The $C^1$ density of nonuniform hyperbolicity in $C^{ r}$ conservative diffeomorphisms

Let $\Diff^{ r}_m(M)$ be the set of $C^{ r}$ volume-preserving diffeomorphisms on a compact Riemannian manifold $M$ ($\dim M\geq 2$). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\Diff^{ r}_m(M), r\geq 1$. We also prove a weaker result for symplectic diffeomorphisms $\mathcal{S}ym^{r}_ω(M), r\geq1$ saying that the symplectic diffeomorphisms with non-zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\mathcal{S}ym^{r}_ω(M), r\geq1$.

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