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 & Continuous Dynamical Systems - A, 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.  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. 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.  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. HuangS. LiS. 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. AdlerA. 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. 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.  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. HuangS. LiS. 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]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[2]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[3]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[4]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[5]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[6]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[8]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[9]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[10]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[11]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[12]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[13]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (88)
  • HTML views (64)
  • Cited by (2)

Other articles
by authors

[Back to Top]