  We obtain a dichotomy for $C^1$-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 Ma\~n\'e.
  A main ingredient in our proof is a generalization to partially hyperbolic
invariant sets of the main result in [Dolgopyat-Wilkinson] that stable
accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
