Purity results for some arithmetically defined measures

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

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