A note on partially hyperbolic systems with mostly expanding centers

Mi, Zeya; Cao, Yongluo; Yang, Dinghui

doi:10.1090/proc/13701

Cited by 12 publications

(21 citation statements)

References 8 publications

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“…In another recent work [22] the authors prove existence and finiteness of physical measures for partially hyperbolic diffeomorphisms f with dominated splitting T M = E cs ⊕ E cu ⊕ E u (with uniform expansion in E u ) satisfying a mixture of mostly contracting and mostly expanding behavior. That is, every Gibbs u-state has positive Lyapunov exponents in the E cu bundle (i.e.…”

Section: Introductionmentioning

confidence: 99%

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: 99%

On mostly expanding diffeomorphisms

Andersson¹,

Vásquez²

2017

Ergod. Th. Dynam. Sys.

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“…Recall that in both works, the authors make effort to show the existence of positive Lebesgue measure set of (weakly) nonuniformly expanding points, then in terms of the previous techniques from [2] or [1] to find SRB measures. In §6, as a byproduct of our result, we provide a proof of the existence of SRB measures for systems considered in [21] (it works for [4] with simpler arguments). We would like to mention that by our new arguments the partially hyperbolic splittings can be restricted to attractors, rather than the whole manifold, which is pivotal there to find the (weakly) non-uniformly expanding points.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

“…It turns out that Gibbs u-states and Gibbs cu-states are crucial candidates of SRB measures and physical measures(e.g. [1,10,12,4,21] In this paper, we deal with the problem of the existence of SRB measures for some systems exhibitting dominated splitting. The key idea is to add small random noise to the deterministic dynamical system and prove that as noise levels tends to zero, the limit of the ergodic stationary measures, called the randomly ergodic limit (see Definition 3.4), has ergodic components to be Gibbs cu-states associated to some sub-bundle E whenever this randomly ergodic limit appears some weak expansion along E. Under some extra assumptions on the other directions of the sub-bundles, one can make these Gibbs cu-states to be SRB measures or physical measures.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In recently, [4,21] showed the existence (and finiteness) of SRB/physical measures on mostly expanding diffeomorphisms. Recall that in both works, the authors make effort to show the existence of positive Lebesgue measure set of (weakly) nonuniformly expanding points, then in terms of the previous techniques from [2] or [1] to find SRB measures.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For the later case, one can apply Theorem A by taking E = E u ⊕ E c and F = E s to get SRB measures from ergodic components of µ. Now we provide a new proof of the existence of SRB/physical measures for systems considered in [21]. Recall that we say f is mostly expanding(resp.…”

Section: Proof Of Theorem Amentioning

confidence: 99%

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SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

Mi¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given C 2 diffeomorphim f with dominated splitting T Λ M = E ⊕ F on an attractor Λ, by considering some suitable random perturbation of f , we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle E, then its ergodic components contain Gibbs cu-states associated to E. With this technique, we show the existence of SRB measures and physical measures for some systems exhibitting dominated splittings and weak hyperbolicity. SRB measures and Gibbs cu-statesLet M be a compact Riemannian manifold, use Leb represent the Lebesgue measure of M . Given a sub-manifold γ ⊂ M , denote by Leb γ the Lebesgue measure on γ induced by the restriction of the Riemannian structure to γ. Let d denote the distance in M , and ρ the distance in the Grassmannian bundle of T M generated by the Riemannian metric. Denote by Diff 2 (M ) the space of C 2 diffeomorphisms on M .Given diffeomorphism f on M , let E be a Df -invariant sub-bundle and µ an f -invariant measure, we define the integrable Lyapunov exponent along E of µ byGiven f ∈ Diff 2 (M ), let Λ be an attractor admitting the dominated splitting T Λ M = E ⊕ ≻ F . Since the distributions E and F are continuous, we may extend

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Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

Cao

2021

Math. Z.

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A note on partially hyperbolic systems with mostly expanding centers

Cited by 12 publications

References 8 publications

On mostly expanding diffeomorphisms

On mostly expanding diffeomorphisms

SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

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