Purity results for some arithmetically defined measures

TL;DR

This paper investigates measures derived from maximal entropy measures on sofic shifts through digital maps involving Pisot numbers, characterizing their continuity and establishing a purity result.

Contribution

It provides a new characterization of the continuity of these measures and proves a purity result for measures associated with Pisot numbers.

Findings

Characterization of measure continuity via automaton structure

Proof of a purity property for these measures

Insights into measures related to Pisot number expansions

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.