Ergodic optimization theory for a class of typical maps

In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems $T:Xto X$ where $X$ is a compact metric space and $T$ is Lipschitz continuous. We show that once $T:Xto X$ satisfies both the {em Anosov shadowing property }({bf ASP}) and the {em Ma~ne-Conze-Guivarch-Bousch property }({bf MCGBP}), the minimizing measures of generic Holder observables are unique and supported on a periodic orbit. Moreover, if $T:Xto X$ is a subsystem of a dynamical system $f:Mto M$ (i.e. $Xsubset M$ 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.