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 $x \in K$ 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.

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