American Institute of Mathematical Sciences

April  2019, 39(4): 2059-2075. doi: 10.3934/dcds.2019086

Q-entropy for general topological dynamical systems

 1 School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, China 2 Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou 215006, Jiangsu, China 3 Department of Applied Mathematics, Chinese Culture University, Yangmingshan, Taipei, 11114, Taiwan

* Corresponding author: Chih-Chang Ho

Received  April 2018 Revised  September 2018 Published  January 2019

The aim of this paper is to extend the $q$-entropy from symbolic systems to a general topological dynamical system. Using a (weak) Gibbs measure as the reference measure, this paper defines $q$-topological entropy and $q$-metric entropy, then studies basic properties of these entropies. In particular, this paper describes the relations between $q$-topological entropy and topological pressure for almost additive potentials, and the relations between $q$-metric entropy and local metric entropy. Although these relations are quite similar to that described in [19], the methods used here need more techniques from the theory of thermodynamic formalism with almost additive potentials.

Citation: Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
References:
 [1] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Dynam. Sys., 16 (1996), 871-927.  doi: 10.1017/S0143385700010117. [2] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Disc. Contin. Dyn. Syst., 16 (2006), 279-305.  doi: 10.3934/dcds.2006.16.279. [3] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17.  doi: 10.1016/j.matpur.2009.04.006. [4] L. Barreira, Almost additive thermodynamic formalism: Some recent developments, Rev. Math. Phys., 22 (2010), 1147-1179.  doi: 10.1142/S0129055X10004168. [5] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X. [6] M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro) (Lecture Notes in Mathematics), Spring-Verlag, Berlin-New York, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [7] Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639. [8] Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Dynam. Sys., 33 (2013), 831-850.  doi: 10.1017/S0143385712000090. [9] V. Climenhaga, Bowen's equation in the non-uniform setting, Dynam. Sys., 31 (2011), 1163-1182.  doi: 10.1017/S0143385710000362. [10] V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141.  doi: 10.4064/fm178-2-3. [11] D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43.  doi: 10.1007/s00220-010-1031-x. [12] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454.  doi: 10.3934/dcds.2006.16.435. [13] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 307-318. [14] Y. Pesin, Dimension Theory in Dynamical Systems, in Contemporary Views and Applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [15] Y. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions, Random and Computational Dynamics, 3 (1995), 137-156. [16] P. Varandas, Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps, J. Stat. Phys., 133 (2008), 813-839.  doi: 10.1007/s10955-008-9639-3. [17] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. [18] M. Yuri, Weak Gibbs measures for certain non-hyperbolic systems, Dynam. Syst., 20 (2000), 1495-1518.  doi: 10.1017/S014338570000081X. [19] Y. Zhao and Y. Pesin, Q-entropy for symbolic dynamical systems, J. Phys. A: Math. Theor., 48 (2015), 494002, 17 pp. doi: 10.1088/1751-8113/48/49/494002. [20] Y. Zhao, L. Zhang and Y. Cao, The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials, Nonlinear Analysis, 74 (2011), 5015-5022.  doi: 10.1016/j.na.2011.04.065.

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References:
 [1] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Dynam. Sys., 16 (1996), 871-927.  doi: 10.1017/S0143385700010117. [2] L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Disc. Contin. Dyn. Syst., 16 (2006), 279-305.  doi: 10.3934/dcds.2006.16.279. [3] L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17.  doi: 10.1016/j.matpur.2009.04.006. [4] L. Barreira, Almost additive thermodynamic formalism: Some recent developments, Rev. Math. Phys., 22 (2010), 1147-1179.  doi: 10.1142/S0129055X10004168. [5] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X. [6] M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro) (Lecture Notes in Mathematics), Spring-Verlag, Berlin-New York, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [7] Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639. [8] Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Dynam. Sys., 33 (2013), 831-850.  doi: 10.1017/S0143385712000090. [9] V. Climenhaga, Bowen's equation in the non-uniform setting, Dynam. Sys., 31 (2011), 1163-1182.  doi: 10.1017/S0143385710000362. [10] V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141.  doi: 10.4064/fm178-2-3. [11] D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43.  doi: 10.1007/s00220-010-1031-x. [12] A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454.  doi: 10.3934/dcds.2006.16.435. [13] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 307-318. [14] Y. Pesin, Dimension Theory in Dynamical Systems, in Contemporary Views and Applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [15] Y. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions, Random and Computational Dynamics, 3 (1995), 137-156. [16] P. Varandas, Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps, J. Stat. Phys., 133 (2008), 813-839.  doi: 10.1007/s10955-008-9639-3. [17] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. [18] M. Yuri, Weak Gibbs measures for certain non-hyperbolic systems, Dynam. Syst., 20 (2000), 1495-1518.  doi: 10.1017/S014338570000081X. [19] Y. Zhao and Y. Pesin, Q-entropy for symbolic dynamical systems, J. Phys. A: Math. Theor., 48 (2015), 494002, 17 pp. doi: 10.1088/1751-8113/48/49/494002. [20] Y. Zhao, L. Zhang and Y. Cao, The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials, Nonlinear Analysis, 74 (2011), 5015-5022.  doi: 10.1016/j.na.2011.04.065.
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