On univoque points for self-similar sets

TL;DR

This paper investigates points with unique codings in self-similar sets, showing they form a graph-directed self-similar set and providing a method to compute their Hausdorff dimension.

Contribution

It establishes that the set of uniquely coded points in interval self-similar sets is a graph-directed self-similar set, generalizing previous results and enabling explicit dimension calculations.

Findings

Unique coding set is a subshift of finite type.

The set of unique coding points is a graph-directed self-similar set.

An explicit algorithm for Hausdorff dimension calculation is provided.

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}.