Multiscale Substitution Tilings

Abstract

We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well-studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from those that arise in the standard setup. We study various structural, geometric, statistical and dynamical aspects of these new objects and establish a wide variety of properties. Among our main results are explicit density formulas and the unique ergodicity of the associated tiling dynamical systems.

arXiv comment49 pages, 14 figures. Revision after the referee report. The lower bound on the discrepancy in chapter 8 has been improved. To appear in the Proceedings of the London Mathematical Society

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Cited by in corpus (2)

• Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings
• Discrepancy and rectifiability of almost linearly repetitive Delone sets