Maximizing entropy for power-free languages

TL;DR

This paper investigates the entropy properties of power-free languages, proving the uniqueness of the measure of maximal entropy in many cases using a novel approach related to Bowen's specification property.

Contribution

It introduces a new method to prove the uniqueness of the measure of maximal entropy for power-free shift spaces, despite the absence of periodic points.

Findings

Power-free languages exhibit positive topological entropy.

Unique measure of maximal entropy exists in many cases.

Lack of periodic points contrasts with typical specification-based results.

Abstract

A power-free language is characterized by the number of symbols used and a limit on how many times a block of symbols can repeat consecutively. For certain values of these parameters, it is known that the number of legal words grows exponentially fast with respect to length. In the terminology of dynamical systems and ergodic theory, this means that the corresponding shift space has positive topological entropy. We prove that in many cases, this shift space has a unique measure of maximal entropy. The proof uses a weak analogue of Bowen's specification property. The lack of any periodic points in power-free shift spaces stands in striking contrast to other applications of specification-based techniques, where the number of periodic points often has exponential growth rate given by the topological entropy.