On Gibbs measures for almost additive potentials

Given an almost additive sequence of continuous functions with bounded variation F=log fₙ\ₙ₌₁∞ on a subshift X over finitely many symbols, we study properties of a function f on X such that limn→∞1/n∫ log fₙ dμ=∫ f dμ for every invariant measure μ on X. Under some conditions we construct a function f on X and study a relation between the property of F and some particular types of f. We also study the case when F is weakly almost additive. As applications we study images of Gibbs measures for continuous functions under one-block factor maps. In order to study conditions for the image to be a Gibbs measure for a continuous function, we investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-mixing property of a factor map.