Multiplicity of measures under factor codes and class degree joinings

TL;DR

This paper introduces the concept of multiplicity for ergodic measures under factor codes between shifts, establishing a relationship between degree and measure multiplicities, and extends these ideas to infinite-to-one codes using class degree joinings.

Contribution

It defines measure multiplicity for factor codes and generalizes the relationship between degree and ergodic measures to infinite-to-one codes with class degree joinings.

Findings

Degree equals the sum of measure multiplicities.

Introduces degree and class degree joinings as key tools.

Extends results to infinite-to-one factor codes.

Abstract

Given a finite-to-one factor code $\pi: X \to Y$ between irreducible sofic shifts and an ergodic $\nu$ on $Y$ with full support, it is known that the fiber $\pi^{-1}_*(\nu)$ has at most $d_\pi$ ergodic measures in it where $d_\pi$ is the degree of $\pi$. We introduce the notion of multiplicity for ergodic measures on $X$ (that depends on $\pi$) and we prove that $d_\pi$ is the sum of the multiplicity of $\mu$ where $\mu$ runs over the ergodic measures in $\pi^{-1}_*(\nu)$. We also build an appropriate generalization to infinite-to-one factor codes in relation to class degree and relatively maximal measures. We also define the notion of degree joining (for finite-to-one factor codes) and class degree joining (for infinite-to-one factor codes) which are the main tool for establishing our results