Diffeomorphisms with positive metric entropy

Artur Avila, Sylvain Crovisier, Amie Wilkinson

Published 2014-08-19Version 1

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. hyperbolic and the splitting into stable and unstable spaces is dominated). We take this dichotomy as a starting point to prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for the $C^1$ topology: "orbitwise" removal of vanishing Lyapunov exponents, linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.