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

TL;DR

This paper demonstrates that in the space of volume-preserving and symplectic diffeomorphisms, those with non-zero Lyapunov exponents on a positive volume set are densely approximable in the $C^1$ topology, highlighting the prevalence of nonuniform hyperbolicity.

Contribution

It establishes $C^1$ density of diffeomorphisms with non-zero Lyapunov exponents on positive volume sets in both volume-preserving and symplectic contexts.

Findings

Density of diffeomorphisms without zero Lyapunov exponents in volume-preserving case

Density of symplectic diffeomorphisms with non-zero Lyapunov exponents on positive volume sets

Extension of nonuniform hyperbolicity results to $C^1$ topology

Abstract

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 $.