Nonuniform hyperbolicity and nonuniform specification
Authors:
Krerley Oliveira and Xueting Tian
Journal:
Trans. Amer. Math. Soc. 365 (2013), 43714392
MSC (2010):
Primary 37A35, 37D05, 37C35
Published electronically:
April 2, 2013
MathSciNet review:
3055699
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Abstract: In this paper we deal with an invariant ergodic hyperbolic measure for a diffeomorphism assuming that is either or and the Oseledec splitting of is dominated. We show that this system satisfies a weaker and nonuniform 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 nonuniform 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.
 1.
Flavio
Abdenur and Lorenzo
J. Díaz, Pseudoorbit shadowing in the 𝐶¹
topology, Discrete Contin. Dyn. Syst. 17 (2007),
no. 2, 223–245. MR 2257429
(2007i:37046)
 2.
A.
M. Blokh, Decomposition of dynamical systems on an interval,
Uspekhi Mat. Nauk 38 (1983), no. 5(233),
179–180 (Russian). MR 718829
(86d:54060)
 3.
Rufus
Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math.
94 (1972), 1–30. MR 0298700
(45 #7749)
 4.
Rufus
Bowen, Equilibrium states and the ergodic theory of Anosov
diffeomorphisms, Lecture Notes in Mathematics, Vol. 470,
SpringerVerlag, BerlinNew York, 1975. MR 0442989
(56 #1364)
 5.
Luis
Barreira and Yakov
Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics
and its Applications, vol. 115, Cambridge University Press, Cambridge,
2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606
(2010c:37067)
 6.
Jérôme
Buzzi, Specification on the
interval, Trans. Amer. Math. Soc.
349 (1997), no. 7,
2737–2754. MR 1407484
(97i:58043), 10.1090/S0002994797018734
 7.
Xiongping
Dai, Exponential closing property and approximation of Lyapunov
exponents of linear cocycles, Forum Math. 23 (2011),
no. 2, 321–347. MR 2787625
(2012d:37118), 10.1515/FORM.2011.011
 8.
Shaobo
Gan, A generalized shadowing lemma, Discrete Contin. Dyn.
Syst. 8 (2002), no. 3, 627–632. MR 1897871
(2003d:37028), 10.3934/dcds.2002.8.627
 9.
Michihiro
Hirayama, Periodic probability measures are dense in the set of
invariant measures, Discrete Contin. Dyn. Syst. 9
(2003), no. 5, 1185–1192. MR 1974422
(2004a:37032), 10.3934/dcds.2003.9.1185
 10.
M.
W. Hirsch, C.
C. Pugh, and M.
Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol.
583, SpringerVerlag, BerlinNew York, 1977. MR 0501173
(58 #18595)
 11.
Boris
Kalinin, Livšic theorem for matrix cocycles, Ann. of
Math. (2) 173 (2011), no. 2, 1025–1042. MR 2776369
(2012b:37082), 10.4007/annals.2011.173.2.11
 12.
Anatole
Katok and Boris
Hasselblatt, Introduction to the modern theory of dynamical
systems, Encyclopedia of Mathematics and its Applications,
vol. 54, Cambridge University Press, Cambridge, 1995. With a
supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
(96c:58055)
 13.
A.
Katok, Lyapunov exponents, entropy and periodic orbits for
diffeomorphisms, Inst. Hautes Études Sci. Publ. Math.
51 (1980), 137–173. MR 573822
(81i:28022)
 14.
Chao
Liang, Geng
Liu, and Wenxiang
Sun, Approximation properties on invariant
measure and Oseledec splitting in nonuniformly hyperbolic
systems, Trans. Amer. Math. Soc.
361 (2009), no. 3,
1543–1579. MR 2457408
(2009m:37085), 10.1090/S0002994708046308
 15.
C. Liang, W. Sun, X. Tian, Ergodic properties of invariant measures for nonuniformly hyperbolic systems, to appear Ergodic Theory Dynamical Systems.
 16.
Gang
Liao, Wenxiang
Sun, and Xueting
Tian, Metric entropy and the number of periodic points,
Nonlinearity 23 (2010), no. 7, 1547–1558. MR 2652470
(2011k:37007), 10.1088/09517715/23/7/002
 17.
Shan
Tao Liao, An existence theorem for periodic orbits, Beijing
Daxue Xuebao 1 (1979), 1–20 (Chinese, with English
summary). MR
560169 (82b:58074)
 18.
Brian
Marcus, A note on periodic points for ergodic toral
automorphisms, Monatsh. Math. 89 (1980), no. 2,
121–129. MR
572888 (81f:28016), 10.1007/BF01476590
 19.
Krerley
Oliveira, Every expanding measure has the
nonuniform specification property, Proc. Amer.
Math. Soc. 140 (2012), no. 4, 1309–1320. MR 2869114
(2012k:37065), 10.1090/S000299392011109857
 20.
Krerley
Oliveira and Marcelo
Viana, Thermodynamical formalism for robust classes of potentials
and nonuniformly hyperbolic maps, Ergodic Theory Dynam. Systems
28 (2008), no. 2, 501–533. MR 2408389
(2009b:37056), 10.1017/S0143385707001009
 21.
V.
I. Oseledec, A multiplicative ergodic theorem. Characteristic
Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat.
Obšč. 19 (1968), 179–210 (Russian). MR 0240280
(39 #1629)
 22.
Mark
Pollicott, Lectures on ergodic theory and Pesin theory on compact
manifolds, London Mathematical Society Lecture Note Series,
vol. 180, Cambridge University Press, Cambridge, 1993. MR 1215938
(94k:58080)
 23.
C.E.
Pfister and W.
G. Sullivan, On the topological entropy of saturated sets,
Ergodic Theory Dynam. Systems 27 (2007), no. 3,
929–956. MR 2322186
(2008f:37036), 10.1017/S0143385706000824
 24.
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
(2004f:37016)
 25.
Karl
Sigmund, Generic properties of invariant measures for Axiom
𝐴\ diffeomorphisms, Invent. Math. 11 (1970),
99–109. MR
0286135 (44 #3349)
 26.
W. Sun, X. Tian, Pesin set, closing lemma and shadowing lemma in nonuniformly hyperbolic systems with limit domination, arXiv:1004.0486.
 27.
Daniel
Thompson, A variational principle for topological pressure for
certain noncompact sets, J. Lond. Math. Soc. (2) 80
(2009), no. 3, 585–602. MR 2559118
(2010k:37040), 10.1112/jlms/jdp041
 28.
X. Tian, Hyperbolic measures with dominated splitting, arXiv:1011.6011v2.
 29.
Paulo
Varandas, Nonuniform specification and large deviations for weak
Gibbs measures, J. Stat. Phys. 146 (2012),
no. 2, 330–358. MR
2873016, 10.1007/s1095501103927
 30.
Zhenqi
Wang and Wenxiang
Sun, Lyapunov exponents of hyperbolic
measures and hyperbolic periodic orbits, Trans.
Amer. Math. Soc. 362 (2010), no. 8, 4267–4282. MR 2608406
(2011m:37030), 10.1090/S0002994710049470
 31.
Kenichiro
Yamamoto, On the weaker forms of the
specification property and their applications, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3807–3814. MR 2529890
(2010f:37027), 10.1090/S0002993909099377
 1.
 F. Abdenur and J. Diaz, Pseudoorbit shadowing in the topology, Discrete Contin. Dyn. Syst. 17 (2007) 223245. MR 2257429 (2007i:37046)
 2.
 A.M. Blokh, Decomposition of dynamical systems on an interval, Russ. Math. Surv. 38, 133134 (1983). MR 718829 (86d:54060)
 3.
 R. Bowen. Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 130. MR 0298700 (45:7749)
 4.
 R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math. 470 (1975). MR 0442989 (56:1364)
 5.
 Luis Barreira and Yakov B. Pesin, Nonuniform hyperbolicity, Cambridge Univ. Press, Cambridge (2007). MR 2348606 (2010c:37067)
 6.
 J. Buzzi, Specification on the interval, Transactions of the American Mathematical Society, 349, 7, 27372754 (1997). MR 1407484 (97i:58043)
 7.
 X. Dai, Exponential closing property and approximation of Lyapunov exponents of linear cocycles, Forum. Math. 23 (2011), no. 2, 321347. MR 2787625 (2012d:37118)
 8.
 S. Gan, A generalized shadowing lemma, Dist. Cont. Dyn. Sys. 8 (2002), 627632. MR 1897871 (2003d:37028)
 9.
 M. Hirayama, Periodic probability measures are dense in the set of invariant measures, Dist. Cont. Dyn. Sys. 9 (2003), 11851192. MR 1974422 (2004a:37032)
 10.
 M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds (Lecture Notes in Math. vol. 583) (Berlin: Spinger), 1977. MR 0501173 (58:18595)
 11.
 B. Kalinin, Livšic Theorem for matrix cocycles, Annals of Mathematics (2) 173 (2011), no. 2, 10251042. MR 2776369 (2012b:37082)
 12.
 A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge Univ. Press, Cambridge (1995). MR 1326374 (96c:58055)
 13.
 A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. 51 (1980), 137173. MR 573822 (81i:28022)
 14.
 C. Liang, G. Liu, W. Sun, Approximation properties on invariant measure and Oseledec splitting in nonuniformly hyperbolic systems, Trans. Amer. Math. Soci. 361 (2009), 15431579. MR 2457408 (2009m:37085)
 15.
 C. Liang, W. Sun, X. Tian, Ergodic properties of invariant measures for nonuniformly hyperbolic systems, to appear Ergodic Theory Dynamical Systems.
 16.
 G. Liao, W. Sun, X. Tian, Metric entropy and the number of periodic points, Nonlinearity 23 (2010) 15471558. MR 2652470 (2011k:37007)
 17.
 S. Liao, An existence theorem for periodic orbits, Acta Sci. Natur. Univ. Pekin. (1979), 120. MR 560169 (82b:58074)
 18.
 B. Marcus, A note on periodic points for ergodic toral automorphisms, Monatsh. Math. 89, 121129 (1980). MR 572888 (81f:28016)
 19.
 K. Oliveira, Every expanding measure has the nonuniform specification property, Proc. Amer. Mah. Soc. 140 (2012), no. 4, 13091320. MR 2869114
 20.
 K. Oliveira, M. Viana, Thermodynamical formalism for robust classes of potentials and nonuniformly hyperbolic maps, Ergodic Theory & Dynamical Systems 28, p. 501533, 2008. MR 2408389 (2009b:37056)
 21.
 V. I. Oseledec, A multiplicative ergodic theorem, Trans. Mosc. Math. Soc. 19 (1968), 197231. MR 0240280 (39:1629)
 22.
 M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, Cambridge Univ. Press, Cambridge (1993). MR 1215938 (94k:58080)
 23.
 C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynamical Systems 27 (2007), 929956. MR 2322186 (2008f:37036)
 24.
 B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents, Mosc. Math. J. 3, no. 1 (2003), 189203. MR 1996808 (2004f:37016)
 25.
 K. Sigmund, Generic properties of invariant measures for axiom A diffeomorphisms, Invention Math. 11 (1970), 99109. MR 0286135 (44:3349)
 26.
 W. Sun, X. Tian, Pesin set, closing lemma and shadowing lemma in nonuniformly hyperbolic systems with limit domination, arXiv:1004.0486.
 27.
 D. Thompson, A variational principle for topological pressure for certain noncompact sets, Journal of the London Mathematical Society 80, no. 3 (2009), 585602. MR 2559118 (2010k:37040)
 28.
 X. Tian, Hyperbolic measures with dominated splitting, arXiv:1011.6011v2.
 29.
 P. Varandas, Nonuniform specification and large deviations for weak Gibbs measures, J. Stat. Phys. 146 (2012), no. 2, 330358. MR 2873016
 30.
 Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic period orbits, Trans. Amer. Math. Soc. 362 (2010), 42674282. MR 2608406 (2011m:37030)
 31.
 K. Yamamoto, On the weaker forms of the specification property and their applications, Proceedings of the American Mathematical Society 137, no. 11 (2009), 38073814. MR 2529890 (2010f:37027)
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Additional Information
Krerley Oliveira
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072090 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:
http://dx.doi.org/10.1090/S000299472013058199
Keywords:
Pesin theory,
nonuniform 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
