Résumé : Ergodic optimization is a subject of find invariant measures that maximize the integral of a given performance function. 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 talk we address this problem using a probabilistic notion of typicallity 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-Good graph.
If time permits, we will also discuss some numerical algorithm on identifying Proper P for the Holder continuous performance functions.
Lieu : Salle 106, bâtiment 1R1