Robust ergodic properties in partially hyperbolic dynamics

Andersson, Martin

doi:10.1090/s0002-9947-09-05027-2

Cited by 26 publications

(39 citation statements)

References 26 publications

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“…(This was noted in [15] and proved in detail in [7]). Moreover it was proved in [7] (see also [17]) that (i) the number of physical measures of mostly contracting diffeomorphisms varies upper semi-continuously with the dynamics, and (ii) physical measures vary continuously in the weak* topology under perturbations that don't change the number of physical measures.…”

Section: Introductionmentioning

confidence: 63%

“…Central to our argument is that the size of local unstable manifolds can be controlled on the sets Λ ℓ (f, σ), uniformly in a neighbourhood of a given mostly expanding diffeomorphism. This was done from scratch in [9,Theorem 4,7] using graph transforms. In this work some further properties of unstable manifolds are needed which are not stated in [9,Theorem 4,7].…”

Section: Unstable Manifolds and Uniform Densitiesmentioning

confidence: 99%

“…This was done from scratch in [9,Theorem 4,7] using graph transforms. In this work some further properties of unstable manifolds are needed which are not stated in [9,Theorem 4,7]. Then, given any ℓ ∈ N, there are a C r neighborhood U of f and numbers r = r(ℓ) > 0, C = C(ℓ) ≥ 0, δ = δ(ℓ) > 0 for which the following holds: (i) Given any g ∈ U and x ∈ Λ ℓ (g, σ), there is a C 1 embedded disk W cu r (g, x) of dimension dim E cu and radius r > 0, centered at x, such that T y W cu r (g, x) = E cu y (g) for every y ∈ W cu r (g, x).…”

Section: Unstable Manifolds and Uniform Densitiesmentioning

confidence: 99%

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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: 63%

Section: Unstable Manifolds and Uniform Densitiesmentioning

confidence: 99%

Section: Unstable Manifolds and Uniform Densitiesmentioning

confidence: 99%

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On mostly expanding diffeomorphisms

Andersson¹,

Vásquez²

2017

Ergod. Th. Dynam. Sys.

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“…Since diffeomorphisms with mostly contracting center form a C 1+ε open set (by Andersson [2]), we may find a C 1+ε neighborhood U ⊂ V such that every g ∈ U has mostly contracting center. By Lemma 2.5, every slice S ′ (g) of S(g) is a skeleton for g. Since #S ′ (g) ≤ #S(g) = S(f ), it follows from Theorem A that the number of physical measures of g ∈ U is not larger than the number of physical measures of f .…”

Section: 2mentioning

confidence: 99%

Geometric and Measure-Theoretical Structures of Maps with Mostly Contracting Center

Dolgopyat

Viana

Yang

2016

Commun. Math. Phys.

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Abstract. We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measure bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each otherthrough explosions of their basins -as the dynamics varies. This theory also allows us to prove that, in the absence of collapses, the basins are continuous functions of the diffeomorphism.

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“…[3,30]). A diffeomorphism f has mostly contracting center if and only if the center Lyapunov exponents of all ergodic Gibbs u-states of f are negative.…”

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confidence: 99%

A robustly transitive diffeomorphism of Kan's type

Cheng¹,

Gan²,

Shi³

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We construct a family of partially hyperbolic skew-product diffeomorphisms on T 3 that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

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Robust ergodic properties in partially hyperbolic dynamics

Cited by 26 publications

References 26 publications

On mostly expanding diffeomorphisms

On mostly expanding diffeomorphisms

Geometric and Measure-Theoretical Structures of Maps with Mostly Contracting Center

A robustly transitive diffeomorphism of Kan's type

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