Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-28T14:45:57.378Z Has data issue: false hasContentIssue false
  • English
  • Français

On the integrability of intermediate distributions for Anosov diffeomorphisms†

Published online by Cambridge University Press:  19 September 2008

M. Jiang ,
Ya B. Pesin  and
R. de la Llave
M. Jiang
Affiliation:
Department of Mathematics, Penn State University, University Park PA 16802, USA (e-mail: jiang@math.psu.edu and pesin@math.psu.edu)
Ya B. Pesin
Affiliation:
Department of Mathematics, Penn State University, University Park PA 16802, USA (e-mail: jiang@math.psu.edu and pesin@math.psu.edu)
R. de la Llave
Affiliation:
Department of Mathematics, University of Texas, Austin TX 78712-1802, USA (e-mail: llave@math.utexas.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 2 , April 1995 , pp. 317 - 331
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BK]Brin, M. I. and Kifer, U.. Dynamics of Markov chains and stable manifolds for random diffeomorphisms. Ergod. Th. & Dynam. Sys. 7 (1987), 351–374.CrossRefGoogle Scholar
[BP]Brin, M. I. and Pesin, Ya. B.. Partial hyperbolic dynamical systems, it Izv. Akad. Nauk. SSSR Ser. Mat. 38 (1974), 170–212. MR 493# 8058. = Math. USSR-Izv. 8 (1974), 177–218.Google Scholar
[Fe]Fenichel, N.. Asymptotic stability with rate conditions. Ind. Math. J. 23 (1974), 1109–1137.CrossRefGoogle Scholar
[FK]Flaminio, L. and Katok, A.. Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori. Ergod. Th. & Dynam. Sys. 11 (1991), 427–441.CrossRefGoogle Scholar
[HPS]Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant manifolds. Bull. Amer. Math. Soc. 76 (1970), 1015–1019.CrossRefGoogle Scholar
[HP]Hirsch, M. W. and Pugh, C. C.. Stable manifolds and hyperbolic sets. Proc. Symp. Pure Math. 14 (1970), 133–63.CrossRefGoogle Scholar
[I]Irwin, M. C.. A new proof of the pseudo stable manifold theorem. J. London Math. Soc. 21(2) (1980), 557–566.CrossRefGoogle Scholar
[LW]de la Llave, R. and Wayne, C. E.. On Irwin's proof of the pseudostable manifold theorem. Available from mp_arc. Math. Zeit. In press.Google Scholar
[L]de la Llave, R.. Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems. Comm. Math. Phys. 150 (1992), 289–320.CrossRefGoogle Scholar
[M]Mather, J. N.. Characterization of Anosov diffeomorphisms. Nederl. Akad. Wetensch. Proc. Ser. A71 = Indag. Math. 30 (1968), 479–483.CrossRefGoogle Scholar
[P]Pesin, Ya. B.. On the existence of invariant fiberings for a diffeomorphism of a smooth manifold. Math. USSR Sbornik 20(2) (1973), 213–222.CrossRefGoogle Scholar
[S]Smale, S.. Differential dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar