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

## 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

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.

Citation: Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & 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.  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
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