Eigenfunctions for substitution tiling systems

Abstract

We prove that for the uniquely ergodic $\mathbb{R}^d$-action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $\theta \gt 1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $\theta$ being a Pisot number.