Multiple codings for self-similar sets with overlaps

Abstract

In this paper we consider a general class $\mathcal{E}$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $\Phi =(E,\{f_i{\}}_{i=1}^m)\in \mathcal{E}$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1{i}_2\dots \in \{1,\dots ,m{\}}^{\mathbb{N}}$, called a coding of $x$, such that $$x=\underset{n\to \infty }{\lim }{f}_{i_1}\circ f_{i_2}\circ \cdots \circ f_{i_n}(0).$$ For $k=1,2,\dots ,{\aleph }_0$ or $2^{{\aleph }_0}$ we investigate the subset ${\mathcal{U}}_k(\Phi )$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that ${\mathcal{U}}_1(\Phi )$ is closed if and only if ${\mathcal{U}}_{{\aleph }_0}(\Phi )$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of ${\mathcal{U}}_k(\Phi )$, and show that the corresponding Hausdorff measure of ${\mathcal{U}}_k(\Phi )$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in ${\mathcal{U}}_k(\Phi )$ and $U_{{\aleph }_0}(\Phi )$.

We add some new results in this version

References in corpus (3)

• Expansions in non-integer bases: lower, middle and top orders
• Subshift of finite type and self-similar sets
• Arithmetic progressions in middle Nth cantor sets

Cited by in corpus (8)

• Projections of cartesian products of the self-similar sets without the irrationality assumption
• Subshift of finite type and self-similar sets
• Multiplication on self-similar sets with overlaps
• Arithmetic representations of real numbers in terms of self-similar sets
• Multiple representations of real numbers on self-similar sets with overlaps
• On arithmetic progressions in self-similar sets
• On a kind of self-similar sets with complete overlaps
• Estimating the Hausdorff dimensions of univoque sets for self-similar sets