Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

VOLUME PRESERVING DYNAMICS WITHOUT GENERICITY AND RELATED TOPICS

  • Choy, Jae-Yoo (Department of Mathematics Kyungpook National University) ;
  • Chu, Hahng-Yun (Department of Mathematics Chungnam National University) ;
  • Kim, Min-Kyu (Department of Mathematics Education Gyeongin National University of Education)
  • Received : 2010.11.04
  • Published : 2012.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let $M$ be a compact manifold with a volume form ${\omega}$ and $f:M{\rightarrow}M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves ${\omega}$. In this paper, we do not assume $f$ is $\mathcal{C}^1$-generic. We prove that $f$ satisfies the chain transitivity and we also show that, on $M$, the $\mathcal{C}^1$-stable shadowability is equivalent to the hyperbolicity.

Keywords

References

  1. M. Arnaud, C. Bonatti, and S. Crovisier, Dynamiques symplectiques generiques, Ergodic Theory Dynam. Systems 25 (2005), no. 5, 1401-1436. https://doi.org/10.1017/S0143385704000975
  2. C. Bonatti and S. Crovisier, Recurrence et genericite, Invent. Math. 158 (2004), no. 1, 33-04.
  3. C. Conley, Isolated Invariant Sets and the Morse Index, C.B.M.S. Regional Lect. 38, A.M.S., 1978.
  4. J. Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177-195. https://doi.org/10.1007/BF01212280
  5. K. Moriyasu, The topological stability of diffeomorphisms, Nagoya Math. J. 123 (1991), 91-102.
  6. H. Nakayama and T. Noda, Minimal sets and chain recurrent sets of projective ows induced from minimal ows on 3-manifolds, Discrete Contin. Dyn. Syst. 12 (2005), no. 4, 629-638. https://doi.org/10.3934/dcds.2005.12.629
  7. X. Wen, S. Gan, and L. Wen, C1-stably shadowable chain components are hyperbolic, J. Differential Equations 246 (2009), no. 1, 340-357. https://doi.org/10.1016/j.jde.2008.03.032

Cited by

  1. Chain Recurrences on Conservative Dynamics vol.54, pp.2, 2014, https://doi.org/10.5666/KMJ.2014.54.2.165