Symplectomorphisms with positive metric entropy

We obtain a dichotomy for C¹-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Mañé. A key ingredient is an analysis of partially hyperbolic sets of positive volume. We generalize to partially hyperbolic invariant sets the main result in Dolgopyat and Wilkinson (Astérisque 287 (2003), 33–60) that stable accessibility is C¹-dense among partially hyperbolic diffeomorphisms.