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


Authors: Chao Liang and Yun Yang
Journal: Proc. Amer. Math. Soc. 145 (2017), 1539-1552
MSC (2010): Primary 37D30, 37D25, 37C25
DOI: https://doi.org/10.1090/proc/13527
Published electronically: December 15, 2016
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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 $ \textup {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 $ \textup {Sym}^{r}_{\omega }(M)$, there is a diffeomorphism $ g\in \textup {Sym}^{r}_{\omega }(M)$ without zero Lyapunov exponents on a set with positive volume.


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Additional Information

Chao Liang
Affiliation: Applied Mathematical Department, Central University of Finance and Economics, Beijing, 100081, People’s Republic of China
Email: chaol@cufe.edu.cn

Yun Yang
Affiliation: Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: yyang@gc.cuny.edu

DOI: https://doi.org/10.1090/proc/13527
Keywords: Nonuniform hyperbolicity, volume-preserving, elliptic periodic points
Received by editor(s): November 21, 2015
Published electronically: December 15, 2016
Additional Notes: The first author was supported by NNSFC(# 11471344) and 2016 NSFC/ICTP Grants(# 11681240278) and Beijing Higher Education Young Elite Teacher Project(YETP0986)
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society