Ergodic optimization for Gauss's continued fraction map

TL;DR

This paper extends ergodic optimization theory to Gauss's continued fraction map, characterizing invariant measures, exploring limit-maximizing measures, and analyzing the typical periodic optimization property for specific function classes.

Contribution

It introduces a new framework for ergodic optimization in the context of Gauss's continued fraction map, including measure characterization and properties of optimization conjectures.

Findings

Characterization of the closure of invariant probability measures.

Failure of the typical periodic optimization conjecture.

Validation of the TPO property for certain function classes.

Abstract

The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for H\"older continuous functions. Although a Ma\~n\'e cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $\alpha$-H\"older essentially compact functions.