Eigenfunctions for substitution tiling systems

Boris Solomyak

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Eigenfunctions for substitution tiling systems

Boris Solomyak

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Abstract

We prove that for the uniquely ergodic ${\bf R}^d$ 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 $\theta>1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $\theta$ being a Pisot number.

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Boris Solomyak. 2005-12-27. Eigenfunctions for substitution tiling systems. https://arxiv.org/abs/math/0512602

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