Ergodic optimization for continuous functions on non-Markov shifts

TL;DR

This paper studies ergodic optimization on non-Markov shifts, revealing a dichotomy in the structure of continuous functions based on the entropy of their maximizing measures, extending previous results to broader classes.

Contribution

It generalizes and unifies prior results on ergodic optimization, applying to a wide range of intrinsically ergodic non-Markov symbolic systems without Bowen's specification property.

Findings

The space of continuous functions splits into two subsets with distinct entropy properties of maximizing measures.

The results apply to various systems including piecewise monotonic maps, coded shifts, and multidimensional beta-transformations.

An example of an intrinsically ergodic subshift with positive obstruction entropy is provided.

Abstract

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space splits into two subsets: one is a $G_\delta$ dense set for which all maximizing measures have `relatively small' entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported ergodic measures with `relatively large' entropy. This result considerably generalizes and unifies the results of Morris (2010) and Shinoda (2018), and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without Bowen's specification property, including any transitive piecewise monotonic interval map, some coded shifts and multidimensional $\beta$-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.