Characteristic measures for language stable subshifts

TL;DR

This paper introduces language stable subshifts, a broad class of symbolic dynamical systems, and proves they support characteristic measures, extending known results about mixing subshifts and zero entropy systems.

Contribution

The paper defines language stable subshifts and demonstrates that they support characteristic measures, expanding the class of systems known to have this property.

Findings

Language stable subshifts form a large, generic class.

All mixing subshifts of finite type support characteristic measures.

Many positive entropy examples are included within this class.

Abstract

We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class of shifts, which we call language stable subshifts, and show that these shifts have characteristic measures. This is a large class that is generic in several senses and contains numerous positive entropy examples.