TY - JOUR

T1 - Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms

AU - Avila, Artur

AU - Bochi, Jairo

PY - 2012

Y1 - 2012

N2 - We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the C 1 topology: Either there is at least one zero Lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure, then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold C r-generically for any r ≥ 1; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the L 1-continuity of Lyapunov exponents.

AB - We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the C 1 topology: Either there is at least one zero Lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure, then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold C r-generically for any r ≥ 1; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the L 1-continuity of Lyapunov exponents.

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U2 - 10.1090/S0002-9947-2012-05423-7

DO - 10.1090/S0002-9947-2012-05423-7

M3 - Article

AN - SCOPUS:84857679649

SN - 0002-9947

VL - 364

SP - 2883

EP - 2907

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -