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Directional Kronecker algebra for $\mathbb {Z}^q$-actions

Part of: Ergodic theory

Published online by Cambridge University Press:  28 January 2022

CHUNLIN LIU  and
LEIYE XU
CHUNLIN LIU*
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (e-mail: leoasa@mail.ustc.edu.cn)
LEIYE XU
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (e-mail: leoasa@mail.ustc.edu.cn)
*
e-mail: lcl666@mail.ustc.edu.cn
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Abstract

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

Keywords

directional sequence entropydirectional Kronecker algebradirectional discrete spectrum system

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37A05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 4 , April 2023 , pp. 1324 - 1350
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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