American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

M. De Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.   Google Scholar

[4]

D. Dou, W. Huang and K. Park, Entropy dimension of measure preserving systems, preprint, arXiv: 1312.7225. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. Glasner and B. Weiss, Quasifactors of ergodic systems with positive entropy, Israel J. Math., 134 (2003), 363-380.  doi: 10.1007/BF02787413.  Google Scholar

[9]

E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686.  doi: 10.2307/2152926.  Google Scholar

[10]

S. Glasner, Quasi-factors in ergodic theory, Israel J. Math., 45 (1983), 198-208.  doi: 10.1007/BF02774016.  Google Scholar

[11]

T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.  doi: 10.1007/s00222-005-0457-9.  Google Scholar

[16]

A. G. Kushntrenko, On metric invariants of entropy type, Russian Math. Surveys., 22 (1967), 57-65.   Google Scholar

[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.  Google Scholar

[18]

A. Saleski, Sequence entropy and mixing, J. Math. Anal. Appl., 60 (1977), 58-66.  doi: 10.1016/0022-247X(77)90047-6.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. De Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.   Google Scholar

[4]

D. Dou, W. Huang and K. Park, Entropy dimension of measure preserving systems, preprint, arXiv: 1312.7225. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. Glasner and B. Weiss, Quasifactors of ergodic systems with positive entropy, Israel J. Math., 134 (2003), 363-380.  doi: 10.1007/BF02787413.  Google Scholar

[9]

E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686.  doi: 10.2307/2152926.  Google Scholar

[10]

S. Glasner, Quasi-factors in ergodic theory, Israel J. Math., 45 (1983), 198-208.  doi: 10.1007/BF02774016.  Google Scholar

[11]

T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.  doi: 10.1007/s00222-005-0457-9.  Google Scholar

[16]

A. G. Kushntrenko, On metric invariants of entropy type, Russian Math. Surveys., 22 (1967), 57-65.   Google Scholar

[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.  Google Scholar

[18]

A. Saleski, Sequence entropy and mixing, J. Math. Anal. Appl., 60 (1977), 58-66.  doi: 10.1016/0022-247X(77)90047-6.  Google Scholar

[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.  Google Scholar

[1]

Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
[4]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[5]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[6]

Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086
[7]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[8]

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
[9]

Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[10]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[11]

L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821
[12]

Tomasz Downarowicz, Olena Karpel. Dynamics in dimension zero A survey. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1033-1062. doi: 10.3934/dcds.2018044
[13]

María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018
[14]

José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301
[15]

Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226
[16]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[17]

Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395
[18]

Nicolas Bedaride. Entropy of polyhedral billiard. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89
[19]

Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[20]

Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (31)
  • HTML views (11)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences