American Institute of Mathematical Sciences

2018, 13: 271-284. doi: 10.3934/jmd.2018021

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

 Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Received  April 30, 2017 Revised  November 22, 2017 Published  December 2018

Fund Project: Supported by the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159

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.

Citation: Jisang Yoo. Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states. Journal of Modern Dynamics, 2018, 13: 271-284. doi: 10.3934/jmd.2018021
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