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

Abstract:

Let $\mathrm {Diff}^{r}_m(M)$ 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 $\mathrm {Diff}^{r}_m(M), r\geq 1$. We also prove a weaker result for symplectic diffeomorphisms $\mathrm {Sym}^{r}_{\omega }(M), r\geq 1$ saying that either the symplectic diffeomorphism $f$ is partially hyperbolic or $C^1$ arbitrarily close to $f$ in $\mathrm {Sym}^{r}_{\omega }(M)$, there is a diffeomorphism $g\in \mathrm {Sym}^{r}_{\omega }(M)$ without zero Lyapunov exponents on a set with positive volume.