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arXiv · 1508.06714

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

Chao Liang·Yun Yang
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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}_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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Chao Liang, Yun Yang. 2015-08-27. The $C^1$ density of nonuniform hyperbolicity in $C^{ r}$ conservative diffeomorphisms. https://arxiv.org/abs/1508.06714

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