Counting generic measures for a subshift of linear growth

TL;DR

This paper establishes the sharp upper bound on the number of ergodic and generic measures for subshifts of linear growth, extending the result to non-minimal cases and applying it to interval exchange transformations.

Contribution

It proves that Boshernitzan's bound is optimal for ergodic and generic measures in subshifts of linear growth, even without minimality assumptions.

Findings

Boshernitzan's bound is sharp for ergodic measures.

The bound applies to nonatomic generic measures.

The result extends to non-minimal systems and interval exchange transformations.

Abstract

In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that Boshernitzan's bound is sharp. We further prove that the same bound holds for the, a priori, larger set of nonatomic generic measures, and that this bound remains valid even if one drops the assumption of minimality. Applying these results to interval exchange transformations, we give an upper bound on the number of nonatomic generic measures of a minimal IET, answering a question recently posed by Chaika and Masur.