Diffeomorphisms with positive metric entropy

Abstract

We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

arXiv comment24 pages, 2 figures. The original arXiv submission has been split into two parts, this new version corresponds to the first part. The second part will be available as "$C^1$ density of stable ergodicity", arXiv:1709.04983