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Bounded complexity, mean equicontinuity and discrete spectrum

Part of:Topological dynamicsErgodic theory

Published online by Cambridge University Press:  07 October 2019

WEN HUANG,
JIAN LIOpen the ORCID record for JIAN LI[Opens in a new window],
JEAN-PAUL THOUVENOT,
LEIYE XU and
XIANGDONG YE
WEN HUANG
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email wenh@mail.ustc.edu.cn, leoasa@mail.ustc.edu.cn, yexd@ustc.edu.cn
JIAN LI
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, P.R. China email lijian09@mail.ustc.edu.cn
JEAN-PAUL THOUVENOT
Affiliation:
Université Paris 6-LPMA, case courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France email jeanpaul.thouvenot@gmail.com
LEIYE XU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email wenh@mail.ustc.edu.cn, leoasa@mail.ustc.edu.cn, yexd@ustc.edu.cn
XIANGDONG YE
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email wenh@mail.ustc.edu.cn, leoasa@mail.ustc.edu.cn, yexd@ustc.edu.cn
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Abstract

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We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$ , the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$ , both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$ ) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$ , if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -mean equicontinuous and if and only if it has discrete spectrum.

Keywords

bounded complexityequicontinuitymean equicontinuitydiscrete spectrum

MSC classification

Primary:37A35: Entropy and other invariants, isomorphism, classification
Secondary:37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021, pp. 494 - 533
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Copyright
© Cambridge University Press, 2019

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