动力熵的范畴论基础
The categorical basis of dynamical entropy
AI总结：
该研究从范畴论视角研究拓扑动力熵，提出合格函子对工具证明紧空间开覆盖的直径与Lebesgue数构成该类函子对，揭示多种迭代复杂度概念收敛到共同拓扑熵是结构假设的结果。
摘要：
理论数学与应用数学的诸多分支都需要可量化的复杂度概念，拓扑动力系统（即度量空间上的连续自映射）就是这类场景之一。针对映射的反复迭代，存在多种可定义的复杂度概念。动力系统理论的基础发现之一是，这些概念存在共同极限，即系统的拓扑熵。我们提出拓扑动力熵的范畴论视角，揭示该共同极限是这些复杂度概念的结构假设的必然结果。研究开发的核心工具之一是“合格函子对（qualifying pair of functors）”，它以类似实分析中夹逼定理的方式保证保极限性。我们证明，紧空间开覆盖的直径与Lebesgue数构成合格函子对。各类复杂度概念被表示为函子，这些函子间的自然变换导致它们共同收敛到该极限。
英文摘要：
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.
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