On the number of ergodic measures for minimal shifts with eventually constant complexity growth

TL;DR

This paper investigates the number of ergodic measures in minimal shifts with eventually constant complexity growth, providing an improved bound and introducing special Rauzy graphs to describe their structure.

Contribution

It establishes a sharper bound on ergodic measures for minimal shifts with eventually constant growth and introduces special Rauzy graphs for their analysis.

Findings

Improved bound on the number of ergodic measures under stronger growth conditions

Introduction of special Rauzy graphs to analyze symbolic dynamics

Explicit description of Rauzy graph relations for all word lengths

Abstract

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length n relates to the one for words of length n+1 for each n = 1,2,3...