Genericity of zero Lyapunov exponents

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We show that, for any compact surface, there is a residual (dense Gδ) set of C¹ 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.