Ergodic optimization theory for a class of typical maps

Abstract

In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems $T:X\to X$ where $X$ is a compact metric space and $T$ is Lipschitz continuous. We show that once $T:X\to X$ satisfies both the {\em Anosov shadowing property }({\bf ASP}) and the {\em Ma\~n\'e-Conze-Guivarc'h-Bousch property }({\bf MCGBP}), the minimizing measures of generic H\"older observables are unique and supported on a periodic orbit. Moreover, if $T:X\to X$ is a subsystem of a dynamical system $f:M\to M$ (i.e. $X\subset M$ and $f|_X=T$) where $M$ is a compact smooth manifold, the above conclusion holds for $C^1$ 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 \cite{YH} for $C^1$-observables is solved consequentially.