $X$. For this purpose, we first we study necessary and sufficient conditions for $\mathcal{F}$ to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map $π: X\rightarrow Y$, 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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Yuki Yayama. 2021-10-03. Relative pressure functions and their equilibrium states. https://arxiv.org/abs/2109.02180

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