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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 |
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 [
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. |
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