Limsup is needed in the definitions of topological entropy via spanning or separation numbers. Issue 3 (2nd July 2020)

• Record Type: Journal Article
• Authors: Just, Winfried; Xin, Ying
• Is Part Of: Dynamical systems. Volume 35:Issue 3(2020)
• Page Start: 430
• Page End: 489
• Publication Date: 2020-07-02
• Subjects: Topological entropy -- separation numbers -- spanning numbers -- subadditivity -- colouring; Primary: 37B40 -- Secondary: 37C35; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2020.1718612
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: 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.