DOI QR코드

DOI QR Code

LIMIT WEAK SHADOWABLE TRANSITIVE SETS OF C1-GENERIC DIFFEOMORPHISMS

  • Lee, Man-Seob (Department of Mathematics Mokwon University) ;
  • Lu, Gang (Department of Mathematics, School of Science ShenYang University of Technology)
  • Received : 2011.06.02
  • Published : 2012.07.31

Abstract

In this paper, we prove that locally maximal transitive set of a $C^1$-generic diffeomorphism is hyperbolic if and only if it is limit weak shadowable.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. F. Abdenur, Generic robustness of spectral decompositions, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 2, 213-224. https://doi.org/10.1016/S0012-9593(03)00008-9
  2. F. Abdenur and L. J. Diaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 223-245.
  3. T. Eirola, O. Nevanlinna, and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 75-92. https://doi.org/10.1080/01630569708816748
  4. K. Lee and M. Lee, Hyperbolic of $C^1$-stably expansive homoclinic classes, Discrete Contin. Dyn. Syst. 27 (2010), no. 3, 1133-1145. https://doi.org/10.3934/dcds.2010.27.1133
  5. K. Lee and X. Wen, Shadowable chain transitive sets of $C^1$-generic diffeomorphisms, to apear in Bull. Korean. Math. Soc.
  6. R. Mane, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503-540. https://doi.org/10.2307/2007021
  7. S. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer 1999.
  8. S. Pilyugin, K. Sakai, and O. Tarakanov, Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing, Discrete Contin. Dyn. Syst. 16 (2006), no. 4, 871-882. https://doi.org/10.3934/dcds.2006.16.871
  9. O. B. Plamenevskaya, Weak shadowing in two-dimensional diffeomorphisms, Vestnik St. Petersburg Univ. Math. 31 (1999), no. 3, 49-56.
  10. K. Sakai, Diffeomorphisms with weak shadowing, Fund. Math. 168 (2001), no. 1, 57-75. https://doi.org/10.4064/fm168-1-2
  11. K. Sakai, A note on weak shadowing, Far East J. Dyn. Syst. 3 (2001), no. 1, 45-49.
  12. K. Sakai, Diffeomorphisms with limit weak shadowing, Trends in Mathematics 7 (2004), 19-27.
  13. D. Yang, Stably weakly shadowing transitive sets and dominated splittings, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2747-2751. https://doi.org/10.1090/S0002-9939-2011-10699-3