Weighted topological and measure-theoretic entropy

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July 2019, 39(7): 3941-3967. doi: 10.3934/dcds.2019159

Weighted topological and measure-theoretic entropy

Tao Wang and Yu Huang ^,

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

^* Corresponding author: Yu Huang

Received May 2018 Revised October 2018 Published April 2019

Fund Project: The second author is supported by National Nature Science Funds of China (11771459)

Feng and Huang in 2016 defined a new notion called weighted topological entropy (pressure) and obtained the corresponding variational principle for compact dynamical systems. In this paper, it was our hope to carry out a further study from the following three aspects:

(1) Inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships.

(2) The classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version.

(3) A formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.

Keywords: Weighted topological entropies, weighted measure-theoretic entropies, weighted conditional entropies, factor maps, entropy formulas.

Mathematics Subject Classification: Primary: 37C45, 37B40, 37A35.

Citation: Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

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]	R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X. 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, The relation between topological entropy and metric entropy, Sov. Math., 11 (1970), 13-16. Google Scholar
[6]	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
[7]	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
[8]	D. J. 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
[9]	D. J. Feng and W. Huang, Variational principle for the weighted topological pressure, J. Math. Pures Appl., 106 (2016), 411-452. doi: 10.1016/j.matpur.2016.02.016. 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. Lond. 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: 10.2307/2036610. Google Scholar
[13]	W. Huang, X. D. Ye and G. H. Zhang, Lowering topological entropy over subsets, Ergodic Theory Dynam. Systems, 30 (2010), 181-209. doi: 10.1017/S0143385709000066. Google Scholar
[14]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173. Google Scholar
[15]	A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR, 119 (1958), 861-864. Google Scholar
[16]	P. Oprocha and G. H. Zhang, Dimensional entropy over sets and fibres, Nonlinearity, 24 (2011), 2325-2346. doi: 10.1088/0951-7715/24/8/009. Google Scholar
[17]	Y. B. Pesin, Dimension Theory in Tynamical Systems: Contemporary views and applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. Google Scholar
[18]	Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63. Google Scholar
[19]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982. Google Scholar
[20]	C. Zhao, E. C. Chen, X. Y. Zhou and Z. Yin, Weighted topological entropy of the set of generic points in topological dynamical systems, J. Dynam. Differential Equations, 30 (2018), 937-955. doi: 10.1007/s10884-017-9575-5. Google Scholar
[21]	X. M. Zhou, A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst., 36 (2016), 4063-4075. doi: 10.3934/dcds.2016.36.4063. 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]	R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X. 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, The relation between topological entropy and metric entropy, Sov. Math., 11 (1970), 13-16. Google Scholar
[6]	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
[7]	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
[8]	D. J. 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
[9]	D. J. Feng and W. Huang, Variational principle for the weighted topological pressure, J. Math. Pures Appl., 106 (2016), 411-452. doi: 10.1016/j.matpur.2016.02.016. 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. Lond. 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: 10.2307/2036610. Google Scholar
[13]	W. Huang, X. D. Ye and G. H. Zhang, Lowering topological entropy over subsets, Ergodic Theory Dynam. Systems, 30 (2010), 181-209. doi: 10.1017/S0143385709000066. Google Scholar
[14]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173. Google Scholar
[15]	A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR, 119 (1958), 861-864. Google Scholar
[16]	P. Oprocha and G. H. Zhang, Dimensional entropy over sets and fibres, Nonlinearity, 24 (2011), 2325-2346. doi: 10.1088/0951-7715/24/8/009. Google Scholar
[17]	Y. B. Pesin, Dimension Theory in Tynamical Systems: Contemporary views and applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. Google Scholar
[18]	Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63. Google Scholar
[19]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982. Google Scholar
[20]	C. Zhao, E. C. Chen, X. Y. Zhou and Z. Yin, Weighted topological entropy of the set of generic points in topological dynamical systems, J. Dynam. Differential Equations, 30 (2018), 937-955. doi: 10.1007/s10884-017-9575-5. Google Scholar
[21]	X. M. Zhou, A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst., 36 (2016), 4063-4075. doi: 10.3934/dcds.2016.36.4063. Google Scholar

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