  We show that, for any compact surface, there is a residual (dense $G_\delta$)
set of $C^1$ area preserving diffeomorphisms which either are Anosov or have
zero Lyapunov exponents a.e. This result was announced by R. Mane, but no proof
was available. We also show that for any fixed ergodic dynamical system over a
compact space, there is a residual set of continuous $SL(2,R)$-cocycles which
either are uniformly hyperbolic or have zero exponents a.e.
