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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])
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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

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