Entropy formulae of conditional entropy in mean metrics

Previous Article
Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems
DCDS Home
This Issue
Next Article
Existence and non-existence results for variational higher order elliptic systems

October 2018, 38(10): 5129-5144. doi: 10.3934/dcds.2018226

Entropy formulae of conditional entropy in mean metrics

Ping Huang ^1, , Ercai Chen ^1,2,, and Chenwei Wang ^3,

1.	School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, China
2.	Center of Nonlinear Science, Nanjing University, Nanjing 210093, Jiangsu, China
3.	College of Mathematical and Physical Sciences, Taizhou University, Taizhou 225300, Jiangsu, China

* Corresponding author: Ercai Chen

Received January 2018 Revised May 2018 Published July 2018

Fund Project: The second author was supported by NNSF of China (11671208, 11431012). The third author was supported by NNSF of China (11401581)

In this paper, we construct the Brin-Katok formula of conditional entropy for invariant measures of continuous maps on a compact metric space by replacing the Bowen metrics with the corresponding mean metrics. Additionally, this paper is also devoted to establishing the Katok's entropy formula of conditional entropy for ergodic measures in the case of mean metrics.

Keywords: Conditional entropy, Brin-Katok formula, Katok' entropy formula, mean metrics, measure decomposition.

Mathematics Subject Classification: Primary: 37B05, 37B20, 37A15.

Citation: Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226

References:

[1]	M. Brin and A. Katok, On local entropy, in Lecture Notes in Mathematics, 1007, Springer, Berlin, (1983), 30–38. doi: 10.1007/BFb0061408. Google Scholar
[2]	Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic Measure, Ergod. Theory Dyn. Syst., 33 (2013), 831-850. doi: 10.1017/S0143385712000090. Google Scholar
[3]	W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function, Discrete Contin. Dyn. Syst., 32 (2012), 487-497. doi: 10.3934/dcds.2012.32.487. Google Scholar
[4]	M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Math., 259, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar
[5]	D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254. doi: 10.1016/j.jfa.2012.07.010. Google Scholar
[6]	E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. Google Scholar
[7]	M. Gröger and T. Jäger, Some remarks on modified power entropy, Dynamics and Numbers, Contem. Math., 669 (2016), 105-122. doi: 10.1090/conm/669. Google Scholar
[8]	B. Hasselblatt and A. Katok, Principal Structures, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, (2002), 1–203. doi: 10.1016/S1874-575X(02)80003-0. Google Scholar
[9]	L. He, J. Lv and L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica, Engl. Ser., 20 (2004), 709-718. doi: 10.1007/s10114-004-0368-5. Google Scholar
[10]	W. Huang, Z. Wang and X. Ye, Measure complexity and Möbius disjointness, arXiv: 1707.06345.Google Scholar
[11]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffemorphisms, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137-173. Google Scholar
[12]	C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Theory Dyn. Syst., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar
[13]	D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar
[14]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., 79, Springer, Berlin, 1982. Google Scholar
[15]	Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear Analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003. Google Scholar
[16]	D. Zheng, E. Chen E and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst. Ser. A(12), 36 (2017), 7191-7206. doi: 10.3934/dcds.2016113. Google Scholar
[17]	X. Zhou, L. Zhou and E. Chen, Brin-Katok formula for the measure theoretic $r-$entropy, C. R. Acad. Sci. Paris, Ser. I., 352 (2014), 473-477. doi: 10.1016/j.crma.2014.04.005. Google Scholar
[18]	X. Zhou, A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 4063-4075. doi: 10.3934/dcds.2016.36.4063. Google Scholar
[19]	Y. Zhu, On local entropy of random transformations, Stoch. Dyn., 8 (2008), 197-207. doi: 10.1142/S0219493708002275. Google Scholar
[20]	Y. Zhu, Two notes on measure-theoretic entropy of random dynamic systems, Acta Math. Sin., 25 (2009), 961-970. doi: 10.1007/s10114-009-7206-8. Google Scholar

show all references

References:

[1]	M. Brin and A. Katok, On local entropy, in Lecture Notes in Mathematics, 1007, Springer, Berlin, (1983), 30–38. doi: 10.1007/BFb0061408. Google Scholar
[2]	Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic Measure, Ergod. Theory Dyn. Syst., 33 (2013), 831-850. doi: 10.1017/S0143385712000090. Google Scholar
[3]	W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function, Discrete Contin. Dyn. Syst., 32 (2012), 487-497. doi: 10.3934/dcds.2012.32.487. Google Scholar
[4]	M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Math., 259, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar
[5]	D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254. doi: 10.1016/j.jfa.2012.07.010. Google Scholar
[6]	E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. Google Scholar
[7]	M. Gröger and T. Jäger, Some remarks on modified power entropy, Dynamics and Numbers, Contem. Math., 669 (2016), 105-122. doi: 10.1090/conm/669. Google Scholar
[8]	B. Hasselblatt and A. Katok, Principal Structures, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, (2002), 1–203. doi: 10.1016/S1874-575X(02)80003-0. Google Scholar
[9]	L. He, J. Lv and L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica, Engl. Ser., 20 (2004), 709-718. doi: 10.1007/s10114-004-0368-5. Google Scholar
[10]	W. Huang, Z. Wang and X. Ye, Measure complexity and Möbius disjointness, arXiv: 1707.06345.Google Scholar
[11]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffemorphisms, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137-173. Google Scholar
[12]	C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Theory Dyn. Syst., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar
[13]	D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar
[14]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., 79, Springer, Berlin, 1982. Google Scholar
[15]	Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear Analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003. Google Scholar
[16]	D. Zheng, E. Chen E and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst. Ser. A(12), 36 (2017), 7191-7206. doi: 10.3934/dcds.2016113. Google Scholar
[17]	X. Zhou, L. Zhou and E. Chen, Brin-Katok formula for the measure theoretic $r-$entropy, C. R. Acad. Sci. Paris, Ser. I., 352 (2014), 473-477. doi: 10.1016/j.crma.2014.04.005. Google Scholar
[18]	X. Zhou, A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 4063-4075. doi: 10.3934/dcds.2016.36.4063. Google Scholar
[19]	Y. Zhu, On local entropy of random transformations, Stoch. Dyn., 8 (2008), 197-207. doi: 10.1142/S0219493708002275. Google Scholar
[20]	Y. Zhu, Two notes on measure-theoretic entropy of random dynamic systems, Acta Math. Sin., 25 (2009), 961-970. doi: 10.1007/s10114-009-7206-8. Google Scholar

[1]	Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[2]	Xiaomin Zhou. A formula of conditional entropy and some applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4063-4075. doi: 10.3934/dcds.2016.36.4063
[3]	Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[4]	Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
[5]	Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[6]	Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[7]	Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[8]	Evgeniy Timofeev, Alexei Kaltchenko. Nearest-neighbor entropy estimators with weak metrics. Advances in Mathematics of Communications, 2014, 8 (2) : 119-127. doi: 10.3934/amc.2014.8.119
[9]	Yunping Jiang. On a question of Katok in one-dimensional case. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1209-1213. doi: 10.3934/dcds.2009.24.1209
[10]	Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[11]	Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1
[12]	Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[13]	L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183
[14]	Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029
[15]	Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187
[16]	Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267
[17]	Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232
[18]	Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[19]	Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016
[20]	Igor Rivin and Jean-Marc Schlenker. The Schlafli formula in Einstein manifolds with boundary. Electronic Research Announcements, 1999, 5: 18-23.

American Institute of Mathematical Sciences