Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures

Uijin Jung

doi:10.3390/e18120428

Abstract

A sliding block code π : X → Y between shift spaces is called fiber-mixing if, for every x and x ′ in X with y = π ( x ) = π ( x ′ ) , there is z ∈ π - 1 ( y ) which is left asymptotic to x and right asymptotic to x ′ . A fiber-mixing factor code from a shift of finite type is a code of class degree 1 for which each point of Y has exactly one transition class. Given an infinite-to-one factor code between mixing shifts of finite type (of unequal entropies), we show that there is also a fiber-mixing factor code between them. This result may be regarded as an infinite-to-one (unequal entropies) analogue of Ashley’s Replacement Theorem, which states that the existence of an equal entropy factor code between mixing shifts of finite type guarantees the existence of a degree 1 factor code between them. Properties of fiber-mixing codes and applications to factors of Gibbs measures are presented.

Highlights

It is well known that for any factor code π : X → Y from an irreducible shift of finite type onto a sofic shift with equal topological entropy, there is a uniform upper bound on the number of preimages of the points in Y
As finding a conjugacy between two shifts of finite type is one of the very difficult problems in the field, finding a factor code of degree 1 has been investigated in many classification problems [1,2,3]
A fiber-mixing factor code from a shift of finite type is a code of class degree 1 for which each point of Y has exactly one transition class; that is, it is a constant-class-to-one code of class degree 1 [7]

Summary

Introduction

It is well known that for any factor code π : X → Y from an irreducible shift of finite type onto a sofic shift with equal topological entropy, there is a uniform upper bound on the number of preimages of the points in Y. In the early 1990s, Ashley showed that if there is a factor code between equal entropy mixing shifts of finite type, there exists a factor code of degree 1 [4]. This was referred to as Replacement Theorem in [5].

Preliminaries and Fiber-Mixing Codes

Existence of Fiber-Mixing Codes

Application