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 -