Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

Baake, Michael; Frank, Natalie Priebe; Grimm, Uwe and Robinson, Jr., E. Arthur (2019). Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction. Studia Mathematica, 247 pp. 109–154.

DOI: https://doi.org/10.4064/sm170613-10-3

Abstract

One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalization analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well.

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About

• Item ORO ID
• 54848
• Item Type
• Journal Item
• ISSN
• 0039-3223
• Extra Information
• 2010 Mathematics Subject Classification. Primary 37A30, 42A38, 37B50; Secondary 37H15, 52C23
• Keywords
• non-Pisot substitutions; tiling dynamics; singular spectrum; Lyapunov exponents
• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Copyright Holders
• © 2018 Instytut Matematyczny PAN
• Depositing User
• Uwe Grimm