Factor maps and invariant measures in symbolic dynamics

Abstract

Factor maps between shift spaces are surjective maps which preserve symbolic dynamical structure. A factor map extends to a map from the set of the invariant measures on the domain onto that of the codomain. The purpose of this thesis is to study how invariant measures behave with respect to factor maps between shift spaces. First, we study when sofic measures are Gibbs measures. Consider a fiber-mixing factor map $\pi: X \to Y$ between two mixing shifts of finite type. We prove that any fully supported Markov measure on $X$ projects to a Gibbs measure on $Y$ under the map $\pi$. In other words, all hidden Markov chains realized by $\pi$ are Gibbs measures. This generalizes a result of Chazottes and Ugalde by guaranteeing that the condition is invariant under conjugacy and symmetric under time reversal. Second, we investigate properties of relatively maximal measures. Given an irreducible shift of finite type $X$, a shift space $Y$, a factor map $\pi : X \to Y$, and a fully supported invariant measure $\nu$ on $Y$, we show that any measure of maximal entropy among the measures in $\pi^{-1}(\nu)$ is fully supported. We also show that for any ergodic fully supported measure $ \nu $ on $ Y $, there is an ergodic fully supported measure in $\pi^{-1}(\nu)$. Finally, we study ranks of semigroups which are related to degrees of factor maps. We prove that if one extends a transitive automaton by adding new states and letters, and there is a word sending all new states to old states, then the rank of the new automaton divides the original rank.