Let K be a self-similar set in ℝ. Generally, if the iterated function system (IFS) of K has some overlaps, then some points in K may have multiple codings. If an x ∈ K has a unique coding, then we call x a univoque point. We denote by 𝒰 (univoque set) the set of points in K having unique codings. In this paper, we shall consider the following natural question: if two self-similar sets are bi-Lipschitz equivalent, then are their associated univoque sets also bi-Lipschitz equivalent. We give a class of self-similar sets with overlaps, and answer the above question affirmatively.

中文翻译：

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自相似集与其单义集之间的共振

让ķ是一个自相似集ℝ. 一般来说，如果迭代函数系统（IFS）ķ有一些重叠，然后有一些点ķ可能有多个编码。如果X ∈ ķ有唯一的编码，那么我们称X一个独特的点。我们表示𝒰（唯一集）中的点集ķ具有独特的编码。在本文中，我们将考虑以下自然问题：如果两个自相似集是bi-Lipschitz 等价的，那么它们关联的univoque 集是否也是bi-Lipschitz 等价的。我们给出一类有重叠的自相似集，并肯定地回答上述问题。