On univoque points for self-similar sets
Volume 228 / 2015
Fundamenta Mathematicae 228 (2015), 265-282
MSC: Primary 37A45; Secondary 37C45.
DOI: 10.4064/fm228-3-4
Abstract
Let $K\subseteq \mathbb {R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of Daróczy, Kátai, Kallós, Komornik and de Vries.
