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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, 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

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