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 ${\theta }_2$ of the substitution matrix satisfies $|{\theta }_2|\ne 1$. If $|{\theta }_2|<1$, then (as is well-known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|{\theta }_2|>1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|{\theta }_2|=1$ is more delicate, and we only obtain some partial results.

20 pages, 1 figure; to appear in Ergodic Theory & Dynamical Systems; minor revision after the referee report