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

TL;DR

This paper proves that generic volume-preserving diffeomorphisms on compact manifolds exhibit a dichotomy: either zero Lyapunov exponents almost everywhere or a dense, ergodic, nonuniformly hyperbolic component with a global dominated splitting.

Contribution

It establishes new generic properties such as the continuity of ergodic decomposition, persistence of invariant sets, and $L^1$-continuity of Lyapunov exponents for volume-preserving diffeomorphisms.

Findings

Either zero Lyapunov exponents almost everywhere or a dense nonuniformly hyperbolic component.

The nonuniformly hyperbolic component, if of positive measure, is essentially dense and admits a global dominated splitting.

New generic properties include continuity of ergodic decomposition and persistence of invariant sets.

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.