  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}_{\omega}(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}_{\omega}(M), r\geq1 $.
