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

Abstract

For $λ\in(0,1/2]$ let $K_λ\subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{λx, λx+1-λ\}$. Given $x\in(0,1/2)$, let $Λ(x)$ be the set of $λ\in(0,1/2]$ such that $x\in K_λ$. In this paper we show that $Λ(x)$ 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 $λ\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_λ$.

Figures (1)

• Figure 1: A defining sequence $\mathscr{W}_\ell=\left\{ (\beta_{k}, \alpha_{k+1}), V_{k,j}: k\ge \ell, j\ge 1 \right\}$ for the Cantor set $C_\ell=\bigcup_{k=\ell}^\infty F_k \cup \{\beta\}$, and for each $k\ge \ell$ a defining sequence $\mathscr{V}_k=\left\{ V_{k,j} \right\}_{j=1}^\infty$ for the Cantor set $F_k$; see (\ref{['eq:defining-sequence-Ek']}) and (\ref{['eq:defining-sequence-C-ell']}) for more explanation.

Theorems & Definitions (34)

• Theorem \oldthetheorem
• Theorem \oldthetheorem
• Corollary \oldthetheorem
• proof
• Theorem \oldthetheorem
• Lemma \oldthetheorem
• proof
• Lemma \oldthetheorem
• proof
• proof : Proof of Theorem \ref{['main:topology']}