Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T16:17:30.705Z Has data issue: false hasContentIssue false

Hyperbolic manifolds with the strongly shadowing property

Published online by Cambridge University Press:  17 April 2009

Keon-Hee Lee
Affiliation:
Department of MathematicsChungnam National UniversityTaejon 305–764Korea e-mail: [email protected]
Jong-Myung Kim
Affiliation:
Department of Mathematics EducationKwandong UniversityKangnung210–701Korea
Rights & Permissions [Opens in a new window]

Abstract

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.

Let f be a C1 diffeomorphism of a compact smooth manifold M and λ ⊂ M a C1 compact invariant submanifold with a hyperbolic structure as a subset of M. We show that the diffeomorphism | λ is Anosov if and only if λ has the strongly shadowing property, and find hyperbolic sets which have the strongly shadowing property.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Franks, J. and Robinson, C., ‘A quasi-Anosov diffeomorphism that is not Anosov’, Trans. Amer. Math. Soc. 223 (1976), 267278.CrossRefGoogle Scholar
[2]Hirsch, M., ‘On invariant subsets of hyperbolic sets’, in Essays in topology and related topics (Springer-Verlag, Berlin, Heidelberg, New York, 1970), pp. 126146.CrossRefGoogle Scholar
[3]Hirsch, M. and Pugh, C., ‘Stable manifolds and hyperbolic sets’, Proc. Sympos. Pure Math. 14 (1970), 125163.CrossRefGoogle Scholar
[4]Kato, K., ‘Stability and the pseudo-orbit tracing property for diffeomorphisms’, Mem. Fac. Sci. Kôchi. Univ. Ser. A Math. 9 (1988), 3758.Google Scholar
[5]Mane, R., ‘Invariant sets of Anosov diffeomorphisms’, Invent. Math. 46 (1978), 147152.CrossRefGoogle Scholar
[6]Ombach, J., ‘Shadowing, expansiveness and hyperbolic homeomorphisms’, J. Austral. Math. Soc. 61 (1996), 5772.CrossRefGoogle Scholar
[7]Palis, J. and Pugh, C., Fifty problems on dynamical systems, Springer Lecture Notes on Mathematics 468 (Springer-Verlag, Berlin, Heidelberg, New York, 1975), pp. 345353.Google Scholar
[8]Zeghib, A., ‘Subsystems of Anosov systems’, Amer J. Math. 117 (1995), 14311448.CrossRefGoogle Scholar