Minimal yet measurable foliations

Ponce, Gabriel; Tahzibi, Ali; Varão, Régis

doi:10.3934/jmd.2014.8.93

Cited by 26 publications

(31 citation statements)

References 16 publications

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“…The above definition was given in [14] in the context of algebraic automorphisms and the existence of such measures in partially hyperbolic diffeomorphism also had been noticed by (see for instance [22], [16]). If µ is virtually hyperbolic, then the central foliation is measurable with respect to µ and conditional measures along center leaves are dirac.…”

Section: Disintegration Of Measuresmentioning

confidence: 78%

“…Observe that the partition into central leaves is not necessarily a measurable partition and we are not allowed apriori to apply Rokhlin disintegration result to this partition. However, the preimage by h of the partition into points is a measurable partition (see [16]). Under the hypothesis of the lemma we just consider the partition into collapse intervals:…”

Section: Proof Of Theorem Amentioning

confidence: 99%

See 1 more Smart Citation

Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part

Crisostomo¹,

Tahzibi²

2019

Nonlinearity

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Section: Disintegration Of Measuresmentioning

confidence: 78%

Section: Proof Of Theorem Amentioning

confidence: 99%

Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part

Crisostomo¹,

Tahzibi²

2019

Nonlinearity

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“…We apply Theorem B of [33] to η and obtain that η has atomic disintegration and one atom per leaf. Although Theorem B of [33] deals with volume measure we point out that the same argument works ipsis litteris by changing volume to η.…”

Section: Proof Of Theorem Bmentioning

confidence: 99%

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

Ponce

Tahzibi²,

Varão

2018

Advances in Mathematics

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We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent.In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization.Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.

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“…In contrast, the center foliation may exhibit a new type of behavior, it may have atomic disintegration: there is a set of full volume which intersects each center leaf in a finite number of points. At first this might sound as a pathological behavior, but it turns out that this is in fact a common behavior for the center foliation [2,16,19].…”

Section: Introductionmentioning

confidence: 99%

Rigidity for partially hyperbolic diffeomorphisms

Varão¹

2017

Ergod. Th. Dynam. Sys.

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In this work we completely classify C ∞ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain C ∞ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and C ∞ center foliation.

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Minimal yet measurable foliations

Cited by 26 publications

References 16 publications

Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part

Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

Rigidity for partially hyperbolic diffeomorphisms

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