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