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

Abstract

Let $\hat{I}{»}_1,\dots ,\hat{I}{»}_n$ be real numbers in $(0,1)$ and $p_1,\dots ,p_n$ be points in ${\mathbb{R}}^d$. Consider the collection of maps $f_j:{\mathbb{R}}^d\to {\mathbb{R}}^d$ given by $$f_j(x)=\hat{I}{»}_{j}x+(1-\hat{I}{»}_j)p_j.$$ It is a well known result that there exists a unique compact set $\hat{I}›\subset {\mathbb{R}}^d$ satisfying $\hat{I}›={\cup }_{j=1}^n{f}_j(\hat{I}›).$ Each $x\in \hat{I}›$ has at least one coding, that is a sequence $(\hat{I}{\mu }_i{)}_{i=1}^{\infty }\in \{1,\dots ,n{\}}^{\mathbb{N}}$ that satisfies ${\lim }_{N\to \infty }{f}_{\hat{I}{\mu }_1}\cdots f_{\hat{I}{\mu }_N}(0)=x.$ We study the size and complexity of the set of codings of a generic $x\in \hat{I}›$ when $\hat{I}›$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $x\in \hat{I}›$ has a continuum of codings. We also show that almost every $x\in \hat{I}›$ has a universal coding. Our work makes no assumptions on the existence of holes in $\hat{I}›$ and improves upon existing results when it is assumed $\hat{I}›$ contains no holes.