Narrow Results

Let $\lambda \in (0,1)$ and $m\ge 3$ an integer. We consider the collection ${𝒜}$ of homogeneous self-similar sets on the line such that every two of copies $f_i(K),f_j(K)$ of the self-similar set $K$ are either separated or overlapped with rank $k$ in $\{2,\dots ,m\}$. For $K\in {𝒜}$ generated by $n$ similitudes, we denote by $n_j$ the number of overlaps with rank $j\in \{2,\dots ,m\}$. The set of points in the self-similar set having a unique coding is called the univoque set and denoted by ${𝒰}$. In this paper, we investigate a uniform method to calculate the Hausdorff dimension of the set ${𝒰}$.

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.