Entropy dimension of topological dynamical systems
Authors:
Dou Dou, Wen Huang and Kyewon Koh Park
Journal:
Trans. Amer. Math. Soc. 363 (2011), 659680
MSC (2000):
Primary 37B99, 54H20
Published electronically:
September 2, 2010
MathSciNet review:
2728582
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References 
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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 MorseHedlund sequence, Sibirsk. Zh. Issled. Oper.
1 (1994), no. 2, 3–7, 103 (Russian, with
Russian summary). MR 1304871
(95h:58075)
 2.
John
Banks, Topological mapping properties defined by digraphs,
Discrete Contin. Dynam. Systems 5 (1999), no. 1,
83–92. MR
1664461 (99j:54038), http://dx.doi.org/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
(95e:54050)
 4.
F.
Blanchard and Y.
Lacroix, Zero entropy factors of topological
flows, Proc. Amer. Math. Soc.
119 (1993), no. 3,
985–992. MR 1155593
(93m:54066), http://dx.doi.org/10.1090/S00029939199311555932
 5.
Mike
Boyle and Douglas
Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102. MR 1355295
(97d:58115), http://dx.doi.org/10.1090/S0002994797016346
 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, http://dx.doi.org/10.1007/354045005X_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
(35 #4369)
 9.
H.
Furstenberg, Recurrence in ergodic theory and combinatorial number
theory, Princeton University Press, Princeton, N.J., 1981. M. B.
Porter Lectures. MR 603625
(82j:28010)
 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
(2007i:37014)
 11.
T.
N. T. Goodman, Topological sequence entropy, Proc. London
Math. Soc. (3) 29 (1974), 331–350. MR 0356009
(50 #8482)
 12.
R.
I. Grigorchuk, On the Milnor problem of group growth, Dokl.
Akad. Nauk SSSR 271 (1983), no. 1, 30–33
(Russian). MR
712546 (85g:20042)
 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
(2004i:37012), http://dx.doi.org/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
(2009g:54084)
 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
(2005g:37012), http://dx.doi.org/10.1090/S0002994704035408
 17.
Wen
Huang and Xiangdong
Ye, Combinatorial lemmas and applications to dynamics, Adv.
Math. 220 (2009), no. 6, 1689–1716. MR 2493178
(2010c:37033), http://dx.doi.org/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
(2003f:37020), http://dx.doi.org/10.1017/S0143385702000585
 19.
Anatole
Katok and JeanPaul
Thouvenot, Slow entropy type invariants and smooth realization of
commuting measurepreserving transformations, Ann. Inst. H.
Poincaré Probab. Statist. 33 (1997), no. 3,
323–338 (English, with English and French summaries). MR 1457054
(98h:28012), http://dx.doi.org/10.1016/S02460203(97)800945
 20.
David
Kerr and Hanfeng
Li, Independence in topological and 𝐶*dynamics, Math.
Ann. 338 (2007), no. 4, 869–926. MR 2317754
(2009a:46126), http://dx.doi.org/10.1007/s002080070097z
 21.
John
Milnor, On the entropy geometry of cellular automata, Complex
Systems 2 (1988), no. 3, 357–385. MR 955558
(90c:54026)
 22.
Donald
Ornstein and Benjamin
Weiss, Entropy is the only finitely observable invariant, J.
Mod. Dyn. 1 (2007), no. 1, 93–105. MR 2261073
(2007j:37006)
 23.
Kyewon
Koh Park, On directional entropy functions, Israel J. Math.
113 (1999), 243–267. MR 1729449
(2000m:37007), http://dx.doi.org/10.1007/BF02780179
 24.
K.
E. Petersen, Disjointness and weak mixing of
minimal sets, Proc. Amer. Math. Soc. 24 (1970), 278–280.
MR
0250283 (40 #3522), http://dx.doi.org/10.1090/S00029939197002502837
 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
(81g:58024)
 26.
N.
Sauer, On the density of families of sets, J. Combinatorial
Theory Ser. A 13 (1972), 145–147. MR 0307902
(46 #7017)
 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
(46 #7018)
 28.
G.
M. Zaslavsky and M.
Edelman, Weak mixing and anomalous kinetics along filamented
surfaces, Chaos 11 (2001), no. 2, 295–305.
MR
1843718 (2002e:37107), http://dx.doi.org/10.1063/1.1355358
 1.
 S. V. Avgustinovich, The number of different subwords of given length in the MorseHedlund sequence. Sibirsk. Zh. Issled. Oper. 1 (1994), no. 2, 37, 103. MR 1304871 (95h:58075)
 2.
 J. Banks, Topological mapping properties defined by digraphs. Discrete Contin. Dyn. Sys. 5 (1999), 8392. MR 1664461 (99j:54038)
 3.
 F. Blanchard, A disjointness theorem involving topological entropy. Bull. Soc. Math. France 121 (1993), 465478. MR 1254749 (95e:54050)
 4.
 F. Blanchard and Y. Lacroix, Zeroentropy factors of topological flows. Proc. Amer. Math. Soc. 119 (1993), 985992. MR 1155593 (93m:54066)
 5.
 M. Boyle and D. Lind, Expansive subdynamics. Trans. Amer. Math. Soc. 349 (1997), 55102. MR 1355295 (97d:58115)
 6.
 Julien Cassaigne, Constructing infinite words of intermediate complexity, Developments in language theory, Lecture Notes in Comput. Sci., vol. 2450, Springer, Berlin, 2003, 173184. MR 2177342
 7.
 D. Dou and K. Park, Examples of entropy generating sequence. preprint.
 8.
 H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 149. MR 0213508 (35:4369)
 9.
 H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, 1981. MR 603625 (82j:28010)
 10.
 Sébastien Ferenczi and K. K. Park, Entropy dimensions and a class of constructive examples. Discrete Cont. Dyn. Syst. 17 (2007), no. 1, 133141. MR 2257422 (2007i:37014)
 11.
 T. N. T. Goodman, Topological sequence entropy. Proc. London Math. Soc. 29 (1974), no. 3, 331350. MR 0356009 (50:8482)
 12.
 R.I. Grigorchuk, On Milnor's problem of group growth. Soviet Math. Dokl. 28 (1983), 2326. MR 712546 (85g:20042)
 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. Li, S. Shao and X. Ye, Null systems and sequence entropy pairs. Ergod. Th. & Dynam. Sys. 23 (2003), 15051523. MR 2018610 (2004i:37012)
 15.
 W. Huang, K. K. Park and X. Ye, Topological disjointness for entropy zero systems. Bull. Soc. Math. France 135 (2007), no. 2, 259282. MR 2430193 (2009g:54084)
 16.
 W. Huang and X. Ye, Dynamical systems disjoint from any minimal system. Trans. Amer. Math. Soc. 357 (2005), 669694. MR 2095626 (2005g:37012)
 17.
 W. Huang and X. Ye, Combinatorial lemmas and applications to dynamics. Adv. Math. 220 (2009), no. 6, 16891716. MR 2493178 (2010c:37033)
 18.
 T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22 (2002), no. 4, 11911199. MR 1926282 (2003f:37020)
 19.
 A. Katok and J.P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measurepreserving transformations. Ann. Inst. H. Poincare Probab. Statist. 33 (1997), no. 3, 323338. MR 1457054 (98h:28012)
 20.
 D. Kerr and H. Li, Independence in topological dynamics. Math. Ann. 338 (2007), no. 4, 869926. MR 2317754
 21.
 J. Milnor, On the entropy geometry of cellular automata. Complex Systems 2 (1988), 357386. MR 955558 (90c:54026)
 22.
 D. Ornstein and B. Weiss, Entropy is the only finitely observable invariant. J. Mod. Dyn. 1 (2007), no. 1, 93105. MR 2261073 (2007j:37006)
 23.
 K. K. Park, On directional entropy functions. Israel J. Math. 113 (1999), 243267. MR 1729449 (2000m:37007)
 24.
 K. Petersen, Disjointness and weak mixing of minimal sets. Proc. Amer. Math. Soc. 24 (1970), 278280. MR 0250283 (40:3522)
 25.
 Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), no. 2, 189197. MR 576270 (81g:58024)
 26.
 N. Sauer, On the density of families of sets. J. Combin. Theory Ser. A 13 (1972), 145147. MR 0307902 (46:7017)
 27.
 S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math. 41 (1972), 247261. MR 0307903 (46:7018)
 28.
 G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces. Chaos 11 (2001), 295305. MR 1843718 (2002e:37107)
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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 442729, 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 442729, South Korea
Email:
kkpark@ajou.ac.kr
DOI:
http://dx.doi.org/10.1090/S000299472010049062
PII:
S 00029947(2010)049062
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 KRF2007313C00044.
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
