Entropy dimension of topological dynamical systems

Dou, Dou; Huang, Wen; Park, Kyewon

doi:10.1090/S0002-9947-2010-04906-2

Entropy dimension of topological dynamical systems
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by Dou Dou, Wen Huang and Kyewon Koh Park PDF

Trans. Amer. Math. Soc. 363 (2011), 659-680 Request permission

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, $\mathcal {D}(X,T)\subset [0,1]$, 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 $K$-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.

References

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
John Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 83–92. MR 1664461, DOI 10.3934/dcds.1999.5.83
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
F. Blanchard and Y. Lacroix, Zero entropy factors of topological flows, Proc. Amer. Math. Soc. 119 (1993), no. 3, 985–992. MR 1155593, DOI 10.1090/S0002-9939-1993-1155593-2
Mike Boyle and Douglas Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102. MR 1355295, DOI 10.1090/S0002-9947-97-01634-6
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, DOI 10.1007/3-540-45005-X_{1}5
D. Dou and K. Park, Examples of entropy generating sequence. preprint.
Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
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, DOI 10.3934/dcds.2007.17.133
T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331–350. MR 356009, DOI 10.1112/plms/s3-29.2.331
R. I. Grigorchuk, On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR 271 (1983), no. 1, 30–33 (Russian). MR 712546
R.I. Grigorchuk and I. Pak, Groups of intermediate growth: An introduction for beginners. http://math.mit.edu/ pak/grigbegin6.pdf.
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, DOI 10.1017/S0143385702001724
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, DOI 10.24033/bsmf.2534
Wen Huang and Xiangdong Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc. 357 (2005), no. 2, 669–694. MR 2095626, DOI 10.1090/S0002-9947-04-03540-8
Wen Huang and Xiangdong Ye, Combinatorial lemmas and applications to dynamics, Adv. Math. 220 (2009), no. 6, 1689–1716. MR 2493178, DOI 10.1016/j.aim.2008.11.009
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, DOI 10.1017/S0143385702000585
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, DOI 10.1016/S0246-0203(97)80094-5
David Kerr and Hanfeng Li, Independence in topological and $C^*$-dynamics, Math. Ann. 338 (2007), no. 4, 869–926. MR 2317754, DOI 10.1007/s00208-007-0097-z
John Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), no. 3, 357–385. MR 955558
Donald Ornstein and Benjamin Weiss, Entropy is the only finitely observable invariant, J. Mod. Dyn. 1 (2007), no. 1, 93–105. MR 2261073, DOI 10.3934/jmd.2007.1.93
Kyewon Koh Park, On directional entropy functions, Israel J. Math. 113 (1999), 243–267. MR 1729449, DOI 10.1007/BF02780179
K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc. 24 (1970), 278–280. MR 250283, DOI 10.1090/S0002-9939-1970-0250283-7
Yves Pomeau and Paul Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74 (1980), no. 2, 189–197. MR 576270
N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A 13 (1972), 145–147. MR 307902, DOI 10.1016/0097-3165(72)90019-2
Saharon Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247–261. MR 307903
G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos 11 (2001), no. 2, 295–305. MR 1843718, DOI 10.1063/1.1355358