On factors of Gibbs measures for almost additive potentials

Abstract

Let $(X, σ_X), (Y, σ_Y)$ be one-sided subshifts with the specification property and $π:X\rightarrow Y$ a factor map. Let $μ$ be a unique invariant Gibbs measure for a sequence of continuous functions $\F=\{\log f_n\}_{n=1}^{\infty}$ on $X$, which is an almost additive potential with bounded variation. We show that $πμ$ is also a unique invariant Gibbs measure for a sequence of continuous functions $\G=\{\log g_n\}_{n=1}^{\infty}$ on $Y$. When $(X, σ_X)$ is a full shift, we characterize $\G$ and $μ$ by using relative pressure. This almost additive potential $\G$ is a generalization of a continuous function found by Pollicott and Kempton in their work on the images of Gibbs measures for continuous functions under factor maps. We also consider the following question: Given a unique invariant Gibbs measure $ν$ for a sequence of continuous functions $\F_2$ on $Y$, can we find an invariant Gibbs measure $μ$ for a sequence of continuous functions $\F_1$ on $X$ such that $πμ=ν$? We show that such a measure exists under a certain condition. If $(X, σ_X)$ is a full shift and $ν$ is a unique invariant Gibbs measure for a function in the Bowen class, then we can find a preimage $μ$ of $ν$ which is a unique invariant Gibbs measure for a function in the Bowen class.