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

Abstract

Let $\Diff^{ r}_m(M)$ be the set of $C^r$ volume-preserving diffeomorphisms on a compact Riemannian manifold $M$ ($\dim M\ge 2$). 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 $\mathcal{S}y{m}_{\omega }^r(M),r\ge 1$ saying that the symplectic diffeomorphisms with non-zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\mathcal{S}y{m}_{\omega }^r(M),r\ge 1$.

13 pages

References in corpus (1)

• Global rigidity of higher rank Anosov algebraic actions

Cited by in corpus (1)

• Extremal exponents of random products of conservative diffeomorphisms