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.

中文翻译：

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通过跨越数或分离数定义拓扑熵时需要 Limsup

摘要 拓扑熵的概念可以根据在 T 时间单位内以分辨率 ϵ 可区分的前向轨迹的数量来概念化。然后，它可以正式定义为涉及覆盖数、分离数或跨度数的上极限的极限。如果使用覆盖数，则上限值将降低为限值。虽然人们普遍认为，当定义基于分离数或跨度数时，后者可能不一定是这种情况，但以前似乎没有实际的反例。在这里，我们通过构建这样的反例来填补文献中的这一空白。