# On the entropy of coverable subshifts

## TL;DR

This paper investigates the conditions under which coverable subshifts in Z^2 have zero topological entropy, providing near-complete characterizations of covers that constrain entropy without enforcing periodicity.

## Contribution

It offers a detailed analysis of covers that determine zero entropy in subshifts, refining previous characterizations and identifying necessary and sufficient conditions.

## Key findings

- Identified conditions that influence zero entropy in coverable subshifts
- Provided necessary and sufficient conditions for zero entropy without periodicity
- Extended understanding of the entropy behavior in coverable subshifts

## Abstract

A coloration w of Z^2 is said to be coverable if there exists a rectangular block q such that w is covered with occurrences of q, possibly overlapping. In this case, q is a cover of w. A subshift is said to have the cover q if each of its points has the cover q. In a previous article, we characterized the covers that force subshifts to be finite (in particular, all configurations are periodic). We also noticed that some covers force subshifts to have zero topological entropy while not forcing them to be finite. In the current paper we work towards characterizing precisely covers which force a subshift to have zero entropy, but not necessarily periodicity. We give a necessary condition and a sufficient condition which are close, but not quite identical.