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 $\mathcal{F}=\{\log f_n{\}}_{n=1}^{\mathrm{\infty }}$ on a subshift $X$ over finitely many symbols, we study properties of a function $f$ on $X$ such that ${\lim }_{n\to \mathrm{\infty }}\frac{1}{n}\int \log f_{n}d\mu =\int fd\mu$ for every invariant measure $\mu$ on $X$. Under some conditions we construct a function $f$ on $X$ explicitly and study a relation between the property of $\mathcal{F}$ 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.