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

TL;DR

This paper investigates the structure of a family of self-similar sets generated by simple iterated function systems, revealing their complex topological and measure-theoretic properties, and demonstrating the abundance of parameters for which multiple points are contained.

Contribution

It characterizes the set of parameters for which a point belongs to the self-similar set, showing it is a zero-measure Cantor set with full Hausdorff dimension, and proves the existence of parameters containing multiple points.

Findings

a set of parameters forms a zero-measure Cantor set with full Hausdorff dimension.

a dense set of parameters contains multiple specified points.

The set of parameters with multiple points has full Hausdorff dimension.

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 $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_\lambda$.