Adapted Metrics for Dominated splittings.

Abstract : A Riemannian metric is adapted to an hyperbolic set of a diffeomorphism if, for this metric, the expansion/contraction of the unstable/stable directions can be 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 on the next bundle. The existence of an adapted metric for a dominated splitting has been asked by Hirsch Pugh and Shub who answer positively to the question in the special case of a dominated splitting in two bundles, one being of dimension 1. 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.
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https://hal.archives-ouvertes.fr/hal-00387460
Contributor : Nicolas Gourmelon <>
Submitted on : Monday, May 25, 2009 - 2:28:19 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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  • HAL Id : hal-00387460, version 1

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Nicolas Gourmelon. Adapted Metrics for Dominated splittings.. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2007, 27 (6), pp.1839--1849. ⟨hal-00387460⟩

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