Limsup is needed in the definitions of topological entropy via spanning or separation numbers

The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ϵ within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spannin...

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Bibliographic Details
Published inDynamical systems (London, England) Vol. 35; no. 3; pp. 430 - 489
Main Authors Just, Winfried, Xin, Ying
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 02.07.2020
Taylor & Francis Ltd
Subjects
colouring
Entropy
Primary: 37B40
Secondary: 37C35
Separation
separation numbers
spanning numbers
subadditivity
Topological entropy
Topology
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Summary:The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ϵ within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the definition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples.
ISSN:1468-9367
1468-9375
DOI:10.1080/14689367.2020.1718612