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

Zero sequence entropy and entropy dimension

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.

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. AdlerA. 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. DouW. 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. HuangS. LiS. 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. AdlerA. 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. DouW. 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. HuangS. LiS. 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.

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