Eigenfunctions for substitution tiling systems

Details Eigenfunctions for substitution tiling systems Eigenfunctions for substitution tiling systems

2005-12-27

arxiv.org/abs/math/0512602v3

Probability and number theory--Kanazawa 2005, 433--454, Adv. Stud. Pure Math., 49, Math. Soc. Japan, Tokyo, 2007

Mathematics

Dynamical Systems

Revised after the referee report. To appear in Advanced Studies in Pure Mathematics, vol.49, Mathematical Society of Japan

Scientific paper

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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