Asymptotically sectional-hyperbolic attractors

B. San Martín; Kendry J. Vivas

doi:10.3934/dcds.2019163

Article Contents

2019, Volume 39, Issue 7: 4057-4071. Doi: 10.3934/dcds.2019163

This issue Previous Article Minimality and gluing orbit property Next Article On the oscillation behavior of solutions to the one-dimensional heat equation

Asymptotically sectional-hyperbolic attractors

B. San Martín and
Kendry J. Vivas^,

Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile

^*Corresponding author.
^*Corresponding author.

Received: August 2018

Revised: November 2018

Published: April 2019

The first author was partially supported by Proyecto FONDECYT 1181183, Chile. The second author was supported by CONICYT grant 21180163, Chile

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Abstract

The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.

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Mathematics Subject Classification: Primary: 54H20, 37B25; Secondary: 37B40.

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Figure 1. A vector field $ X $ without non-atomic SRB-like measures

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Figure 2. The cube $ Q $ and the cells $ T_l $ and $ T_r $

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Figure 3. The flow in the resulting $ 3 $-cell

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Figure 4. The flow from the cusp triangles to $ S^+ $ and $ S^- $

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Figure 6. The trapping region $ U $

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