Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

ASYMPTOTIC AVERAGE SHADOWING PROPERTY ON A CLOSED SET

  • Lee, Manseob (Department of Mathematics Mokwon University) ;
  • Park, Junmi (Department of Mathematics Chungnam National University)
  • Published : 2012.02.15
⟨ Previous Next ⟩

Abstract

Let $f$ be a difeomorphism of a closed $n$ -dimensional smooth manifold M, and $p$ be a hyperbolic periodic point of $f$. Let ${\Lambda}(p)$ be a closed set which containing $p$. In this paper, we show that (i) if $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$, then ${\Lambda}(p)$ is the chain component which contains $p$. (ii) suppose $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$. Then if $f|_{\Lambda(p)}$ has the $C^{1}$-stably shadowing property then it is hyperbolic.

Keywords

References

  1. C. Bonatti and S. Crovisier, Recurrence and genericity, Invent. Math. 158 (2004), 33-104.
  2. S. Corvisier, Periodic orbit and chain transitive sets of $C^{1}$-diffeomorphisms, Publ. Math. Inst. Hautes tudes Sci. 104 (2006), 87-141. https://doi.org/10.1007/s10240-006-0002-4
  3. R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal. 67 (2007), 1680-1689. https://doi.org/10.1016/j.na.2006.07.040
  4. B. Honary and A. Zamani Bahabadi, Asymptotic average shadowing property on compact metric spaces, Nonlinear Anal. 69 (2008), 2857-2863. https://doi.org/10.1016/j.na.2007.08.058
  5. K. Lee, K. Moriyasu and K. Sakai, $C^{1}$-stable shadowing diffeomorphisms, Discrete Contin. Dyn. Syst. 22 (2008), 683-697. https://doi.org/10.3934/dcds.2008.22.683
  6. K. Lee and X. Wen, Shadowable chain transitive sets of $C^{1}$-generic diffeomorphisms, preprint.
  7. K. Sakai, $C^{1}$-stably shadowable chain components, Ergod. Th. & Dynam. Syst. 28 (2008), 987-1029.
  8. X. Wen, S. Gan and L. Wen, $C^{1}$-stably shadowable chain components are hyperbolic, J. Diff. Equa. 246 (2009), 340-357. https://doi.org/10.1016/j.jde.2008.03.032