Qualitative Theory of Dynamical Systems

, Volume 15, Issue 2, pp 309–326 | Cite as

A Dichotomy in Area-Preserving Reversible Maps



In this paper we study R-reversible area-preserving maps \(f:M\rightarrow M\) on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that \(R\circ f=f^{-1}\circ R\) where \(R:M\rightarrow M\) is an isometric involution. We obtain a \(C^1\)-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits, thus establishing the stability conjecture in this setting. Along the paper we derive the \(C^1\)-Closing Lemma for reversible maps and other perturbation toolboxes.


Reversing symmetry Area-preserving map Closing Lemma  Elliptic point 

Mathematics Subject Classification

Primary 37D20 37C20 Secondary 37C27 34D30 



The authors are grateful to Maria Carvalho (CMUP) for enlightening discussions and for several suggestions that improved the quality of the paper. A. Rodrigues has been funded by the European Regional Development Fund within the program COMPETE and by the Portuguese Government through FCT –Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013. A. Rodrigues has benefited from the FCT grant SFRH/BPD/84709/2012.


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© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal
  2. 2.Centro de Matemática da Universidade do PortoPortoPortugal

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