Ergodic Optimization for Continuous Functions on the Dyck-Motzkin Shifts

Abstract

Ergodic optimization aims to describe properties of 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 with non-unique maximal entropy measures. We show that the space of continuous functions on any Dyck-Motzkin shift contains two disjoint subsets: one is a dense \(G_\delta \) set with empty interior for which any maximizing measure is not mixing and has zero entropy; the other is a dense set of functions for which there exist uncountably many, fully supported maximizing measures that are Bernoulli. Key ingredients of a proof of this result are the density of closed orbit measures in the space of ergodic measures and the path connectedness of the space of ergodic measures of any Dyck-Motzkin shift.