Entropy formulae of conditional entropy in mean metrics

Ping Huang; Ercai Chen; Chenwei Wang

doi:10.3934/dcds.2018226

Article Contents

2018, Volume 38, Issue 10: 5129-5144. Doi: 10.3934/dcds.2018226

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

Entropy formulae of conditional entropy in mean metrics

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
* Corresponding author: Ercai Chen

Received: December 31, 2017

Revised: April 30, 2018

Published: September 2018

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[6]	E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.
[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.
[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.
[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.
[10]	W. Huang, Z. Wang and X. Ye, Measure complexity and Möbius disjointness, arXiv: 1707.06345.
[11]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffemorphisms, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137-173.
[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.
[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.
[14]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., 79, Springer, Berlin, 1982.
[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.
[16]	D. Zheng, E. Chen E and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst. Ser. A, 36 (2017), 7191-7206. doi: 10.3934/dcds.2016113.
[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.
[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.
[19]	Y. Zhu, On local entropy of random transformations, Stoch. Dyn., 8 (2008), 197-207. doi: 10.1142/S0219493708002275.
[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.