Open sets of partially hyperbolic skew products having a unique SRB measure

Obata, Davi

doi:10.1016/j.aim.2023.109136

Advances in Mathematics

2023

DOI: 10.1016/j.aim.2023.109136

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Open sets of partially hyperbolic skew products having a unique SRB measure

Davi Obata

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2024

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Cited by 3 publications

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Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

Crovisier,

Obata,

Poletti

2024

Commun. Math. Phys.

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Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

Crovisier,

Obata,

Poletti

2024

Commun. Math. Phys.

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On invariant holonomies between centers

SAGHIN

2024

Ergod. Th. Dynam. Sys.

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We prove that for $C^{1+\theta }$ , $\theta $ -bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are $C^1$ , and the derivative depends continuously on the points and on the map. Also for $C^{1+\theta }$ , $\theta $ -bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.

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Shub’s example revisited

LIANG,

SAGHIN,

YANG

et al. 2024

Ergod. Th. Dynam. Sys.

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For a class of robustly transitive diffeomorphisms on ${\mathbb T}^4$ introduced by Shub [Topologically transitive diffeomorphisms of $T^4$ . Proceedings of the Symposium on Differential Equations and Dynamical Systems (Lecture notes in Mathematics, 206). Ed. D. Chillingworth. Springer, Berlin, 1971, pp. 39–40], satisfying an additional bunching condition, we show that there exists a $C^2$ open and $C^r$ dense subset ${\mathcal U}^r$ , $2\leq r\leq \infty $ , such that any two hyperbolic points of $g\in {\mathcal U}^r$ with stable index $2$ are homoclinically related. As a consequence, every $g\in {\mathcal U}^r$ admits a unique homoclinic class associated to the hyperbolic periodic points with index $2$ , and this homoclinic class coincides with the whole ambient manifold. Moreover, every $g\in {\mathcal U}^r$ admits at most one measure of maximal entropy, and every $g\in {\mathcal U}^{\infty }$ admits a unique measure of maximal entropy.

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Open sets of partially hyperbolic skew products having a unique SRB measure

Cited by 3 publications

References 45 publications

Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

On invariant holonomies between centers

Shub’s example revisited

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