Ergodic optimization of prevalent super-continuous functions

TL;DR

This paper investigates the typicality of performance functions that maximize time averages at periodic orbits in hyperbolic dynamical systems, using prevalence in infinite-dimensional spaces and wavelet approximations.

Contribution

It proves that property P is prevalent among regular functions for the one-sided shift on two symbols using wavelet-based finite-dimensional approximations.

Findings

Property P is prevalent in certain function spaces.

Wavelet approximation reduces the problem to a finite-dimensional max cycle mean problem.

The approach applies to hyperbolic dynamical systems with strong regularity.

Abstract

Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph.