On factors of Gibbs measures for almost additive potentials

Abstract

Let $(X,{\sigma }_X),(Y,{\sigma }_Y)$ be one-sided subshifts with the specification property and $\pi :X\to Y$ 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.

33 pages, To appear in Ergodic Theory and Dynamical Systems

References in corpus (5)

• Lyapunov spectrum of asymptotically sub-additive potentials
• On the topological pressure of random bundle transformations in sub-additive case
• Almost-additive thermodynamic formalism for countable Markov shifts
• On the preservation of Gibbsianness under symbol amalgamation
• Projection of Markov measures may be Gibbsian

Cited by in corpus (3)

• Additive, almost additive and asymptotically additive potential sequences are equivalent
• On Gibbs measures for almost additive sequences associated to some relative pressure functions
• Sequential Gibbs Measures and Factor Maps