We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.

中文翻译：

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封闭关系非零熵的充分条件

我们引入了紧凑度量空间上封闭关系的回报和对齐集合的概念，然后使用它们来获得这种关系具有非零熵的非平凡充分条件。此外，我们用 Li-Yorke 和 DC2 混沌描述了非零熵的有限关系。