Adapted Metrics for Dominated splittings.

Nicolas Gourmelon (1, 2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Dominated splitting; partially hyperbolic; adapted metric; Banach bundle; linear cocycle

Domaines: Systèmes dynamiques [math.DS]

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