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Register a new userErgodic optimization theory for Axiom A flows
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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.
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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.
For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat (1998) for geodesic flows on compact surfaces (for general potentials)and transitive Anosov flows on compact manifolds with C^1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, decay of correlations for Holder continuous potentials.
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.
Let ${T^t}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $mu$ be an ergodic measure of maximal entropy. We show that either ${T^t}$ is Bernoulli, or ${T^t}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
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