Nonuniform Hyperbolicity, Global Dominated Splittings and Generic Properties of Volume-Preserving Diffeomorphisms

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 non-zero 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 $r \geq 1$, namely: the continuity of the ergodic decomposition, the persistence of invariant sets, and the $L^1$-continuity of Lyapunov exponents.