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

Q-entropy for general topological dynamical systems

Yun Zhao 1,2, , Wen-Chiao Cheng 3, and  Chih-Chang Ho 3,,

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.

Keywords: q-entropy, Gibbs measure, Hentschel-Procaccia spectrum, topological pressure, bounded distortion.
Mathematics Subject Classification: Primary: 37D35, 37C45; Secondary: 28D20.
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.

show all references

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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