Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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STABLY PERIODIC SHADOWING AND DOMINATED SPLITTING

  • Lee, Keonhee (Department of Mathematics Chungnam University) ;
  • Lee, Manseob (Department of Mathematics Mokwon University) ;
  • Ahn, Jiweon (Department of Mathematics Chungnam University)
  • Received : 2011.07.11
  • Accepted : 2011.08.25
  • Published : 2011.12.30
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Abstract

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we introduce the notion of $C^1$-stably periodic shadowing property for a closed f-invariant set, and prove that for a transitive set ${\Lambda}$, if f has the $C^1$-stably periodic shadowing property on ${\Lambda}$, then ${\Lambda}$ admits a dominated splitting.

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References

  1. C. Bonatti, L. J. Diaz and E. Pujals, A $C^{1}$-generic dichotomy for diffeomorphisms : weak forms of hyperbolicity or infinitely many sinks or sources, Annal. of Math. 158 (2003), 355-418. https://doi.org/10.4007/annals.2003.158.355
  2. C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergod. Th. & Dynam. Sys. 26 (2006), 1307-1337. https://doi.org/10.1017/S0143385706000253
  3. J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301-308. https://doi.org/10.1090/S0002-9947-1971-0283812-3
  4. N. Gourmelon, A Frank's Lemma that preserves invariant manifolds, preprint.
  5. S. Hayashi, Diffomorphisms in $F^{1}$(M) satisfy Axiom A, Ergod. Th. & Dynam. Sys. 12 (1992), 233-253.
  6. M. Lee, Transitive sets with dominated splitting, J. Chungcheong Math. Soc. 23 (2010), 65-71.
  7. M. Lee, Dominated splitting with stably expansive, to apear in J. of the KSME, Series B.
  8. K. Lee and M. Lee, Stably inverse shadowable transiteve sets and dominated splitting, to appear in Proc. Amer. Math. Soc.
  9. K. Lee, G. Lu and X. Wen, $C^{1}$-stably weak shadowing property of chain tran- sitive sets, preprint.
  10. R. Mane, An ergodic closing lemma, Ann. Math. 116 (1982), 503-540. https://doi.org/10.2307/2007021
  11. A. V. Osipov, S. Y. Pilyugin, and S. B. Tikhomirov, Periodic shadowing and $\Omega$­-stability, Regular and Chaotic Dynamics, 15 (2010), 404-417. https://doi.org/10.1134/S1560354710020255
  12. D. Yang, Stably weakly shadowing transiteve sets and dominated splitting, Proc. Amer. Math. Soc. 139 (2011), 2747-2751. https://doi.org/10.1090/S0002-9939-2011-10699-3