# Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states

## TL;DR

This paper proves that infinite-to-one factor codes can be decomposed into simpler components and shows that equilibrium states on the target space uniquely lift to measures of maximal relative entropy on the source space, generalizing previous results.

## Contribution

It introduces a structure theorem for infinite-to-one factor codes and establishes the uniqueness of lifted equilibrium states for regular potentials.

## Key findings

- Factor maps decompose into finite-to-one and class degree one codes
- Unique lift of equilibrium states for regular potentials
- Generalization of finite-to-one case to infinite-to-one codes

## Abstract

We show that an arbitrary factor map $\pi:X \to Y$ on an irreducible subshift of finite type is a composition of a finite-to-one factor code and a class degree one factor code. Using this structure theorem on infinite-to-one factor codes, we then prove that any equilibrium state $\nu$ on $Y$ for a potential function of sufficient regularity lifts to a unique measure of maximal relative entropy on $X$. This answers a question raised by Boyle and Petersen (for lifts of Markov measures) and generalizes the earlier known special case of finite-to-one factor codes.