Every expanding measure has the nonuniform specification property

Author:
Krerley Oliveira

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1309-1320

MSC (2010):
Primary 37D25, 37A99

Published electronically:
July 29, 2011

MathSciNet review:
2869114

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Abstract: Exploring abundance and nonlacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the length of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property. As consequences, we (re)obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents, and we prove that any expanding measure is the limit of Dirac measures on periodic points.

**1.**José Ferreira Alves and Vítor Araújo,*Random perturbations of nonuniformly expanding maps*, Astérisque**286**(2003), xvii, 25–62 (English, with English and French summaries). Geometric methods in dynamics. I. MR**2052296****2.**José F. Alves, Christian Bonatti, and Marcelo Viana,*SRB measures for partially hyperbolic systems whose central direction is mostly expanding*, Invent. Math.**140**(2000), no. 2, 351–398. MR**1757000**, 10.1007/s002220000057**3.**Ludwig Arnold,*Random dynamical systems*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1723992****4.**B. Saussol, S. Troubetzkoy, and S. Vaienti,*Recurrence and Lyapunov exponents*, Mosc. Math. J.**3**(2003), no. 1, 189–203, 260 (English, with English and Russian summaries). MR**1996808****5.**Rufus Bowen,*Periodic orbits for hyperbolic flows*, Amer. J. Math.**94**(1972), 1–30. MR**0298700****6.**Rufus Bowen,*Some systems with unique equilibrium states*, Math. Systems Theory**8**(1974/75), no. 3, 193–202. MR**0399413****7.**Chao Liang, Geng Liu, and Wenxiang Sun,*Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems*, Trans. Amer. Math. Soc.**361**(2009), no. 3, 1543–1579. MR**2457408**, 10.1090/S0002-9947-08-04630-8**8.**A. Katok,*Lyapunov exponents, entropy and periodic orbits for diffeomorphisms*, Inst. Hautes Études Sci. Publ. Math.**51**(1980), 137–173. MR**573822****9.**Krerley Oliveira,*Equilibrium states for non-uniformly expanding maps*, Ergodic Theory Dynam. Systems**23**(2003), no. 6, 1891–1905. MR**2032493**, 10.1017/S0143385703000257**10.**Krerley Oliveira and Marcelo Viana,*Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps*, Ergodic Theory Dynam. Systems**28**(2008), no. 2, 501–533. MR**2408389**, 10.1017/S0143385707001009**11.**V. I. Oseledec,*A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems*, Trudy Moskov. Mat. Obšč.**19**(1968), 179–210 (Russian). MR**0240280****12.**Ya. B. Pesin.

Characteristic Lyapunov exponents and smooth ergodic theory.*Russian Math. Surveys*, 32(4):55-114, 1977.**13.**V. Pinheiro.

Expanding measures.

Preprint, 2008, http://arxiv.org/abs/0811.2545.

To appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.**14.**V. A. Pliss,*The position of the separatrices of saddle-point periodic motions of systems of second order differential equations*, Differencial′nye Uravnenija**7**(1971), 1199–1225, 1340–1341 (Russian). MR**0285770****15.**W. Sun and X. Tian.

Pesin set, closing lemma and shadowing lemma in non-uniformly hyperbolic systems with limit domination.

Preprint, 2010, arXiv:1004.0486.**16.**P. Varandas.

Non-uniform specification and large deviations for weak Gibbs measures, arXiv:0906.3350v2, February 2010.

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Additional Information

**Krerley Oliveira**

Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072-090 Maceió, AL, Brazil

Email:
krerley@gmail.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10985-7

Keywords:
Periodic points,
Lyapunov exponents.

Received by editor(s):
July 28, 2010

Received by editor(s) in revised form:
December 22, 2010, and December 27, 2010

Published electronically:
July 29, 2011

Additional Notes:
This work was partially supported by CNPq, CAPES, FAPEAL, INCTMAT and PRONEX

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.