Abstract
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.
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Acknowledgments
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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Bessa, M., Rodrigues, A.A.P. A Dichotomy in Area-Preserving Reversible Maps. Qual. Theory Dyn. Syst. 15, 309–326 (2016). https://doi.org/10.1007/s12346-015-0155-y
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DOI: https://doi.org/10.1007/s12346-015-0155-y