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Null systems and sequence entropy pairs

Published online by Cambridge University Press:  23 September 2003

W. HUANG
Affiliation:
Graduate School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn, lisimin@ms.u-tokyo.ac.jp)
S. M. LI
Affiliation:
Graduate School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn, lisimin@ms.u-tokyo.ac.jp) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153, Japan
S. SHAO
Affiliation:
Graduate School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn, lisimin@ms.u-tokyo.ac.jp)
X. D. YE
Affiliation:
Graduate School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn, lisimin@ms.u-tokyo.ac.jp)

Abstract

A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.

Type
Research Article
Copyright
2003 Cambridge University Press

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