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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

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