Lipschitz classification of self-similar sets with overlaps
Let \(I=[0,1]\) be a unit interval, \(n \geq 3\) be an integer and let \(0<r<1 / n\). Denote by \(\mathcal{C}_{n, r, \ell}\) the collection of self-similar sets \(K\) satisfying: (1) \(K=\bigcup_{i=1}^{n} S_{i}(K)\) where \(S_{i}(x)=r x+b_{i}\) for \(i=1, \ldots, n\). (2) \(0=b_{1}<b_{2}<\cdots<b_{n}=1-r\). (3) If \(S_{i}(I) \cap S_{i+1}(I) \neq \emptyset\) for some \(1 \leq i<n\), then the intersection either is a one-point set (touching) or satisfies \(\left|S_{i}(I) \cap S_{i+1}(I)\right|=r^{2}\) (overlapping). (4) \(\ell=\#\left\{i \in\{1, \ldots, n-1\}:\left|S_{i}(I) \cap S_{i+1}(I)\right|=r^{2}\right\}\) (the number of overlaps). In this short note, by applying a matrix rearrangeable condition introduced in [\textit{J. J. Luo}, J. Lond. Math. Soc., II. Ser. 99, No. 2, 428--446 (2019; Zbl 1502.28007)], the authors prove that \textbf{Theorem 1.1} Any two self-similar sets in \(\mathcal{C}_{n, r, \ell}\) are Lipschitz equivalent. This theorem generalized the result of \textit{H. Rao} et al. [C. R., Math., Acad. Sci. Paris 342, No. 3, 191--196 (2006; Zbl 1086.28007)] onto the self-similar sets with overlapping structure. i.e., they extend the condition \(S_{i}(I) \cap S_{j}(I)\) contains at most one point for \(i\neq j\) to \((3),(4)\).
scientific article; zbMATH DE number 7356432
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