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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 |
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 [
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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