# American Institute of Mathematical Sciences

January  2017, 37(1): 435-448. doi: 10.3934/dcds.2017018

## Zero sequence entropy and entropy dimension

 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China 2 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210023, China

* Corresponding author

Received  January 2016 Revised  August 2016 Published  November 2016

Let $(X, T)$ be a topological dynamical system and $M(X)$ the set of all Borel probability measures on $X$ endowed with the weak$^*$ -topology. In this paper, it is shown that for a given sequence $S$ , a homeomorphism $T$ of $X$ has zero topological sequence entropy if and only if so does the induced homeomorphism $T$ of $M(X)$ . This extends the result of Glasner and Weiss [9,Theorem A] for topological entropy and also the result of Kerr and Li [15,Theorem 5.10]for null systems. Moreover, it turns out that the upper entropy dimension of $(X, T)$ is equal to that of $(M(X), T)$ . We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.

Citation: Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
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