TY - JOUR

T1 - Uniform (projective) hyperbolicity or no hyperbolicity

T2 - A dichotomy for generic conservative maps

AU - Bochi, Jairo

AU - Viana, Marcelo

PY - 2002

Y1 - 2002

N2 - We show that the Lyapunov exponents of volume preserving C1 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 C1-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 [4].

AB - We show that the Lyapunov exponents of volume preserving C1 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 C1-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 [4].

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U2 - 10.1016/S0294-1449(01)00094-4

DO - 10.1016/S0294-1449(01)00094-4

M3 - Article

AN - SCOPUS:0347146025

VL - 19

SP - 113

EP - 123

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 1

ER -