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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
DOI: https://doi.org/10.1090/S0002-9947-2010-04906-2
Published electronically: September 2, 2010
MathSciNet review: 2728582
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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.


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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 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.

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