Mixing properties and entropy bounds of a family of Pisot random substitutions

TL;DR

This paper studies a family of Pisot random substitutions, analyzing their combinatorial, topological, and entropy properties, revealing recognisable words, non-topological mixing, and explicit entropy bounds based on parameters.

Contribution

It introduces a new two-parameter family of random substitutions and characterizes their recognisability, mixing properties, and entropy bounds, advancing understanding of Pisot substitution dynamics.

Findings

Recognisable words exist at every level.

Subshifts are not topologically mixing.

Explicit bounds for topological entropy are provided.

Abstract

We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts they define are not topologically mixing. We then show that they satisfy a weaker mixing property using a numeration system arising from a sequence of lengths of inflated words. Moreover, we provide explicit bounds for the corresponding topological entropy in terms of the defining parameters $n$ and $p$.