Fusion: a general framework for hierarchical tilings of $\R^d$

Details Fusion: a general framework for hierarchical tilings of $\R^d$ Fusion: a general framework for hierarchical tilings of $\R^d$

2011-01-25

arxiv.org/abs/1101.4930v2

Mathematics

Dynamical Systems

45 pages. Revision includes new subsection on pure point spectrum

Scientific paper

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing. We also exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum.

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