On Gibbs measures for almost additive sequences associated to some relative pressure functions

Given a weakly 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 lim n→ ∞(1/n)∫ log fₙ\,dμ =∫ f\,dμ for every invariant measure μ on X . Under some conditions, we construct a function f on X explicitly, and study a relation between the property of F and some particular types of f . As applications, we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.