On the role of limsup in the definition of topological entropy via spanning or separation numbers. Part I: Basic examples

TL;DR

This paper investigates the use of limsup in defining topological entropy through spanning and separation numbers, providing the first known counterexamples where the limit superior does not reduce to a limit.

Contribution

It constructs the first explicit counterexamples demonstrating that the limit superior in topological entropy definitions via separation or spanning numbers does not always simplify to a limit.

Findings

Counterexamples show limsup does not always equal the limit in these definitions

Clarifies the distinction between covering, separation, and spanning number-based entropy

Enhances understanding of the mathematical properties of topological entropy

Abstract

The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution $\varepsilon$ 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.