Symplectomorphisms with positive metric entropy

We obtain a dichotomy for C1$C<^>1$-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 Mane. 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 (Asterisque 287 (2003), 33-60) that stable accessibility is C1$C<^>1$-dense among partially hyperbolic diffeomorphisms.