SOME SYSTEMS WITH <i>C</i><sub>1</sub> REGULARITY AND ONLY NEGATIVE LYAPUNOV EXPONENTS

Congcong Qu

doi:10.11948/20200327

2021 Volume 11 Issue 3

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Congcong Qu. SOME SYSTEMS WITH C1 REGULARITY AND ONLY NEGATIVE LYAPUNOV EXPONENTS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1600-1609. doi: 10.11948/20200327

Citation:

Congcong Qu. SOME SYSTEMS WITH C₁ REGULARITY AND ONLY NEGATIVE LYAPUNOV EXPONENTS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1600-1609. doi: 10.11948/20200327

SOME SYSTEMS WITH C₁ REGULARITY AND ONLY NEGATIVE LYAPUNOV EXPONENTS

Congcong Qu^1, ,

1.
Department of mathematics, Soochow University, Shizi Street No.1, Suzhou City, China

Corresponding author: Email: congcongqu@foxmail.com(C. Qu)

Abstract

Abstract

In this paper, we prove that for a $C^{1}$ diffeomorphism preserving an ergodic measure $\mu$ with only negative Lyapunov exponents, the support set of $\mu$ is a periodic orbit. For a skew product system preserving an ergodic measure with only negative fiberwise exponents, whose fiber maps are $C^{1}$ diffeomorphisms, we get that for almost all fibers, the disintegration of this measure on fibers is supported on finitely many points.
- Negative Lyapunov exponents /
- skew product systems /
- periodic point
MSC: 37D25

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References

[1]	F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^{1}$-generic diffeomorphisms, Israel J. Math., 2011, 183, 1-60. doi: 10.1007/s11856-011-0041-5 CrossRef Google Scholar
[2]	A. Avila, A. Kocsard and X. Liu, Livšic theorem for diffeomorphism cocycles, Geom. Funct. Anal., 2018, 28, 943-964. doi: 10.1007/s00039-018-0454-y CrossRef Google Scholar
[3]	L. Backes, Periodic approximation of Oseledets subspaces for semi-invertible cocycles, Dyn. Syst., 2018, 33, 480-496. doi: 10.1080/14689367.2017.1388768 CrossRef Google Scholar
[4]	L. Barreira and Y. Pesin, Nonuniform hyperbolicity, encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, 2007. Google Scholar
[5]	L. Backes and M. Poletti, A Livšic theorem for matrix cocycles over non-uniformly hyperbolic systems, J. Dynam. Differential Equations, 2019, 31, 1825-1838. doi: 10.1007/s10884-018-9691-x CrossRef Google Scholar
[6]	A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge, 1995. Google Scholar
[7]	A. Kocsard and R. Potrie, Livšic theorem for low-dimensional diffeomorphism cocycles, Comment. Math. Helv., 2016, 91, 39-64. doi: 10.4171/CMH/377 CrossRef Google Scholar
[8]	C. Liang, G. Liu and W. Sun, Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems, Trans. Amer. Math. Soc., 2009, 361, 1543-1579. Google Scholar
[9]	C. Liang, G. Liao and W. Sun, A note on approximation properties of the Oseledets splitting, Proc. Amer. Math. Soc., 2014, 142, 3825-3838. doi: 10.1090/S0002-9939-2014-12093-4 CrossRef Google Scholar
[10]	V. Oseledets, A multiplicative ergodic theorem, Trans. Moscow. Math. Soc., 1968, 19, 197-231. Google Scholar
[11]	M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1993. Google Scholar
[12]	D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 2001, 219, 481-487. doi: 10.1007/s002200100420 CrossRef Google Scholar
[13]	M. Viana, Lectures on Lyapunov exponents, Cambridge University Press, Cambridge, 2014. Google Scholar
[14]	P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. Google Scholar
[15]	R. Zou and Y. Cao, Livšic theorem for matrix cocycles over non-uniformly hyperbolic systems, Stoch. Dyn., 2019, 19(02), 1950010. doi: 10.1142/S0219493719500102 CrossRef Google Scholar