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On factors of Gibbs measures for almost additive potentials

Published online by Cambridge University Press:  11 August 2014

YUKI YAYAMA
YUKI YAYAMA*
Affiliation:
Departamento de Ciencias Básicas, Universidad del Bío-Bío, Av. Andrés Bello, s/n Casilla 447 Chillán, Chile email yyayama@ubiobio.cl
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016 , pp. 276 - 309
Copyright
© Cambridge University Press, 2014 

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