拓扑熵、均维数与弱等价流
Topological entropy, mean dimension, and weakly equivalent flows
AI总结：
本文通过拓扑方法重新审视弱等价流的拓扑熵和均维数理论，提出新的方法改进Ohno的理论并加强其结论。
摘要：
在本文中，我们重新审视了一个关于弱等价流拓扑熵的漂亮理论，该理论最初由Ohno于1980年研究。我们将发展一种新的方法，比Ohno提供的测度论方法更直接和基础，用于弱等价流的理论，并且作为创新点，我们以高度统一的过程研究了拓扑熵和均维数与这些对象的关系。特别是，对于没有固定点的弱等价流，我们用一种显著不同的方法恢复了Ohno关于拓扑熵关系的定理，并且在均维数框架内进行了类比；而对于有固定点的弱等价流，我们的技术改进了Ohno构造中的过程，并加强了Ohno的例子，以考虑动力系统的拓扑复杂性。本质上，我们的方法是拓扑的。为此，我们首先对拓扑熵的定义进行了修改，通过将状态空间内的覆盖集与外部的有限集相关联，这导致了一个微小但关键的不同，通过映射实现，并进一步证明这最终导致与拓扑熵相同的值。虽然在一定程度上，修改后的量通常不与经典定义中的量一致，但它具有一些基本的组合特征，不仅灵活地应用于拓扑熵，还直接适用于均维数的上下文。使用这种替代方法，我们得以重新建立并丰富Ohno的理论。
英文摘要：
In this paper, we mainly revisit a nice theory for topological entropy of weakly equivalent flows, which was originally investigated by Ohno in 1980. We will develop a new approach, being more straightforward and elementary than the measure-theoretic one provided by Ohno, to the theory for weak equivalence of flows, and as a novelty, we study both topological entropy and mean dimension with a highly unified process in relation to such objects. In particular, for weakly equivalent flows without fixed points we recover Ohno's theorem for topological entropy relation with a substantially different method, and moreover, carry out an analogue within the framework of mean dimension; while for weakly equivalent flows with fixed points, our technique refines the procedure suggested in Ohno's construction, and strengthens Ohno's example with a view towards topological complexity of dynamical systems. Essentially, our method is topological. For this purpose, we first introduce a modification to the definition of topological entropy, by relating a spanning set inside the state space to a finite set outside, which comes to be a tiny but key difference, via a map, and further, show that it leads eventually to the same value as the topological entropy. Although to an intermediate extent, the modified quantity generally does not coincide with the one appearing in the classical definition, it has some basic combinatorial feature, which not only applies flexibly to topological entropy, but also adapts directly to the mean dimension context. Using this alternative, we are allowed to re-establish and enrich Ohno's theory.
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