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Topological mixing for substitutions on two letters

Richard Kenyon·Lorenzo Sadun·Boris Solomyak
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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.

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Richard Kenyon, Lorenzo Sadun, Boris Solomyak. 2005-07-22. Topological mixing for substitutions on two letters. https://arxiv.org/abs/math/0406503

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