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

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