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arXiv · 1501.00961

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.

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Jairo Bochi, Yiwei Zhang. 2015-11-06. Ergodic optimization of prevalent super-continuous functions. https://arxiv.org/abs/1501.00961

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