Purity results for some arithmetically defined measures

Abstract

We study measures that are obtained as push-forwards of measures of maximal entropy on sofic shifts under digital maps $(x_k{)}_{k\in \mathbb{N}}\mapsto {\sum }_{k\in \mathbb{N}}{x}_k{\beta }^{-k}$, where $\beta >1$ is a Pisot number. We characterise the continuity of such measures in terms of the underlying automaton and show a purity result.