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在一篇奠基性论文中引入[Trans. Am. Math. Soc., 1965]，它是一个非负数，用于衡量系统的不确定性或混乱程度。与正熵系统相比，零熵系统的认识尚不充分。为区分零熵系统，Huang和Ye [Adv. Math., 2009]引入了拓扑动力系统的最大模式熵概念，其分析核心是Sauer–Shelah型引理。本文对该引理的一个加强版给出了更简洁且更具概念性的证明，并讨论了其在动力系统、组合数学以及通信复杂性近期突破之间的惊人联系。此外，我们通过证明一个新的Sauer–Shelah型引理，改进了Huang和Ye关于零维系统最大模式熵的主要结果之一；该引理统一并增强了VC维、Natarajan维和Steele维的若干极值结论。