On univoque points for self-similar sets

Simon Baker; Karma Dajani; Kan Jiang

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arXiv · 1406.3263

On univoque points for self-similar sets

Simon Baker·Karma Dajani·Kan Jiang

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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 of this, 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 \cite{MW}. 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 \cite{DKK}, \cite{K1}, \cite{K2}, and \cite{MK}.

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Simon Baker, Karma Dajani, Kan Jiang. 2014-06-12. On univoque points for self-similar sets. https://arxiv.org/abs/1406.3263

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