DOI QR코드

DOI QR Code

C1-STABLY SHADOWABLE CONSERVATIVE DIFFEOMORPHISMS ARE ANOSOV

  • Bessa, Mario
  • Received : 2012.01.23
  • Published : 2013.09.30

Abstract

In this short note we prove that if a symplectomorphism $f$ is $C^1$-stably shadowable, then $f$ is Anosov. The same result is obtained for volume-preserving diffeomorphisms.

Keywords

Anosov maps;shadowing;uniform hyperbolicity

References

  1. A. Arbieto and T. Catalan, Hyperbolicity in the Volume Preserving Scenario, Ergod. Th. & Dynam. Sys. (at press) DOI: http://dx.doi.org/10.1017/etds.2012.111. https://doi.org/10.1017/etds.2012.111
  2. A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, With an appendix by David Diica and Yakov Simpson-Weller., Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1399-1417.
  3. A. Avila, On the regularization of conservative maps, Acta Math. 205 (2010), no. 1, 5-18. https://doi.org/10.1007/s11511-010-0050-y
  4. M. Bessa and J. Rocha, A remark on the topological stability of symplectomorphisms, Appl. Math. Lett. 25 (2012), no. 2, 163-165. https://doi.org/10.1016/j.aml.2011.08.007
  5. C. Bonatti, L. Diaz, and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), no. 2, 355-418. https://doi.org/10.4007/annals.2003.158.355
  6. V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 5, 641-661. https://doi.org/10.1016/j.anihpc.2005.06.002
  7. K. Moriyasu The topological stability of diffeomorphisms, Nagoya Math. J. 123 (1991), 91-102. https://doi.org/10.1017/S0027763000003664
  8. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. https://doi.org/10.1090/S0002-9947-1965-0182927-5
  9. J. Moser and E. Zehnder, Notes on dynamical systems, Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005.
  10. S. Newhouse, Quasi-eliptic periodic points in conservative dynamical systems, Amer. J. Math. 99 (1977), no. 5, 1061-1087. https://doi.org/10.2307/2374000
  11. S. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer-Verlag, Berlin, 1999.
  12. K. Sakai, $C^1$-stably shadowable chain components, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 987-1029.
  13. P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 231-244, Lecture Notes in Math., 668, Springer, Berlin, 1978.
  14. E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), pp. 828-854. Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977.

Cited by

  1. STABLE WEAK SHADOWABLE SYMPLECTOMORPHISMS ARE PARTIALLY HYPERBOLIC vol.29, pp.2, 2014, https://doi.org/10.4134/CKMS.2014.29.2.285
  2. Shadowing, expansiveness and specification for C1-conservative systems vol.35, pp.3, 2015, https://doi.org/10.1016/S0252-9602(15)30005-9
  3. Symplectic diffeomorphisms with limit shadowing vol.10, pp.02, 2017, https://doi.org/10.1142/S1793557117500681