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

A formula of conditional entropy and some applications

Xiaomin Zhou 1,

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.
Keywords: Measure-theoretical entropies, conditional entropies., measure decomposition.
Mathematics Subject Classification: Primary: 37B05, 37B20, 37A1.
Citation: Xiaomin Zhou. A formula of conditional entropy and some applications. Discrete & 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.  Google Scholar

[2]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99.   Google Scholar

[3]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[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.  Google Scholar

[5]

E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576.  Google Scholar

[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.  Google Scholar

[17]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99.   Google Scholar

[3]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[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.  Google Scholar

[5]

E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576.  Google Scholar

[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.  Google Scholar

[17]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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