The topological entropy of a topological dynamical system, introduced in a foundational paper by Adler, Konheim and McAndrew [Trans. Am. Math. Soc., 1965], is a nonnegative number that measures the uncertainty or disorder of the system. Comparing with positive entropy systems, zero entropy systems are much less understood. In order to distinguish between zero entropy systems, Huang and Ye [Adv. Math., 2009] introduced the concept of maximal pattern entropy of a topological dynamical system. At the heart of their analysis is a Sauer-Shelah type lemma. In the present paper, we provide a shorter and more conceptual proof of a strengthening of this lemma, and discuss its surprising connection between dynamical system, combinatorics and a recent breakthrough in communication complexity. We also improve one of the main results of Huang and Ye on the maximal pattern entropy of zero-dimensional systems, by proving a new Sauer-Shelah type lemma, which unifies and enhances various extremal results on VC-dimension, Natarajan dimension and Steele dimension.

翻译：拓扑动力系统的拓扑熵由Adler、Konheim和McAndrew在其奠基性论文[Trans. Am. Math. Soc., 1965]中引入，是一个衡量系统不确定性或混乱度的非负数。与正熵系统相比，零熵系统的性质远未被充分理解。为区分不同零熵系统，Huang与Ye [Adv. Math., 2009]提出了拓扑动力系统最大模式熵的概念，其分析核心依赖于一类Sauer-Shelah型引理。本文给出了该引理强化版本的更简洁且更具概念性的证明，并探讨了动力系统、组合学与通信复杂性领域近期突破之间的惊人联系。此外，通过证明一个新的Sauer-Shelah型引理——该引理统一并改进了关于VC维、Natarajan维与Steele维的若干极值结果，我们改进了Huang与Ye关于零维系统最大模式熵的主要结论之一。