On factors of Gibbs measures for almost additive potentials

Let $(X, \sigma_X), (Y, \sigma_Y)$ be one-sided subshifts with the specification property and $\pi:X\rightarrow Y$ 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.