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VOL. 49 | 2007 Eigenfunctions for substitution tiling systems
Boris Solomyak

Editor(s) Shigeki Akiyama, Kohji Matsumoto, Leo Murata, Hiroshi Sugita

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.

Information

Published: 1 January 2007
First available in Project Euclid: 27 January 2019

zbMATH: 1139.37009
MathSciNet: MR2405614

Digital Object Identifier: 10.2969/aspm/04910433

Subjects:
Primary: 37B50
Secondary: 52C23

Rights: Copyright © 2007 Mathematical Society of Japan

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