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There are no proper topological hyperbolic homoclinic classes for area-preserving maps

Part of: Connections with other structures, applications Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  12 November 2019

Mário Bessa  and
Maria Joana Torres
Mário Bessa
Affiliation:
Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001 Covilhã, Portugal (bessa@ubi.pt)
Maria Joana Torres
Affiliation:
CMAT and Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4700-057 Braga, Portugal (jtorres@math.uminho.pt)
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Abstract

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

Keywords

shadowingexpansivenesstopological dynamicshomoclinic classes

MSC classification

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 54H20: Topological dynamics
Secondary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 217 - 228
Copyright
Copyright © Edinburgh Mathematical Society 2019

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