The C¹ density of nonuniform hyperbolicity in Cʳ conservative diffeomorphisms

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Let ʳₘ(M) be the set of Cʳ volume-preserving diffeomorphisms on a compact Riemannian manifold M (M≥ 2). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are C¹ dense in ʳₘ(M), r≥ 1. We also prove a weaker result for symplectic diffeomorphisms Symʳω(M), r≥1 saying that the symplectic diffeomorphisms with non-zero Lyapunov exponents on a set of positive volume are C¹ dense in Symʳω(M), r≥1.