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

Avila, Artur; Bochi, Jairo

doi:10.1090/S0002-9947-2012-05423-7

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

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 $r \ge 1$; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the $L^1$-continuity of Lyapunov exponents.

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