Decompositions of factor codes and embeddings between shift spaces with unequal entropies

TL;DR

This paper explores how factor codes and embeddings between shift spaces can be decomposed into intermediate steps with entropies densely covering the range between the original spaces, providing new insights into their structure.

Contribution

It introduces a method to decompose factor codes and embeddings between shift spaces with controlled entropy properties, including finite type shifts.

Findings

Decompositions produce dense entropy intervals between original shift spaces.

Finite type shifts can be used as intermediate steps in decompositions.

Characterization of entropy sets for embeddings into irreducible sofic shifts.

Abstract

Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.