Statistical stability for diffeomorphisms with dominated splitting

Vásquez, Carlos H.

doi:10.1017/s0143385706000721

Cited by 26 publications

(21 citation statements)

References 23 publications

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“…Yet there doesn't seem to be any good reason why it would be intrinsic to non-uniform expansion, or that one can find a universal mechanism that gives rise to it. In the setting of partially hyperbolic diffeomorphisms, a similar result was proved in [27]. Briefly speaking, consider a partially hyperbolic diffeomorphisms f : M → M with a Dfinvariant splitting T M = E s ⊕ E c and satisfying the NUE-condition on the center direction (here E u = {0} is taken trivial).…”

Section: Introductionmentioning

confidence: 73%

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On mostly expanding diffeomorphisms

Andersson¹,

Vásquez²

2017

Ergod. Th. Dynam. Sys.

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We study how physical measures vary with the underlying dynamics in the open class of C r , r > 1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.Résumé. Nousétudions comment les mesures physiques varient avec la dynamique sousjacente, dans la classe ouverte des difféomorphismes C r , r > 1, fortement partiellement hyperboliques pour lesquelles les exposants de Lyapunov centraux de tout u-état de Gibbs sont positifs. Lorsque transitifs, de tels difféomorphismes possédent une unique mesure physique qui persiste et varie continment avec la dynamique.Un des ingrédients principaux de la preuve est un nouveau lemme de type Pliss qui, appliqué dans le contexte adéquate, implique que la fréquence des temps hyperboliques est proche de un. Une autre nouveauté est l'introduction d'une nouvelle caractérisation des cu-états de Gibbs. Chacun de ses deux aspects ayant leur propre intérêt.Le cas non transitif est aussi traité : dans ce contexte, le nombre de mesures physiques est une fonction semi-continue supérieure du difféomorphisme, et les mesures physiques varient continument sous des hypothèses naturelles.

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Section: Introductionmentioning

confidence: 73%

“…f is mostly contracting along E cs ). All results in the present work can be extended to the setting in [22] using our results and following the ideas in the proof of Theorem C in [27].…”

Section: Introductionmentioning

confidence: 91%

On mostly expanding diffeomorphisms

Andersson¹,

Vásquez²

2017

Ergod. Th. Dynam. Sys.

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“…It also holds in some non-uniform settings without critical behavior, in particular for maps with dominated splitting satisfying robustly a separation condition between the positive and negative Lyapunov exponents [247] (see also [7]). …”

Section: Statistical Stabilitymentioning

confidence: 86%

Chaos and Ergodic Theory

Buzzi¹

2012

Mathematics of Complexity And Dynamical Systems

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“…It merely reflects the strong definition of statistical stability considered. Should one have settled with the weaker form of statistical stability suggested in [Vás07], one would obtain (quite trivially) that all mostly contracting systems are statistically stable -not only an open and dense set.…”

Section: Corollary 18 Any Ergodic Diffeomorphism In MC Is Automaticmentioning

confidence: 99%

Robust ergodic properties in partially hyperbolic dynamics

Andersson

2009

Trans. Amer. Math. Soc.

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Abstract. We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti and Viana (2000) about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C 2 -open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations.

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Statistical stability for diffeomorphisms with dominated splitting

Cited by 26 publications

References 23 publications

On mostly expanding diffeomorphisms

On mostly expanding diffeomorphisms

Chaos and Ergodic Theory

Robust ergodic properties in partially hyperbolic dynamics

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