We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations of Mane's Theorem and Utz's Theorem about the finite dimensionality of the phase space of an expansive system, and the finiteness of a space supporting a positively expansive homeomorphism, respectively. We also discuss the notion of entropy in this setting calculating it for the non-Hausdorff shifts.

中文翻译：

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格子上的膨胀系统

我们在分布格及其自同构的设置中研究膨胀动力系统，当格是由包含排序的相空间的拓扑时，紧致度量空间同胚的膨胀的通常概念是特殊情况，自同构是一个由同胚诱导，将开集映射到开集。我们在此上下文中分别证明了 Mane 定理和 Utz 定理关于膨胀系统相空间的有限维数和支持正膨胀同胚的空间的有限性的推广。我们还讨论了在此设置中为非 Hausdorff 移位计算熵的概念。