On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Abstract

Let $\lambda _{1},\ldots ,\lambda _{n}$ be real numbers in $(0,1)$ and $p_{1},\ldots ,p_{n}$ be points in $\mathbb {R}^{d}$. Consider the collection of maps $f_{j}:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ given by

$$\begin{aligned} f_{j}(x)=\lambda _{j} x +\left( 1-\lambda _{j}\right) p_{j}. \end{aligned}$$

It is a well known result that there exists a unique nonempty compact set $\Lambda \subset \mathbb {R}^{d}$ satisfying $\Lambda =\cup _{j=1}^{n} f_{j}(\Lambda ).$ Each $x\in \Lambda$ has at least one coding, that is a sequence $(\epsilon _{i})_{i=1}^{\infty }\in \{1,\ldots ,n\}^{\mathbb {N}}$ that satisfies $\lim _{N\rightarrow \infty }f_{\epsilon _{1}}\ldots f_{\epsilon _{N}} (0)=x.$ We study the size and complexity of the set of codings of a generic $x\in \Lambda$ when $\Lambda$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $x\in \Lambda$ has a continuum of codings. We also show that almost every $x\in \Lambda$ has a universal coding. Our work makes no assumptions on the existence of holes in $\Lambda$ and improves upon existing results when it is assumed $\Lambda$ contains no holes.