  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.
