Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors

TL;DR

This paper establishes new sufficient conditions for intrinsic ergodicity in shift spaces and their factors, demonstrating that beta-shifts and S-gap shifts are intrinsically ergodic, and introduces a novel specification property for non-uniform symbolic systems.

Contribution

It provides the first comprehensive criteria ensuring intrinsic ergodicity for shift spaces and their factors, including beta-shifts and S-gap shifts, using a new version of the specification property.

Findings

Every subshift factor of a beta-shift is intrinsically ergodic.

Every subshift factor of an S-gap shift is intrinsically ergodic.

Introduces a new specification property adapted to non-uniform symbolic spaces.

Abstract

We give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be intrinsically ergodic, along with sufficient conditions for every subshift factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that every subshift factor of a $\beta$-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.