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

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

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 ([email protected])
Maria Joana Torres
Affiliation:
CMAT and Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4700-057 Braga, Portugal ([email protected])
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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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References

1.Akin, E., Hurley, M. and Kennedy, J., Dynamics of topologically generic homeomorphisms, Mem. Am. Math. Soc. 164 (2003), 783.Google Scholar
2.Aoki, N., Topological dynamics, in Topics in general topology (ed. Morita, K. and Nagata, J.-I.), Volume 41, Chapter 15, pp. 625–740 (North-Holland Mathematical Library, 1989).CrossRefGoogle Scholar
3.Hiraide, K., Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117–162.Google Scholar
4.Lewowicz, J., Expansive homeomorphisms of surfaces, Bol. Soc. Bras. de Mat. 20 (1989), 113–133.10.1007/BF02585472CrossRefGoogle Scholar
5.Newhouse, S., Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125–150.CrossRefGoogle Scholar
6.Newhouse, S., Topics in conservative dynamics, in Regular and chaotic motions in dynamic systems (ed. G. Velo and A. S. Wightman), NATO Advanced Study Institutes Series, Volume 118, pp. 103–184 (D. Reidel Publishing Company, Dordrecht-Holland, 1985).10.1007/978-1-4684-1221-5_4CrossRefGoogle Scholar
7.Palis, J. and de Melo, W., Geometric theory of dynamical systems. An introduction (Springer-Verlag, New York–Berlin, 1982).CrossRefGoogle Scholar
8.Palis, J. and Smale, S., Structural stability theorems, Proceedings of Symposia in Pure Mathematics, Volume 14 (American Mathematical Society, Providence, RI, 1970).CrossRefGoogle Scholar
9.Shub, M., Global stability of dynamical systems (Springer-Verlag, 1987).CrossRefGoogle Scholar
10.Smale, S., Diffeomorphisms with many periodic points, in Differential and combinatorial topology, a symposium in honor of Marston Morse, pp. 63–80 (Princeton University Press, Princeton, NJ, 1965).10.1515/9781400874842-006CrossRefGoogle Scholar
11.Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar