Partially hyperbolic sets with a dynamically minimal lamination

Luiz Felipe Nobili França

doi:10.3934/dcds.2018114

Article Contents

2018, Volume 38, Issue 6: 2717-2729. Doi: 10.3934/dcds.2018114

This issue Previous Article Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem Next Article Asymptotic properties of various stochastic Cucker-Smale dynamics

Partially hyperbolic sets with a dynamically minimal lamination

Luiz Felipe Nobili França

Departamento de Matemática - Instituto Tecnológico de Aeronáutica (ITA) - Praça Marechal Eduardo Gomes, 50 - Vila das Acacias, São José dos Campos, CEP 12228-900, SP, Brazil

Received: February 28, 2017

Revised: November 30, 2017

Published: May 2018

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Abstract

We study partially hyperbolic sets of $C^1$-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely.

We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to $C^1$-generic/robustly transitive attractors with one-dimensional center bundle.

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Mathematics Subject Classification: 37B20, 37B29, 37C20, 37C70, 37D10, 37D30.

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