October  2017, 37(10): 5407-5431. doi: 10.3934/dcds.2017235

Topological entropy of level sets of empirical measures for non-uniformly expanding maps

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil

Received  November 2016 Revised  May 2017 Published  June 2017

In this article we obtain a variational principle for saturated sets for maps with some non-uniform specification properties. More precisely, we prove that the topological entropy of saturated sets coincides with the smallest measure theoretical entropy among the invariant measures in the accumulation set. Using this fact we provide lower bounds for the topological entropy of the irregular set and the level sets in the multifractal analysis of Birkhoff averages for continuous observables. The topological entropy estimates use as tool a non-uniform specification property on topologically large sets, which we prove to hold for open classes of non-uniformly expanding maps. In particular we prove some multifractal analysis results for C1-open classes of non-uniformly expanding local diffeomorphisms and Viana maps [1,33].

Citation: Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.

[2]

J. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32. doi: 10.1017/S0143385702000019.

[3]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[4]

L. BarreiraY. Pesin and J. Schmeling, Multifractal spectra and multifractal rigidity for horseshoes, J. Dynam. Control Systems, 3 (1997), 33-49. doi: 10.1007/BF02471761.

[5]

T. Bomfim and P. Varandas, Multifractal analysis for weak Gibbs measures: From large deviations to irregular sets, Ergodic Theory Dynam. Systems, 37 (2017), 79-102. doi: 10.1017/etds.2015.46.

[6]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[7]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.2307/1995452.

[8]

M. Brin and A. Katok, On local entropy. Geometric dynamics, (Rio de Janeiro, 1981), Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408.

[9]

V. Climenhaga, Multifractal formalism derived from thermodynamics, Electron. Res. Announc. Math. Sci., 17 (2010), 1-11. doi: 10.3934/era.2010.17.1.

[10]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976.

[11]

Y. Dong, P. Oprocha and X. Tian, On the irregular points for systems with the shadowing property, Ergodic Theory and Dynamical Systems to appear. doi: 10.1017/etds.2016.126.

[12]

Y. Dong and X. Tian, Multifractal Analysis of The New Level Sets, arXiv: 1510.06514.

[13]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, Journal of Functional Analysis, 263 (2012), 2228-2254. doi: 10.1016/j.jfa.2012.07.010.

[14]

K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergod. Th. Dynam. Sys., 29 (2009), 919-940. doi: 10.1017/S0143385708080462.

[15]

T. Jordan and M. Rams, Multifractal analysis of weak Gibbs measures for non-uniformly expanding $C^1$ maps, Ergod. Th. Dynam. Sys., 31 (2011), 143-164. doi: 10.1017/S0143385709000960.

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137-173.

[17]

C. LiangG. LiaoW. Sun and X. Tian, Variational equalities of Entropy in Nonuniformly Hyperbolic Systems, Trans. Amer. Math. Society, 369 (2017), 3127-3156. doi: 10.1090/tran/6780.

[18]

K. MoriyasuK. Sakai and K. Yamamoto, Regular maps with the specification property, Discrete and Cointinuous Dynam. Sys., 33 (2013), 2991-3009. doi: 10.3934/dcds.2013.33.2991.

[19]

K. Oliveira and X. Tian, Non-uniform hyperbolicity and non-uniform specification, Transactions of the American Mathematical Society, 365 (2013), 4371-4392. doi: 10.1090/S0002-9947-2013-05819-9.

[20]

L. Olsen, A multifractal formalism, Adv. in Math., 116 (1995), 82-196. doi: 10.1006/aima.1995.1066.

[21]

L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. Lond. Math. Soc., 67 (2003), 103-122. doi: 10.1112/S0024610702003630.

[22]

Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242.

[23]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.

[24]

V. Pinheiro and P. Varandas, Thermodynamic formalism of expanding measures: Intrinsic ergodicity, Preprint, 2016.

[25]

N. SumiP. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170. doi: 10.1090/proc/12830.

[26]

F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys., 203 (1999), 593-612. doi: 10.1007/s002200050627.

[27]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst., 25 (2010), 25-51. doi: 10.1080/14689360903156237.

[28]

D. Thompson, Irregular sets, the β-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1.

[29]

X. Tian, Nonexistence of Lyapunov exponents for matrix cocycles, Annales de I'Institut Henri Poincaré (B) Probabilités et Statistiques, 53 (2017), 493-502. doi: 10.1214/15-AIHP733.

[30]

M. Todd, Multifractal analysis for multimodal maps, Preprint Arxiv, 2008.

[31]

P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, Journal of Statistical Physics, 146 (2012), 330-358. doi: 10.1007/s10955-011-0392-7.

[32]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Ann. I. H. Poincaré -Analyse Non-Lineaire, 27 (2010), 555-593. doi: 10.1016/j.anihpc.2009.10.002.

[33]

M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96.

[34]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.

[35]

X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems, Nonlinearity, 26 (2013), 1975-1997. doi: 10.1088/0951-7715/26/7/1975.

show all references

References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.

[2]

J. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32. doi: 10.1017/S0143385702000019.

[3]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[4]

L. BarreiraY. Pesin and J. Schmeling, Multifractal spectra and multifractal rigidity for horseshoes, J. Dynam. Control Systems, 3 (1997), 33-49. doi: 10.1007/BF02471761.

[5]

T. Bomfim and P. Varandas, Multifractal analysis for weak Gibbs measures: From large deviations to irregular sets, Ergodic Theory Dynam. Systems, 37 (2017), 79-102. doi: 10.1017/etds.2015.46.

[6]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[7]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.2307/1995452.

[8]

M. Brin and A. Katok, On local entropy. Geometric dynamics, (Rio de Janeiro, 1981), Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408.

[9]

V. Climenhaga, Multifractal formalism derived from thermodynamics, Electron. Res. Announc. Math. Sci., 17 (2010), 1-11. doi: 10.3934/era.2010.17.1.

[10]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976.

[11]

Y. Dong, P. Oprocha and X. Tian, On the irregular points for systems with the shadowing property, Ergodic Theory and Dynamical Systems to appear. doi: 10.1017/etds.2016.126.

[12]

Y. Dong and X. Tian, Multifractal Analysis of The New Level Sets, arXiv: 1510.06514.

[13]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, Journal of Functional Analysis, 263 (2012), 2228-2254. doi: 10.1016/j.jfa.2012.07.010.

[14]

K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergod. Th. Dynam. Sys., 29 (2009), 919-940. doi: 10.1017/S0143385708080462.

[15]

T. Jordan and M. Rams, Multifractal analysis of weak Gibbs measures for non-uniformly expanding $C^1$ maps, Ergod. Th. Dynam. Sys., 31 (2011), 143-164. doi: 10.1017/S0143385709000960.

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137-173.

[17]

C. LiangG. LiaoW. Sun and X. Tian, Variational equalities of Entropy in Nonuniformly Hyperbolic Systems, Trans. Amer. Math. Society, 369 (2017), 3127-3156. doi: 10.1090/tran/6780.

[18]

K. MoriyasuK. Sakai and K. Yamamoto, Regular maps with the specification property, Discrete and Cointinuous Dynam. Sys., 33 (2013), 2991-3009. doi: 10.3934/dcds.2013.33.2991.

[19]

K. Oliveira and X. Tian, Non-uniform hyperbolicity and non-uniform specification, Transactions of the American Mathematical Society, 365 (2013), 4371-4392. doi: 10.1090/S0002-9947-2013-05819-9.

[20]

L. Olsen, A multifractal formalism, Adv. in Math., 116 (1995), 82-196. doi: 10.1006/aima.1995.1066.

[21]

L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. Lond. Math. Soc., 67 (2003), 103-122. doi: 10.1112/S0024610702003630.

[22]

Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242.

[23]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.

[24]

V. Pinheiro and P. Varandas, Thermodynamic formalism of expanding measures: Intrinsic ergodicity, Preprint, 2016.

[25]

N. SumiP. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170. doi: 10.1090/proc/12830.

[26]

F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys., 203 (1999), 593-612. doi: 10.1007/s002200050627.

[27]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst., 25 (2010), 25-51. doi: 10.1080/14689360903156237.

[28]

D. Thompson, Irregular sets, the β-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1.

[29]

X. Tian, Nonexistence of Lyapunov exponents for matrix cocycles, Annales de I'Institut Henri Poincaré (B) Probabilités et Statistiques, 53 (2017), 493-502. doi: 10.1214/15-AIHP733.

[30]

M. Todd, Multifractal analysis for multimodal maps, Preprint Arxiv, 2008.

[31]

P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, Journal of Statistical Physics, 146 (2012), 330-358. doi: 10.1007/s10955-011-0392-7.

[32]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Ann. I. H. Poincaré -Analyse Non-Lineaire, 27 (2010), 555-593. doi: 10.1016/j.anihpc.2009.10.002.

[33]

M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96.

[34]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.

[35]

X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems, Nonlinearity, 26 (2013), 1975-1997. doi: 10.1088/0951-7715/26/7/1975.

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