On a class of self-similar sets which contain finitely many common points

Abstract

For $\lambda \in (0,1/2]$ let $K_{\lambda }\subset \mathbb{R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\lambda x+1-\lambda \}$. Given $x\in (0,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,1/2]$ such that $x\in K_{\lambda }$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\dots ,y_p\in (0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,1/2]$ such that $y_1,\dots ,y_p\in K_{\lambda }$.

21 pages, 1 figure, final version

References in corpus (1)

• How likely can a point be in different Cantor sets