# American Institute of Mathematical Sciences

July  2016, 36(7): 4063-4075. doi: 10.3934/dcds.2016.36.4063

## A formula of conditional entropy and some applications

 1 Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, School of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received  June 2015 Revised  November 2015 Published  March 2016

In this paper we establish a formula of conditional entropy and give two examples of applications of the formula.
Citation: Xiaomin Zhou. A formula of conditional entropy and some applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4063-4075. doi: 10.3934/dcds.2016.36.4063
##### References:
 [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9. [2] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99. [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [4] M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro,,(1981),in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [5] E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22. [6] T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), 217-247. doi: 10.4064/fm172-3-2. [7] M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, 2011. doi: 10.1007/978-0-85729-021-2. [8] C. Fang, W. Huang, Y. Yi and P. Zhang, Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628. doi: 10.1017/S0143385710000982. [9] 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. [10] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [11] T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180. doi: 10.1112/blms/3.2.176. [12] L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688. doi: 10.1090/S0002-9939-1969-0247030-3. [13] W. Huang, Stable sets and $\epsilon$-stable sets in positive-entropy systems, Comm. Math. Phys., 279 (2008), 535-557. doi: 10.1007/s00220-008-0430-8. [14] W. Huang, J. Li and X.D. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394. doi: 10.1016/j.jfa.2014.01.005. [15] F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576. [16] P. D. Liu, A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 593-603. doi: 10.1017/S0143385704000586. [17] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [18] Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3. [19] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, (Russian) Dokl. Akad. Nauk. SSSR (N.S.), 119 (1958), 861-864.

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##### References:
 [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9. [2] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99. [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [4] M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro,,(1981),in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [5] E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22. [6] T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), 217-247. doi: 10.4064/fm172-3-2. [7] M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, 2011. doi: 10.1007/978-0-85729-021-2. [8] C. Fang, W. Huang, Y. Yi and P. Zhang, Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628. doi: 10.1017/S0143385710000982. [9] 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. [10] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [11] T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180. doi: 10.1112/blms/3.2.176. [12] L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688. doi: 10.1090/S0002-9939-1969-0247030-3. [13] W. Huang, Stable sets and $\epsilon$-stable sets in positive-entropy systems, Comm. Math. Phys., 279 (2008), 535-557. doi: 10.1007/s00220-008-0430-8. [14] W. Huang, J. Li and X.D. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394. doi: 10.1016/j.jfa.2014.01.005. [15] F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576. [16] P. D. Liu, A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 593-603. doi: 10.1017/S0143385704000586. [17] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [18] Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3. [19] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, (Russian) Dokl. Akad. Nauk. SSSR (N.S.), 119 (1958), 861-864.
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