Diffeomorphisms with positive metric entropy

Abstract

We obtain a dichotomy for C1-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 Ma\~n\'e.

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