On a class of selfsimilar sets which contain finitely many common points
Abstract
For $\lambda\in(0,1/2]$ let $K_\lambda\subset\mathbb R$ be a selfsimilar 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 $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda\in(0,1/2]$ such that $y_1,\ldots, y_p$ are common points of $K_\lambda$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.10014
 Bibcode:
 2021arXiv210910014W
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Classical Analysis and ODEs;
 Primary: 28A78;
 Secondary: 28A80;
 37B10
 EPrint:
 19 pages, 1 figure