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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.
In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $mc C^{0,a}$ ($ain(0,1
We survey the impact of the Poincare recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory.
This survey is an update of the 2008 version, with recent developments and new references.
We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the
The main purpose of this paper is to present a kneading theory for two-dimensional triangular maps. This is done by defining a tensor product between the polynomials and matrices corresponding to the one-dimensional basis map and fiber map. We also d