Topological mixing for substitutions on two letters

Abstract

We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $θ_2$ of the substitution matrix satisfies $|θ_2|\ne 1$. If $|θ_2|<1$, then (as is well-known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|θ_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|θ_2|=1$ is more delicate, and we only obtain some partial results.