Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps
type: Item
instance of: scholarly article
title: Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps
zbMATH Open document ID: 1125.37308
DOI: 10.1016/S0294-1449(01)00094-4
author: J. Bochi
author: Marcelo Viana
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2002-09-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78536
review text: We show that the Lyapunov exponents of volume preserving \(C^1\) diffeomorphisms of a compact manifold are continuous at a given diffeomorphism if and only if the Oseledets splitting is either dominated or trivial. It follows that for a \(C^1\)-residual subset of volume preserving diffeomorphisms the Oseledets splitting is either dominated or trivial. We obtain analogous results in the setting of symplectic diffeomorphisms, where the conclusion is actually stronger: dominated splitting is replaced by partial hyperbolicity. We also obtain versions of these results for continuous cocycles with values in some matrix groups. In the text we give the precise statements of these results and the ideas of the proofs. The complete proofs will appear in [A sharp dichotomy for conservative systems: zero Lyapunov exponents or projective hyperbolicity, in preparation].
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D30
zbMATH DE Number: 1785834
MaRDI profile type: Publication
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Wikidata QID: Q125366756
OpenAlex ID: W2050860224
description: scientific article; zbMATH DE number 1785834