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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems


Authors: Chao Liang, Geng Liu and Wenxiang Sun
Journal: Trans. Amer. Math. Soc. 361 (2009), 1543-1579
MSC (2000): Primary 37C40, 37D25, 37H15, 37A35
Published electronically: October 21, 2008
MathSciNet review: 2457408
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Abstract: We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization.

In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.


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Additional Information

Chao Liang
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China – and – Applied Mathematical Department, The Central University of Finance and Economics, Beijing 100081, People’s Republic of China
Email: Chaol@math.pku.edu.cn, Chaol@cufe.edu.cn

Geng Liu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China – and – China Institute of Advanced Study, The Central University of Finance and Economics, Beijing 100081, People’s Republic of China
Email: yaseplay@gmail.com

Wenxiang Sun
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: sunwx@math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04630-8
PII: S 0002-9947(08)04630-8
Keywords: Independence number, invariant measure, mean angle
Received by editor(s): March 2, 2007
Published electronically: October 21, 2008
Additional Notes: The first and second authors were supported by NNSFC(# 10671006)
The third author was supported by NNSFC (# 10231020, # 10671006), the National Basic Research Program of China (973 Program)(# 2006CB805900) and the Doctoral Education Foundation of China
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.