Ergodic optimization is the study of problems relating to maximizing invariant measures and maximum ergodic averages. In ergodic optimization theory, one important problem is the typically periodic optimization (TPO) conjecture. This conjecture was proposed by Mañé[6], Hunt, Ott and Yuan in the 1990s, which reveals the principle of least action in dynamical system settings. To be more precise, TPO indicates that when the dynamical system is suitably hyperbolic and the observable is suitably regular, then the maximizing measure is "genetically" supported on a periodic orbit with relatively low period. In the setting of uniformly open expanding maps with Lipschitz/Holder observables, TPO was obtained in topological genetic sense by Contreras in 2016. In this talk, I will report a numer recent progress on understanding TPO conjecture both in probabilistic and topological sense, and for more general uniformly and non-uniformly hyperbolic systems.

Meeting ID: 852 4277 3200 Passcode: 103121