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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 and Continuous Dynamical Systems, 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.

[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(12), 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.

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.

[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(12), 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.

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