摘要： 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 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. If time permits, I will also discuss the applications of the above combinatorial optimization methods in dealing with (non)-Sturiman maximizing (open) problems for doubling maps.
The context of this talk has been presented as the first part of my collaborator Prof. Jairo Bochi’s 45 minutes short talk at ICM in Rio de Janeiro, 2018.
张一威先后在英国埃克塞特大学(英国罗素集团成员，泰晤士排名全英第七)、智利天主教大学(南美前三名校)和波兰科学院数学所跟随Mark Holland教授，Peter Ashwin教授、Juan Rivera-Leterlier教授和Feliks Przytycki教授从事动力系统现代理论与应用的研究工作。 自2009年, 共发表包括Int. Mathem. Res. Notes, Ergodic Theory Dyn. Syst., Biophys. J，Astrophys. J.等在内的国际著名期刊11篇。 并已特约邀请人身份参加2018年国际数学家大会动力系统方向卫星会议，日本京都大学数理解析所动力系统主题2017年会，美国工业数学会2017年会等多个大型国际学术大会，并在会议上做特邀报告。