Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-24T06:49:50.353Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterisations of Ω-stability and structural stability via inverse shadowing

Published online by Cambridge University Press:  17 April 2009

Taeyoung Choi ,
Keonhee Lee  and
Yong Zhang
Taeyoung Choi
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon, 305–764, Korea e-mail: shadowcty@hanmail.net, khlee@math.cnu.ac.kr
Keonhee Lee
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon, 305–764, Korea e-mail: shadowcty@hanmail.net, khlee@math.cnu.ac.kr
Yong Zhang
Affiliation:
Department of Mathematics, Suzhou University, Suzhou, China e-mail: yongzhang@suda.edu.cn
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give characterisations of Ω-stable diffeomorphisms and structurally stable diffeomorphisms via the notions of weak inverse shadowing and orbital inverse shadowing, respectively. More precisely, it is proved that the C1 interior of the set of diffeomorphisms with the weak inverse shadowing property coincides with the set of Ω-stable diffeomorphisms and the C1 interior of the set of diffeomorphisms with the orbital inverse shadowing property coincides with the set of structurally stable diffeomorphisms.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 2 , October 2006 , pp. 185 - 196
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Choi, T., Kim, S. and Lee, K., ‘Weak inverse shadowing and genericity’, Bull. Korean Soc. 43 (2006), 43–52.CrossRefGoogle Scholar
[2]Corless, R. and Pilyugin, S., ‘Approximate and real trajectories for generic dynamical systems’, J. Math. Anal. Appl. 189 (1995), 409–423.CrossRefGoogle Scholar
[3]Diamond, p., Lee, K. and Han, Y., ‘Bishadowing and hyperbolicity’, Internat. J. Bifur. Chaos appl. Sci. Engrg. 12 (2002), 1779–1788.CrossRefGoogle Scholar
[4]Hayashi, S., ‘Diffeomorphisms in F l(M) satisfy Axiom A’, Ergodic Theory Dynamical Systems 12 (1992), 233–253.CrossRefGoogle Scholar
[5]Kloeden, P., Ombach, J. and Porkrovskii, A., ‘Continuous and inverse shadowing’, Funct. Differ. Equ. 6 (1999), 137–153.Google Scholar
[6]Lee, K., ‘Continuous inverse shadowing and hyperbolicity’, Bull. Austral. Math. Soc. 67 (2003), 15–26.CrossRefGoogle Scholar
[7]Lee, K. and Park, J., ‘Inverse shadowing of cirlce maps’, Bull. Austral. Math. Soc. 69 (2004), 353–359.Google Scholar
[8]Lee, K. and Sakai, K., ‘Various shadowing properties and their equivalence’, Disc. Contin. Dyn. Syst. 13 (2005), 533–539.CrossRefGoogle Scholar
[9]Mané, R., ‘Contribution to stability conjecture’, Topology 17 (1978), 383–396.CrossRefGoogle Scholar
[10]Moriyasu, K., ‘The topological stability of diffeomorphisms’, Nagoya Math. J. 123 (1991), 91–203.CrossRefGoogle Scholar
[11]Moriyasu, K., Sakai, K. and Sumi, N., ‘Vector fields with topological stability’, Trans. Amer. Math. Soc. 353 (2001), 3391–3408.CrossRefGoogle Scholar
[12]Pilyugin, S., ‘Inverse shadowing by continuous methods’, Disc. Contin. Dyn. Syst. 8 (2002), 29–38.CrossRefGoogle Scholar
[13]Pilyugin, S., Rodionova, A. and Sakai, K., ‘Orbital and weak shadowing properties’, Disc. Contin. Dyn. Sys. 9 (2003), 287–308.CrossRefGoogle Scholar
[14]Sakai, K., ‘Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds’, Osaka J. Math. 31 (1994), 373–386.Google Scholar
[15]Sakai, K., ‘Diffeomorphism with persistency’, Proc. Amer. Math. Soc. 124 (1996), 2249–2254.CrossRefGoogle Scholar
[16]Sakai, K., ‘Diffeomorphism with weak shadowing’, Fund. Math. 168 (2001), 57–75.CrossRefGoogle Scholar