Countable Sofic Shifts with a Periodic Direction

TL;DR

This paper investigates whether all countable multidimensional sofic shifts have countable SFT covers, providing counterexamples and establishing conditions under which such covers exist.

Contribution

It introduces explicit counterexamples to the equal entropy cover problem and formulates conditions for countable SFT covers in certain shift spaces.

Findings

Counterexamples to the equal entropy cover problem

Necessary and sufficient conditions for countable SFT covers in a restricted class

Identification of limitations in existing cover theories

Abstract

As a variant of the equal entropy cover problem, we ask whether all multidimensional sofic shifts with countably many configurations have SFT covers with countably many configurations. We answer this question in the negative by presenting explicit counterexamples. We formulate necessary conditions for a vertically periodic shift space to have a countable SFT cover, and prove that they are sufficient in a natural (but quite restricted) subclass of shift spaces.