Subshift of finite type and self-similar sets

Karma Dajani, Kan Jiang

Published 2015-09-16Version 1

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $\beta$-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. This application generalizes a result of de Vries and Komornik \cite{MK}. Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the attractor can be identified with a subshift of finite type. The third application partially answers a problem posed by Barrera \cite{Barrera}.