Ergodicity and partial hyperbolicity on the 3-torus

Hammerlindl, Andy; Ures, Raúl

doi:10.1142/s0219199713500387

Cited by 25 publications

(26 citation statements)

References 30 publications

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“…Finally, by [10] the diffeomorphisms obtained in Theorem 1.1 are ergodic and so almost every point have the same Lyapunov exponent. For diffeomorphisms obtained in Theorem 1.2, with vanishing average of central exponent, without proving the ergodicity we obtain vanishing Lyapunov exponent almost everywhere.…”

Section: Theorem 11mentioning

confidence: 92%

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Central Lyapunov exponent of partially hyperbolic diffeomorphisms of 𝕋³

Ponce¹,

Tahzibi²

2014

Proc. Amer. Math. Soc.

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Abstract. In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on T 3 such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point.

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Section: Theorem 11mentioning

confidence: 92%

“…We mention that by a recent result of A. Hammerlindl and R. Ures [10], a non-ergodic volume preserving isotopic to Anosov diffeomorphism on T 3 , if it exists, should have zero central Lyapunov exponent almost everywhere. The ergodicity of the diffeomorphisms with zero central exponent in Theorem 1.2 is an open problem.…”

Section: Theorem 11mentioning

confidence: 92%

Central Lyapunov exponent of partially hyperbolic diffeomorphisms of 𝕋³

Ponce¹,

Tahzibi²

2014

Proc. Amer. Math. Soc.

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“…So let us prove the dichotomy. If the Lebesgue measure of C does not vanish, then we claim that f is accessible, otherwise by [17] it would be topologically conjugate to its linearization, hence the set C = ∅, which is an absurd. Thus f is accessible, therefore Kolmogorov.…”

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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“…Proof of Corollary B. By Hammerlindl, Ures [20,Theorem 7.2], every C 2 volume-preserving partially hyperbolic diffeomorphism g ∈ D(A) whose integrated center Lyapunov exponent λ c (g) is different from zero is ergodic. Thus, since the map g → λ c (g) is continuous, every volume-preserving g in a neighborhood of f is ergodic.…”

Section: It Has Been Show By Potriementioning

confidence: 99%

Measure-theoretical properties of center foliations

Viana¹,

Yang²

2017

Modern Theory of Dynamical Systems

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Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous.

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Ergodicity and partial hyperbolicity on the 3-torus

Abstract: We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic. arXiv:1907.04755v1 [math.DS]

Cited by 25 publications

References 30 publications

Central Lyapunov exponent of partially hyperbolic diffeomorphisms of 𝕋³

Central Lyapunov exponent of partially hyperbolic diffeomorphisms of 𝕋³

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

Measure-theoretical properties of center foliations

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