Strong Approximations of Shifts and the Characteristic Measures Problem

TL;DR

This paper explores conditions under which symbolic systems support characteristic measures invariant under all automorphisms, including measures of maximal entropy, expanding understanding of invariant measures in dynamical systems.

Contribution

It provides sufficient conditions for the existence of characteristic measures in symbolic systems, applicable to a large class of shifts and their factors.

Findings

Conditions for characteristic measures are satisfied by a dense G_delta set of shifts.

Characteristic measures can often be chosen as measures of maximal entropy.

The class of systems satisfying these conditions is closed under taking factors.

Abstract

Every symbolic system supports a Borel measure that is invariant under the shift, but it is not known if every such systems supports a measure that is invariant under all of its automorphisms; known as a characteristic measure. We give sufficient conditions to find a characteristic measure, additionally showing when it can be taken to be a measure of maximal entropy. The class of systems to which these sufficient conditions apply is large, containing a dense $G_{\delta}$ set in the space of all shifts on a given alphabet, and is also large in the sense that it is closed under taking factors. We also investigate natural systems to which these sufficient conditions apply.