On factors of Gibbs measures for almost additive potentials

• Published in: arXiv.org 2014-07
• Main Author: Yayama, Yuki
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Invariants; Nominations; Tornadoes
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1309.7703

Let \((X, \sigma_X), (Y, \sigma_Y)\) be one-sided subshifts with the specification property and \(\pi:X\rightarrow Y\) a factor map. Let \(\mu\) 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 \(\pi\mu\) is also a unique invariant Gibbs measure for a sequence of continuous functions \(\G=\{\log g_n\}_{n=1}^{\infty}\) on \(Y\). When \((X, \sigma_X)\) is a full shift, we characterize \(\G\) and \(\mu\) 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 \(\nu\) for a sequence of continuous functions \(\F_2\) on \(Y\), can we find an invariant Gibbs measure \(\mu\) for a sequence of continuous functions \(\F_1\) on \(X\) such that \(\pi\mu=\nu\)? We show that such a measure exists under a certain condition. If \((X, \sigma_X)\) is a full shift and \(\nu\) is a unique invariant Gibbs measure for a function in the Bowen class, then we can find a preimage \(\mu\) of \(\nu\) which is a unique invariant Gibbs measure for a function in the Bowen class.