Global dynamics for symmetric planar maps

Begoña Alarcón; Sofia B. S. D. Castro; Isabel S. Labouriau

doi:10.3934/dcds.2013.33.2241

Article Contents

2013, Volume 33, Issue 6: 2241-2251. Doi: 10.3934/dcds.2013.33.2241

This issue Previous Article Next Article Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle

Global dynamics for symmetric planar maps

1.
Department of Mathematics, University of Oviedo, Calvo Sotelo s/n, 33007 Oviedo
2.
Centro de Matemática and Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto
3.
Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto

Received: January 31, 2012

Revised: September 30, 2012

Published: June 2013

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Abstract

We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.

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Mathematics Subject Classification: Primary: 37C80; Secondary: 37B99, 37C70.

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