关于limsup在通过生成数或分离数定义拓扑熵中的作用。第一部分：基本实例
On the role of limsup in the definition of topological entropy via spanning or separation numbers. Part I: Basic examples
摘要：
拓扑熵的概念可以根据在$T$时间单位内在分辨率$\varepsilon$上可区分的前向轨迹的数量来概念化。然后它可以被正式定义为一个极限上的极限，涉及到覆盖数，或分离数，或跨越数。如果使用了覆盖数，则上限将减少到一个极限。虽然人们普遍认为，后者可能不一定是这种情况下，定义是基于分离或跨越数，没有实际的反例似乎已经知道。在这里，我们填补了这个空白，在文献中构建这样的反例。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
