Class Degree and Relative Maximal Entropy

TL;DR

This paper introduces the class degree as a generalization of the degree for factor codes between shifts of finite type and sofic shifts, providing bounds on ergodic measures with maximal entropy.

Contribution

It defines the class degree for any factor code on a shift of finite type and establishes an invariant upper bound on ergodic measures with maximal entropy in fibers.

Findings

The class degree matches the degree for finite-to-one codes.

An invariant upper bound on ergodic measures with maximal entropy is established.

A uniform distribution property for ergodic measures of relative maximal entropy is proved.

Abstract

Given a factor code $\pi$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $\pi$ is finite-to-one there is an invariant called the degree of $\pi$ which is defined the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $\nu$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $\nu$ and have maximal entropy among all measures in the fibre $\pi^{-1}\{\nu\}$. We show that this bound and the class degree of the code agree when $\nu$ is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy.