Generic area-preserving reversible diffeomorphisms

Let M be a surface and R : M → M an area-preserving C^∞ diffeomorphism which is an involution and whose set of fixed points is a submanifold with dimension one. We will prove that C¹ - generically either an area-preserving R-reversible diffeomorphism, is Anosov, or, for μ-almost every x ∈ M, the Lyapunov exponents at x vanish or else the orbit of x belongs to a compact hyperbolic set with an empty interior. We will also describe a nonempty C¹-open subset of area-preserving R-reversible diffeomorphisms where for C¹ - generically each map is either Anosov or its Lyapunov exponents vanish from almost everywhere.