Ergodic optimization for continuous functions on the Dyck-Motzkin shifts

TL;DR

This paper investigates ergodic optimization on Dyck-Motzkin shifts, revealing a complex structure of maximizing measures and the path connectedness of ergodic measure space, highlighting the rich dynamics of these non-intrinsically ergodic systems.

Contribution

It characterizes the space of continuous functions on Dyck-Motzkin shifts, showing a dichotomy in maximizing measures and establishing the path connectedness of ergodic measures.

Findings

Dense G_delta set of functions with zero entropy maximizing measures

Closure of functions with uncountably many Bernoulli measures

Path connectedness of ergodic measure space

Abstract

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not intrinsically ergodic. We show that the space of continuous functions on any Dyck-Motzkin shift splits into two subsets: one is a dense $G_\delta$ set with empty interior for which any maximizing measure has zero entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported measures that are Bernoulli. One key ingredient of a proof of this result is the path connectedness of the space of ergodic measures of the Dyck-Motzkin shift.