## Non-uniform hyperbolicity and non-uniform specification

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- by Krerley Oliveira and Xueting Tian PDF
- Trans. Amer. Math. Soc.
**365**(2013), 4371-4392 Request permission

## 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
- 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). - © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3055699