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}\to\mathbb{R}^{d} $ given by $$f_{j}(x)=\lambda_{j} x +(1-\lambda_{j})p_{j}.$$ It is a well known result that there exists a unique 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 $\lim_{N\to\infty}f_{\epsilon_{1}}\cdots f_{\epsilon_{N}} (0)=x.$ 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.