June  2013, 33(6): 2241-2251. doi: 10.3934/dcds.2013.33.2241

Global dynamics for symmetric planar maps

1. 

Department of Mathematics, University of Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Spain

2. 

Centro de Matemática and Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal

3. 

Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Received  February 2012 Revised  October 2012 Published  December 2012

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.
Citation: Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau. Global dynamics for symmetric planar maps. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241
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show all references

References:
[1]

B. Alarcón, Rotation numbers for planar attractors of equivariant homeomorphisms,, Preprint CMUP 2012-23 and , (): 2012.   Google Scholar

[2]

Journal of Singularities, 6 (2012), 1-14. doi: 10.5427/jsing.2012.6a.  Google Scholar

[3]

Nonlinear Anal., 69 (2008), 140-150. doi: 10.1016/j.na.2007.05.005.  Google Scholar

[4]

J.Difference Equ. Appl., 14 (2008), 421-428. doi: 10.1080/10236190701698155.  Google Scholar

[5]

Springer-Verlag, New York, 2002.  Google Scholar

[6]

Houston J. Math, 11 (1985), 455-469.  Google Scholar

[7]

Nonlinear Anal., 35 (1999), 343-354. doi: 10.1016/S0362-546X(97)00715-3.  Google Scholar

[8]

A. van den Essen, Conjectures and problems surrounding the Jacobian conjecture,, in, 429 ().   Google Scholar

[9]

Ergod. Th. & Dynam. Sys., 12 (1992), 217-226. doi: 10.1017/S0143385700006702.  Google Scholar

[10]

2, Applied Mathematical Sciences 69, Springer Verlag, 1985. doi: 10.1007/978-1-4612-4574-2.  Google Scholar

[11]

Academic Press, San Diego, 1974.  Google Scholar

[12]

Astérisque 292, 2004.  Google Scholar

[13]

J.Dyn and Diff Equations, 10 (1998), 275-302. doi: 10.1023/A:1022618000699.  Google Scholar

[14]

R. Ortega, Topology of the plane and periodic differential equations,, 2008. Available from: , ().   Google Scholar

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