Topological Mixing of Random Substitutions

TL;DR

This paper studies topological mixing in random substitutions, providing a criterion for primitive cases and showing non-mixing in certain Pisot and Fibonacci cases, extending previous deterministic results.

Contribution

It introduces a simple criterion for topological mixing in primitive random substitutions and analyzes non-mixing in recognisable Pisot and Fibonacci cases.

Findings

Primitive random substitutions with eigenvalue > 1 are topologically mixing if the criterion is met.

Recognisable Pisot random substitutions are not topologically mixing.

The random Fibonacci substitution subshift is not topologically mixing.

Abstract

We investigate topological mixing of compatible random substitutions. For primitive random substitutions on two letters whose second eigenvalue is greater than one in modulus, we identify a simple, computable criterion which is equivalent to topological mixing of the associated subshift. This generalises previous results on deterministic substitutions. In the case of recognisable, irreducible Pisot random substitutions, we show that the associated subshift is not topologically mixing. Without recognisability, we rely on more specialised methods for excluding mixing and we apply these methods to show that the random Fibonacci substitution subshift is not topologically mixing.