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.
Categories: math.DS