On Factor Maps that Send Markov Measures to Gibbs Measures

Abstract

Let X and Y be mixing shifts of finite type. Let p be a factor map from X to Y that is fiber-mixing, i.e., given x, (x) over bar is an element of X with pi((x) over bar) = y is an element of Y, there is z is an element of pi(-1)(y) that is left asymptotic to x and right asymptotic to (x) over bar. We show that any Markov measure on X projects to a Gibbs measure on Y under pi (for a Holder continuous potential). In other words, all hidden Markov chains (i.e. sofic measures) realized by pi are Gibbs measures. In 2003, Chazottes and Ugalde gave a sufficient condition for a sofic measure to be a Gibbs measure. Our sufficient condition generalizes their condition and is invariant under conjugacy and time reversal. We provide examples demonstrating our result.