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Transactions of the American Mathematical Society

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Entropy dimension of measure preserving systems


Authors: Dou Dou, Wen Huang and Kyewon Koh Park
Journal: Trans. Amer. Math. Soc. 371 (2019), 7029-7065
MSC (2010): Primary 37A35, 37A05, 28D20
DOI: https://doi.org/10.1090/tran/7542
Published electronically: September 24, 2018
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Abstract: The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight for analyzing entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate joinings among entropy zero systems and prove the disjointness property among some classes of entropy zero systems using dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.


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Additional Information

Dou Dou
Affiliation: Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, People’s Republic of China
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: Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 130-722, Korea
Email: kkpark@kias.re.kr

DOI: https://doi.org/10.1090/tran/7542
Keywords: Entropy dimension, dimension set, uniform dimension, joining
Received by editor(s): May 5, 2017
Received by editor(s) in revised form: January 9, 2018, and February 13, 2018
Published electronically: September 24, 2018
Additional Notes: The first author is supported by NNSF of China (Grant Nos. 10901080, 11271191, 11790274).
The second author is supported by NNSF of China (Grant Nos. 11225105 and 11431012).
The third author is supported in part by NRF 2010-0020946.
Article copyright: © Copyright 2018 American Mathematical Society