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Register a new userErgodic Theory: Recurrence
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We survey the impact of the Poincare recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory.
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We discuss an invertible version of Furstenbergs `Ergodic CP Shift Systems. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.
This survey is an update of the 2008 version, with recent developments and new references.
Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.
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 $mc C^{0,a}$ ($ain(0,1]$) or $mc C^1$ the minimizing measure is unique and is supported on a periodic orbit.
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.
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