Entropy dimension of topological dynamical systems

Authors:
Dou Dou, Wen Huang and Kyewon Koh Park

Journal:
Trans. Amer. Math. Soc. **363** (2011), 659-680

MSC (2000):
Primary 37B99, 54H20

Published electronically:
September 2, 2010

MathSciNet review:
2728582

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, , of a topological dynamical system to study the complexity of its factors. We construct a minimal example whose dimension set consists of one number. This implies the property that every nontrivial open cover has the same entropy dimension. This notion for zero entropy systems corresponds to the -mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated.

**1.**S. V. Avgustinovich,*The number of different subwords of given length in the Morse-Hedlund sequence*, Sibirsk. Zh. Issled. Oper.**1**(1994), no. 2, 3–7, 103 (Russian, with Russian summary). MR**1304871****2.**John Banks,*Topological mapping properties defined by digraphs*, Discrete Contin. Dynam. Systems**5**(1999), no. 1, 83–92. MR**1664461**, 10.3934/dcds.1999.5.83**3.**François Blanchard,*A disjointness theorem involving topological entropy*, Bull. Soc. Math. France**121**(1993), no. 4, 465–478 (English, with English and French summaries). MR**1254749****4.**F. Blanchard and Y. Lacroix,*Zero entropy factors of topological flows*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 985–992. MR**1155593**, 10.1090/S0002-9939-1993-1155593-2**5.**Mike Boyle and Douglas Lind,*Expansive subdynamics*, Trans. Amer. Math. Soc.**349**(1997), no. 1, 55–102. MR**1355295**, 10.1090/S0002-9947-97-01634-6**6.**Julien Cassaigne,*Constructing infinite words of intermediate complexity*, Developments in language theory, Lecture Notes in Comput. Sci., vol. 2450, Springer, Berlin, 2003, pp. 173–184. MR**2177342**, 10.1007/3-540-45005-X_15**7.**D. Dou and K. Park,*Examples of entropy generating sequence.*preprint.**8.**Harry Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**0213508****9.**H. Furstenberg,*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625****10.**Sébastien Ferenczi and Kyewon Koh Park,*Entropy dimensions and a class of constructive examples*, Discrete Contin. Dyn. Syst.**17**(2007), no. 1, 133–141. MR**2257422****11.**T. N. T. Goodman,*Topological sequence entropy*, Proc. London Math. Soc. (3)**29**(1974), 331–350. MR**0356009****12.**R. I. Grigorchuk,*On the Milnor problem of group growth*, Dokl. Akad. Nauk SSSR**271**(1983), no. 1, 30–33 (Russian). MR**712546****13.**R.I. Grigorchuk and I. Pak,*Groups of intermediate growth: An introduction for beginners.*http://math.mit.edu/ pak/grigbegin6.pdf.**14.**W. Huang, S. M. Li, S. Shao, and X. D. Ye,*Null systems and sequence entropy pairs*, Ergodic Theory Dynam. Systems**23**(2003), no. 5, 1505–1523. MR**2018610**, 10.1017/S0143385702001724**15.**Wen Huang, Kyewon Koh Park, and Xiangdong Ye,*Topological disjointness from entropy zero systems*, Bull. Soc. Math. France**135**(2007), no. 2, 259–282 (English, with English and French summaries). MR**2430193****16.**Wen Huang and Xiangdong Ye,*Dynamical systems disjoint from any minimal system*, Trans. Amer. Math. Soc.**357**(2005), no. 2, 669–694 (electronic). MR**2095626**, 10.1090/S0002-9947-04-03540-8**17.**Wen Huang and Xiangdong Ye,*Combinatorial lemmas and applications to dynamics*, Adv. Math.**220**(2009), no. 6, 1689–1716. MR**2493178**, 10.1016/j.aim.2008.11.009**18.**Teturo Kamae and Luca Zamboni,*Sequence entropy and the maximal pattern complexity of infinite words*, Ergodic Theory Dynam. Systems**22**(2002), no. 4, 1191–1199. MR**1926282**, 10.1017/S0143385702000585**19.**Anatole Katok and Jean-Paul Thouvenot,*Slow entropy type invariants and smooth realization of commuting measure-preserving transformations*, Ann. Inst. H. Poincaré Probab. Statist.**33**(1997), no. 3, 323–338 (English, with English and French summaries). MR**1457054**, 10.1016/S0246-0203(97)80094-5**20.**David Kerr and Hanfeng Li,*Independence in topological and 𝐶*-dynamics*, Math. Ann.**338**(2007), no. 4, 869–926. MR**2317754**, 10.1007/s00208-007-0097-z**21.**John Milnor,*On the entropy geometry of cellular automata*, Complex Systems**2**(1988), no. 3, 357–385. MR**955558****22.**Donald Ornstein and Benjamin Weiss,*Entropy is the only finitely observable invariant*, J. Mod. Dyn.**1**(2007), no. 1, 93–105. MR**2261073****23.**Kyewon Koh Park,*On directional entropy functions*, Israel J. Math.**113**(1999), 243–267. MR**1729449**, 10.1007/BF02780179**24.**K. E. Petersen,*Disjointness and weak mixing of minimal sets*, Proc. Amer. Math. Soc.**24**(1970), 278–280. MR**0250283**, 10.1090/S0002-9939-1970-0250283-7**25.**Yves Pomeau and Paul Manneville,*Intermittent transition to turbulence in dissipative dynamical systems*, Comm. Math. Phys.**74**(1980), no. 2, 189–197. MR**576270****26.**N. Sauer,*On the density of families of sets*, J. Combinatorial Theory Ser. A**13**(1972), 145–147. MR**0307902****27.**Saharon Shelah,*A combinatorial problem; stability and order for models and theories in infinitary languages*, Pacific J. Math.**41**(1972), 247–261. MR**0307903****28.**G. M. Zaslavsky and M. Edelman,*Weak mixing and anomalous kinetics along filamented surfaces*, Chaos**11**(2001), no. 2, 295–305. MR**1843718**, 10.1063/1.1355358

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37B99,
54H20

Retrieve articles in all journals with MSC (2000): 37B99, 54H20

Additional Information

**Dou Dou**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, People’s Republic of China – and – Department of Mathematics, Ajou University, Suwon 442-729, South Korea

Email:
doumath@163.com

**Wen Huang**

Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

Email:
wenh@mail.ustc.edu.cn

**Kyewon Koh Park**

Affiliation:
Department of Mathematics, Ajou University, Suwon 442-729, South Korea

Email:
kkpark@ajou.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-2010-04906-2

Keywords:
Entropy dimension,
dimension set,
u.d. system

Received by editor(s):
March 3, 2008

Received by editor(s) in revised form:
August 29, 2008

Published electronically:
September 2, 2010

Additional Notes:
The second author was supported by NNSF of China, 973 Project and FANEDD (Grant No 200520).

The third author was supported in part by KRF-2007-313-C00044.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.