Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T08:13:00.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Adapted metrics for dominated splittings

Published online by Cambridge University Press:  01 December 2007

NIKOLAZ GOURMELON
NIKOLAZ GOURMELON*
Affiliation:
I.M.B., UMR 5584 du CNRS, B.P. 47 870, 21078 Dijon Cedex, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1839 - 1849
Copyright
Copyright © Cambridge University Press 2007

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

[1]Bonatti, C., Diaz, L. J. and Pujals, E. R.. A C 1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. 158(2) (2000), 355–418.CrossRefGoogle Scholar
[2]Bonatti, C., Diaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. Springer, Berlin, 2004.Google Scholar
[3]Brin, M. and Pesin, Ya.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk. SSSR 38 (1974), 177–212.Google Scholar
[4]Hayashi, R. S.. Connecting invariant manifolds and the solution of the C 1-stability and Ω-stability conjectures for flows. Ann. of Math. 145 (1997), 81–137.CrossRefGoogle Scholar
[5]Hisch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[6]Liao, S. D.. On the stability conjecture. Chinese Ann. Math. 1 (1980), 9–30.Google Scholar
[7]Mané, R.. An ergodic closing lemma. Ann. of Math. 116 (1982), 503–540.CrossRefGoogle Scholar
[8]Pliss, V.. On a conjecture due to smale. Differ. Uravn. 8 (1972), 262–268.Google Scholar
[9]Palis, J. and Smale, S.. Structural Stability Theorems (Proceedings of the Institute on Global Analysis, XIV). American Mathematical Society, Providence, RI, 1970, pp. 223–232.Google Scholar
[10]Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. 3 (2000), 961–1023.CrossRefGoogle Scholar