Qualitative Theory of Dynamical Systems

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

A Dichotomy in Area-Preserving Reversible Maps

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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.

Keywords

Reversing symmetry Area-preserving map Closing Lemma  Elliptic point 

Mathematics Subject Classification

Primary 37D20 37C20 Secondary 37C27 34D30 

Notes

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.

References

  1. 1.
    Anosov, D.V., Zhuzhoma, E.V.: Closing lemmas. Differ. Equs. 48(13), 1653–1699 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arbieto, A., Matheus, C.: A pasting lemma and some applications for conservative systems. With an appendix by David Diica and Yakov Simpson-Weller. Ergodic Theory Dyn. Syst. 27, 1399–1417 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bessa, M., Duarte, P.: Abundance of elliptic dynamics on conservative \(3\)-flows. Dyn. Syst. Int. J. 23(4), 409–424 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bessa, M., Carvalho, M., Rodrigues, A.: Generic area-preserving reversible diffeomorphisms. Nonlinearity 28, 1695–1720 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bessa, M., Carvalho, M., Rodrigues, A.: Linear reversible Anosov diffeomorphisms on the two-torus. https://cmup.fc.up.pt/main/preprints (2015)
  6. 6.
    Birkhoff, G.D.: The restricted problem of three bodies. Rend. Circ. Mat. Palermo 39, 265–334 (1915)CrossRefMATHGoogle Scholar
  7. 7.
    Bochi, J.: Genericity of zero Lyapunov exponents. Ergodic Theory Dyn. Syst. 22, 1667–1696 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bochi, J., Viana, M.: Lyapunov Exponents: How Frequently are Dynamical System Hyperbolic? Modern Dynamical Systems and applications, 217–297. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  9. 9.
    Bonatti, C., Crovisier, S.: Récurrence et généricité. Invent. Math. 158(1), 33–104 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bonatti, C., Díaz, L.J., Pujals, E.: A \(C^1\)-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. Math. 158, 355–418 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bonatti, C., Gourmelon, N., Vivier, T.: Perturbations of the derivative along periodic orbits. Ergodic Theory Dyn. Syst. 26(5), 1307–1337 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Field, M., Melbourne, I., Nicol, M.: Symmetric attractors for diffeomorphisms and flows. Proc. Lond. Math. Soc. 72, 657–669 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Franks, J.: Necessary conditions for the stability of diffeomorphisms. Trans. Am. Math. Soc. 158, 301–308 (1971)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Franks, J.: Anosov diffeomorphisms, global analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif.). Amer. Math. Soc., Providence, R.I. 1070, pp. 61–93 (1968)Google Scholar
  15. 15.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, Berlin (2000)MATHGoogle Scholar
  16. 16.
    Hurewicz, W., Wallman, H.: Dimension Theory, Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton (1941)MATHGoogle Scholar
  17. 17.
    Kumicak, J., de Hemptinne, X.: The dynamics of thermodynamics. Physica D 112, 258–274 (1988)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lamb, J., Roberts, J.: Time-reversal symmetry in dynamical systems: a survey. Physica D 112, 1–39 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lamb, J., Stenkin, O.: Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits. Nonlinearity 17, 1217–1244 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Meiss, J.D.: Symplectic maps, variational principles, and transport. Rev. Modern Phys. 64(3), 795–848 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Moser, J., Zehnder, E.: Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, Providence (2005)Google Scholar
  22. 22.
    Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math. 99, 1061–1087 (1977)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Prigogine, I.: Why irreversibility? The formulation of classical and quantum mechanics for nonintegrable systems. Int. J. Bifur. Chaos Appl. Sci. Eng 5(1), 3–16 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pugh, C.: The closing lemma. Am. J. Math. 89(4), 956–1009 (1967)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pugh, C.: An improved closing lemma and a general density theorem. Am. J. Math. 89(4), 1010–1021 (1967)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pugh, C., Robinson, C.: The \(C^1\) closing lemma, including hamiltonians. Ergodic Theory Dyn. Syst. 3, 261–313 (1983)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pujals, H., Sambarino, M.: On the dynamics of dominated splitting. Ann. Math. 169(3), 675–740 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Roberts, J.A.G., Quispel, G.R.W.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63–177 (1992)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Webster, K.: Bifurcations in reversible systems with application to the Michelson system, PhD. Thesis, Imperial College of London (2005)Google Scholar
  30. 30.
    Zehnder, E.: Homoclinic points near elliptic fixed points. Comun. Pure Appl. Math. 26, 131–182 (1973)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zehnder, E.: Note on smoothing symplectic and volume-preserving diffeomorphisms. In: Proceedings of III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976, volume 597 Lecture Notes in Math, pp. 828–854, Springer, Berlin (1977)Google Scholar

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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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