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Non-uniform hyperbolicity and non-uniform specification


Authors: Krerley Oliveira and Xueting Tian
Journal: Trans. Amer. Math. Soc. 365 (2013), 4371-4392
MSC (2010): Primary 37A35, 37D05, 37C35
DOI: https://doi.org/10.1090/S0002-9947-2013-05819-9
Published electronically: April 2, 2013
MathSciNet review: 3055699
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Abstract: In this paper we deal with an invariant ergodic hyperbolic measure $ \mu $ for a diffeomorphism $ f,$ assuming that $ f$ is either $ C^{1+\alpha }$ or $ C^1$ and the Oseledec splitting of $ \mu $ is dominated. We show that this system $ (f,\mu )$ satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers.

Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the main theorem of ``Recurrence and Lyapunov exponents'' by Saussol, Troubetzkoy and Vaienti that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any such measure is the weak limit of averages of Dirac measures at periodic points, as in a paper by Sigmund. One can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points whenever the dynamic is positive expansive and every measure with pressure close to the topological pressure is hyperbolic.


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Additional Information

Krerley Oliveira
Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, 57072-090 Maceó, AL, Brazil
Email: krerley@gmail.com

Xueting Tian
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: xuetingtian@fudan.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2013-05819-9
Keywords: Pesin theory, non-uniform specification property, Lyapunov exponents, hyperbolic measures, (exponentially) shadowing property, dominated splitting, quantitative recurrence
Received by editor(s): June 15, 2011
Received by editor(s) in revised form: January 13, 2012
Published electronically: April 2, 2013
Additional Notes: The first author was supported by CNPq, CAPES, FAPEAL, INCTMAT and PRONEX
The second author was the corresponding author and was supported by CAPES and China Postdoctoral Science Foundation (No. 2012M510578).
Article copyright: © Copyright 2013 American Mathematical Society