Every expanding measure has the nonuniform specification property

Oliveira, Krerley

doi:10.1090/S0002-9939-2011-10985-7

Every expanding measure has the nonuniform specification property
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Proc. Amer. Math. Soc. 140 (2012), 1309-1320 Request permission

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.

References

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
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, DOI 10.1007/s002220000057
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
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, DOI 10.17323/1609-4514-2003-3-1-189-203
Rufus Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30. MR 298700, DOI 10.2307/2373590
Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 399413, DOI 10.1007/BF01762666
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, DOI 10.1090/S0002-9947-08-04630-8
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
Krerley Oliveira, Equilibrium states for non-uniformly expanding maps, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1891–1905. MR 2032493, DOI 10.1017/S0143385703000257
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, DOI 10.1017/S0143385707001009
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
Ya. B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys, 32(4):55–114, 1977.
V. Pinheiro. Expanding measures. Preprint, 2008, http://arxiv.org/abs/0811.2545. To appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.
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
W. Sun and X. Tian. Pesin set, closing lemma and shadowing lemma in $C^1$ non-uniformly hyperbolic systems with limit domination. Preprint, 2010, arXiv:1004.0486.
P. Varandas. Non-uniform specification and large deviations for weak Gibbs measures, arXiv:0906.3350v2, February 2010.