Relative pressure functions and their equilibrium states

TL;DR

This paper investigates conditions for the existence of functions related to relative pressure in subshifts, characterizing equilibrium states and continuous compensation functions, with applications to invariant measures and Gibbs states.

Contribution

It provides new necessary and sufficient conditions for asymptotically additive sequences and characterizes equilibrium states for relative pressure functions in subshifts.

Findings

Characterization of asymptotically additive sequences via periodic points

Conditions for existence of continuous compensation functions

Application to projections of invariant weak Gibbs measures

Abstract

For a subshift $(X, \sigma_X)$ 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 $\lim_{n\rightarrow\infty}(1/{n})\int \log f_n d \mu=\int h d \mu$ for every invariant measure $\mu$ 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 $\pi: X\rightarrow Y$, where $(X, \sigma_X)$ is an irreducible shift of finite type and $(Y, \sigma_Y)$ 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 $\pi\mu$ of an invariant weak Gibbs measure $\mu$ for a continuous function on an irreducible shift of finite type.