Topological mixing for substitutions on two letters

Details Topological mixing for substitutions on two letters Topological mixing for substitutions on two letters

2004-06-24

arxiv.org/abs/math/0406503v2

Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1919--1934

Mathematics

Dynamical Systems

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

Scientific paper

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.

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