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

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.

69 pages; Slightly modified the wording in Lemma 1, Lemma 10, and the derivation of Theorem 4 from Lemma 36 at page 60 so as to increase precision and eliminate sources of possible confusion; Corrected one typo in (PY2+) at page 34 that would likely have led to misreading of the condition