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Korean Mathematical Society (대한수학회)
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C1-STABLE INVERSE SHADOWING CHAIN COMPONENTS FOR GENERIC DIFFEOMORPHISMS

  • Published : 2009.01.31
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Abstract

Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

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References

  1. F. Abdenur and L. J. Díaz, Pseodo-orbit shadowing in the $C^1$-topology, Discrete Contin. Dyn. Syst. 17 (2007), 223–245.
  2. C. Bonatti and S. Crovisier, R$\acute{e}$currence et g$\acute{e}n\acute{e}ricit\acute{e}$, Invent. Math. 158 (2004), 33–104.
  3. R. Corless and S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), 409–423. https://doi.org/10.1006/jmaa.1995.1027
  4. J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301–308. https://doi.org/10.2307/1995906
  5. S. Hayashi, Diffeomorphisms in $F^1$-(M) satisfy Axiom A, Ergod. Th. & Dynam. Sys. 12 (1992), 233–253.
  6. M. Hurley, Bifurcations and chain recurrence, Ergodic Theory & Dynam. Sys. 3 (1983), 231–240.
  7. K. Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc. 67 (2003), 15–26. https://doi.org/10.1017/S0004972700033487
  8. K. Lee and M. Lee, Hyperbolicity of $C^1$-stably expansive homoclinic classes, to apear Discrete Contin. Dyn Syst.
  9. K. Lee, K. Moriyasu, and K. Sakai, $C^1$-stable shadowing diffeomorphisms, Discrete Contin. Dyn. Syst. 22 (2008), 683–697.
  10. R. Mane, An ergodic closing lemma, Ann of Math. 116 (1982), 503–540. https://doi.org/10.2307/2007021
  11. J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics 35, Cambridge University Press, 1993.
  12. S. Pilyugin, Inverse shadowing by continuous methods, Discrete Contin. Dyn. Syst. 8 (2002), 29–38. https://doi.org/10.3934/dcds.2002.8.29
  13. S. Pilyugin, A. A. Rodionova, and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst. 9 (2003), 287–308. https://doi.org/10.3934/dcds.2003.9.287
  14. C. Robinson, Dynamical Systems: stability, symbolic dynamics, and chaos (2-nd Ed.): Studies in Advanced Mathematics, CRC Press 1999.
  15. K. Sakai, Quasi-Anosov diffeomorphisms and pseudo-orbit tracing property, Nagoya Math. J. 111 (1988), 111–114.
  16. K. Sakai, $C^1$-stably shadowable chain components, Ergod. Th. & Dynam. Sys. 28 (2008), 987–1029.

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