Ergodic Optimization of Prevalent Super-continuous Functions

Jairo Bochi, Yiwei Zhang

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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.

Original languageEnglish (US)
Pages (from-to)5988-6017
Number of pages30
JournalInternational Mathematics Research Notices
Volume2016
Issue number19
DOIs
StatePublished - 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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