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Topology and topological sequence entropy

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Abstract

Let X be a compact metric space and T: X → X be continuous. Let h* (T) be the supremum of topological sequence entropies of T over all subsequences of ℤ+ and S(X) be the set of the values h*(T) for all continuous maps T on X. It is known that {0} ⊆ S(X) ⊆ {0, log 2, log 3, …} ∪ {∞}. Only three possibilities for S(X) have been observed so far, namely S(X) = {0}, S(X) = {0, log 2, ∞} and S(X) = {0, log 2, log 3, …}∪{∞}.

In this paper we completely solve the problem of finding all possibilities for S(X) by showing that in fact for every set {0} ⊆ A ⊆ {0, log 2, log 3, …} ∪ {∞} there exists a one-dimensional continuum XA with S(XA) = A. In the construction of XA we use Cook continua. This is apparently the first application of these very rigid continua in dynamics.

We further show that the same result is true if one considers only homeomorphisms rather than continuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open.

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Acknowledgements

The first author was supported by the Slovak Research and Development Agency (Grant No. APVV-15-0439) and by VEGA (Grant No. 1/0786/15). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11371339 and 11431012). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11871188 and 11671094). This project was started when the first two authors visited Max Planck Institute for Mathematics, Bonn, in 2009. Much of the work on the project was done also when the first author visited University of Science and Technology of China, Hefei in 2011 and 2016 and when the third author visited Matej Bel University in 2018. The authors thank all these institutions for the warm hospitality and financial supports. The authors thank Jerzy Krzempek for providing them with the reference [37], where the existence of Cook continua in the plane is proved. This enabled the authors to simplify the geometry of their constructions. Sincere thanks of the authors go to Hanfeng Li for his comments on the preliminary version of the paper, which resulted in the homeomorphism case in Subsection 8.2, the group actions case in Subsection 8.3, and an open problem in Subsection 9.6. The authors also thank the anonymous referees for their helpful comments and suggestions that improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Banská Bystrica, 974 01, Slovakia

    L’ubomír Snoha

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Xiangdong Ye

  3. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Ruifeng Zhang

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  1. L’ubomír Snoha
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  2. Xiangdong Ye
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  3. Ruifeng Zhang
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Correspondence to Ruifeng Zhang.

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Snoha, L., Ye, X. & Zhang, R. Topology and topological sequence entropy. Sci. China Math. 63, 205–296 (2020). https://doi.org/10.1007/s11425-019-9536-7

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