Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stably inverse shadowable transitive sets and dominated splitting

Authors: Keonhee Lee and Manseob Lee
Journal: Proc. Amer. Math. Soc. 140 (2012), 217-226
MSC (2000): Primary 37D30; Secondary 37C50
Published electronically: May 19, 2011
MathSciNet review: 2833534
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a diffeomorphism of a closed $ n$-dimensional smooth manifold. In this paper, we show that if $ f$ has the $ C^1$-stably inverse shadowing property on a transitive set, then it admits a dominated splitting.

References [Enhancements On Off] (What's this?)

  • 1. C. Bonatti, L. J. Díaz 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), 187-222. MR 2018925 (2007k:37032)
  • 2. C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergod. Th. & Dynam. Syst. 26 (2006), 1307-1337. MR 2266363 (2007i:37062)
  • 3. T. Choi, K. Lee and Y. Zhang, Characterisations of $ \Omega$-stability and structural satbility via inverse shadowing, Bull. Austral. Math. Soc. 74 (2006), 185-196. MR 2260487 (2007h:37028)
  • 4. J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301-308. MR 0283812 (44:1042)
  • 5. N. Gourmelon, A Frank's lemma that preserves invariant manifolds, preprint (arXiv:0912.1121).
  • 6. S. Hayashi, Diffeomorphisms in $ \mathcal{F}^1(M)$ satisfy Axiom A, Ergod. Th. & Dynam. Syst. 12 (1992), 233-253. MR 1176621 (94d:58081)
  • 7. K. Lee and J. Park, Inverse shadowing of circle maps, Bull. Austral. Math. Soc. 69 (2004), 353-359. MR 2066653 (2005c:37073)
  • 8. K. Lee, G. Lu and X. Wen, $ C^1$-stably weak shadowing property of chain transitive sets, preprint.
  • 9. M. Lee, $ C^1$-stably inverse shadowing chain components for generic diffeomorphisms, Commun. Korean Math. Soc. 24 (2009), 127-144. MR 2488815 (2009m:37065)
  • 10. R. Mãné, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), 503-540. MR 678479 (84f:58070)
  • 11. D. Yang, Stably weakly shadowing transitive sets and dominated splitting, preprint (arXiv:1003.2104v1).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37D30, 37C50

Retrieve articles in all journals with MSC (2000): 37D30, 37C50

Additional Information

Keonhee Lee
Affiliation: Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea

Manseob Lee
Affiliation: Department of Mathematics, Mokwon University, Daejeon, 302-729, Republic of Korea

Keywords: Dominated splitting, genericity, inverse shadowing, transitive set.
Received by editor(s): June 15, 2010
Received by editor(s) in revised form: October 7, 2010, and November 3, 2010
Published electronically: May 19, 2011
Additional Notes: The second author is the corresponding author.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society