Singular cw-expansive flows

Alfonso Artigue

doi:10.3934/dcds.2017126

Article Contents

2017, Volume 37, Issue 6: 2945-2956. Doi: 10.3934/dcds.2017126

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Singular cw-expansive flows

Alfonso Artigue

Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay

Received: August 31, 2016

Revised: December 31, 2016

Published: May 2017

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Abstract

We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply expansivity. We also construct a singular Axiom A vector field on a three-manifold being singular cw-expansive and with a Lorenz attractor and a Lorenz repeller in its non-wandering set.

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Mathematics Subject Classification: Primary: 37E35; Secondary: 54H20.

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Figure 1. Classical geometric model of the Lorenz attractor.

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Figure 2. Topological model of the Lorenz attractor.

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Figure 3. The cylinder $B'$ is transverse to the flow.

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Figure 4. The genus two surface S transverse to the flow and containing the Lorenz attractor.

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Figure 5. A singular point of the stable foliation appears in $G$.

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