Zero sequence entropy and entropy dimension

Previous Article
Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation
DCDS Home
This Issue
Next Article
Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations

January 2017, 37(1): 435-448. doi: 10.3934/dcds.2017018

Zero sequence entropy and entropy dimension

Yixiao Qiao ^1, and Xiaoyao Zhou ^2,,

1.	School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
2.	School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210023, China

* Corresponding author

Received January 2016 Revised August 2016 Published November 2016

Let $(X, T)$ be a topological dynamical system and $M(X)$ the set of all Borel probability measures on $X$ endowed with the weak$^*$ -topology. In this paper, it is shown that for a given sequence $S$ , a homeomorphism $T$ of $X$ has zero topological sequence entropy if and only if so does the induced homeomorphism $T$ of $M(X)$ . This extends the result of Glasner and Weiss [9,Theorem A] for topological entropy and also the result of Kerr and Li [15,Theorem 5.10]for null systems. Moreover, it turns out that the upper entropy dimension of $(X, T)$ is equal to that of $(M(X), T)$ . We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.

Keywords: Zero sequence entropy, entropy dimension, quasi-factor.

Mathematics Subject Classification: Primary:54H2.

Citation: Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

References:

[1]	L. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.
[2]	W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92. doi: 10.1007/BF01585664.
[3]	M. De Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.
[4]	D. Dou, W. Huang and K. Park, Entropy dimension of measure preserving systems, preprint, arXiv: 1312.7225.
[5]	D. Dou, W. Huang and K. Park, Entropy dimension of topological dynamical systems, Trans. Amer. Math. Soc., 363 (2011), 659-680. doi: 10.1090/S0002-9947-2010-04906-2.
[6]	S. Ferenczi and K. Park, Entropy dimensions and a class of constructive examples, Discrete Contin. Dyn. Sys., 17 (2007), 133-141. doi: 10.3934/dcds.2007.17.133.
[7]	F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy Ergodic Theory Dynam. Systems. doi: 10.1017/etds. 2015.83.
[8]	E. Glasner and B. Weiss, Quasifactors of ergodic systems with positive entropy, Israel J. Math., 134 (2003), 363-380. doi: 10.1007/BF02787413.
[9]	E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686. doi: 10.2307/2152926.
[10]	S. Glasner, Quasi-factors in ergodic theory, Israel J. Math., 45 (1983), 198-208. doi: 10.1007/BF02774016.
[11]	T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350. doi: 10.1112/plms/s3-29.2.331.
[12]	W. Huang, S. Li, S. Shao and X. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523. doi: 10.1017/S0143385702001724.
[13]	W. Huang and X. Ye, Combinatorial lemmas and applications to dynamics, Adv. Math., 220 (2009), 1689-1716. doi: 10.1016/j.aim.2008.11.009.
[14]	P. Hulse, On the sequence entropy of transformations with quasi-discrete spectrum, J. London Math. Soc., 20 (1979), 128-136. doi: 10.1112/jlms/s2-20.1.128.
[15]	D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686. doi: 10.1007/s00222-005-0457-9.
[16]	A. G. Kushntrenko, On metric invariants of entropy type, Russian Math. Surveys., 22 (1967), 57-65.
[17]	K. R. Parthasarathy, Probability Measures on Metric Spaces Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967. doi: 10.1016/B978-1-4832-0022-4.50006-5.
[18]	A. Saleski, Sequence entropy and mixing, J. Math. Anal. Appl., 60 (1977), 58-66. doi: 10.1016/0022-247X(77)90047-6.
[19]	P. Walters, An Introduction to Ergodic Theory Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4612-5775-2.

show all references

References:

[1]	L. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.
[2]	W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92. doi: 10.1007/BF01585664.
[3]	M. De Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.
[4]	D. Dou, W. Huang and K. Park, Entropy dimension of measure preserving systems, preprint, arXiv: 1312.7225.
[5]	D. Dou, W. Huang and K. Park, Entropy dimension of topological dynamical systems, Trans. Amer. Math. Soc., 363 (2011), 659-680. doi: 10.1090/S0002-9947-2010-04906-2.
[6]	S. Ferenczi and K. Park, Entropy dimensions and a class of constructive examples, Discrete Contin. Dyn. Sys., 17 (2007), 133-141. doi: 10.3934/dcds.2007.17.133.
[7]	F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy Ergodic Theory Dynam. Systems. doi: 10.1017/etds. 2015.83.
[8]	E. Glasner and B. Weiss, Quasifactors of ergodic systems with positive entropy, Israel J. Math., 134 (2003), 363-380. doi: 10.1007/BF02787413.
[9]	E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686. doi: 10.2307/2152926.
[10]	S. Glasner, Quasi-factors in ergodic theory, Israel J. Math., 45 (1983), 198-208. doi: 10.1007/BF02774016.
[11]	T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350. doi: 10.1112/plms/s3-29.2.331.
[12]	W. Huang, S. Li, S. Shao and X. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523. doi: 10.1017/S0143385702001724.
[13]	W. Huang and X. Ye, Combinatorial lemmas and applications to dynamics, Adv. Math., 220 (2009), 1689-1716. doi: 10.1016/j.aim.2008.11.009.
[14]	P. Hulse, On the sequence entropy of transformations with quasi-discrete spectrum, J. London Math. Soc., 20 (1979), 128-136. doi: 10.1112/jlms/s2-20.1.128.
[15]	D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686. doi: 10.1007/s00222-005-0457-9.
[16]	A. G. Kushntrenko, On metric invariants of entropy type, Russian Math. Surveys., 22 (1967), 57-65.
[17]	K. R. Parthasarathy, Probability Measures on Metric Spaces Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967. doi: 10.1016/B978-1-4832-0022-4.50006-5.
[18]	A. Saleski, Sequence entropy and mixing, J. Math. Anal. Appl., 60 (1977), 58-66. doi: 10.1016/0022-247X(77)90047-6.
[19]	P. Walters, An Introduction to Ergodic Theory Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4612-5775-2.

[1]	Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[2]	Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339
[3]	Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
[4]	Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[5]	José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[6]	Iztok Banič, Goran Erceg, Judy Kennedy. Closed relations with non-zero entropy that generate no periodic points. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 5137-5166. doi: 10.3934/dcds.2022089
[7]	Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086
[8]	Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[9]	Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[10]	Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135
[11]	Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[12]	L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821
[13]	Tomasz Downarowicz, Olena Karpel. Dynamics in dimension zero A survey. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1033-1062. doi: 10.3934/dcds.2018044
[14]	María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations and Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018
[15]	José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301
[16]	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
[17]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[18]	Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395
[19]	Nicolas Bedaride. Entropy of polyhedral billiard. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89
[20]	Karl Petersen, Ibrahim Salama. Entropy on regular trees. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4453-4477. doi: 10.3934/dcds.2020186

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Yixiao Qiao Xiaoyao Zhou
on Google Scholar
Yixiao Qiao Xiaoyao Zhou

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Yixiao Qiao Xiaoyao Zhou
on Google Scholar
Yixiao Qiao Xiaoyao Zhou

[Back to Top]