On factors of Gibbs measures for almost additive potentials

Abstract

Let$(X,{\it\sigma}_{X}),(Y,{\it\sigma}_{Y})$be one-sided subshifts and${\it\pi}:X\rightarrow Y$a factor map. Suppose that$X$has the specification property. Let${\it\mu}$be a unique invariant Gibbs measure for a sequence of continuous functions${\mathcal{F}}=\{\log f_{n}\}_{n=1}^{\infty }$on$X$, which is an almost additive potential with bounded variation. We show that${\it\pi}{\it\mu}$is a unique invariant Gibbs measure for a sequence of continuous functions${\mathcal{G}}=\{\log g_{n}\}_{n=1}^{\infty }$on$Y$. When$(X,{\it\sigma}_{X})$is a full shift, we characterize${\mathcal{G}}$and${\it\mu}$by using relative pressure. This${\mathcal{G}}$is a generalization of a continuous function found by Pollicott and Kempton in their work on factors of Gibbs measures for continuous functions. We also consider the following question: given a unique invariant Gibbs measure${\it\nu}$for a sequence of continuous functions${\mathcal{F}}_{2}$on$Y$, can we find an invariant Gibbs measure${\it\mu}$for a sequence of continuous functions${\mathcal{F}}_{1}$on$X$such that${\it\pi}{\it\mu}={\it\nu}$? We show that such a measure exists under a certain condition. In particular, if$(X,{\it\sigma}_{X})$is a full shift and${\it\nu}$is a unique invariant Gibbs measure for a function in the Bowen class, then there exists a preimage${\it\mu}$of ${\it\nu}$which is a unique invariant Gibbs measure for a function in the Bowen class.