2019/04/03 by Huang, Wen, Lian, Zeng, Ma, Xiao +2 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.01915
In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems T:X→ X where X is a compact metric space and T is Lipschitz continuous. We show that once T:X→ X satisfies both the \em Anosov shadowing property (\bf ASP) and the \em Mañé-Conze-Guivarc'h-Bousch property (\bf MCGBP), the minimizing measures of generic Hölder observables are unique and supported on a periodic orbit. Moreover, if T:X→ X is a subsystem of a dynamical system f:M→ M (i.e. X⊂ M and f|X=T) where M is a compact smooth manifold, the above conclusion holds for C1 observables. Note that a broad class of classical dynamical systems satisfies both ASP and MCGBP, which includes \em Axiom A attractors, Anosov diffeomorphisms and \em uniformly expanding maps. Therefore, the open problem proposed by Yuan and Hunt in \citeYH for C1-observables is solved consequentially.