In this paper, we consider a class of self-similar sets, denoted by $ \mathcal{A} $, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by $ U_1 $. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of $ U_1 $, is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition $ U_1 $ is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

中文翻译：

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明确集合之间的同构和bi-Lipschitz等价

在本文中，我们考虑一类由$ \ mathcal {A} $表示的自相似集，并研究具有唯一编码的自相似集中的点集。我们称这样的集合为唯一集合，并用$ U_1 $表示。我们分析了单义集之间的同构和bi-Lipschitz等价。就$ U_1 $的维数而言，本文的主要结果是给出几个等价条件，这些条件描述了在懒散映射下，两个唯一集的闭合相对于最大值的唯一度量在理论上是同构的熵。此外，我们证明了在$ U_1 $闭合的条件下，单义集之间的同构和bi-Lipschitz等价具有共振现象。