Fusion: a general framework for hierarchical tilings of \mathbb{R }^d |
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Authors: | Natalie Priebe Frank Lorenzo Sadun |
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Institution: | 1. Department of Mathematics, Vassar College, Poughkeepsie, NY, 12604, USA 2. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA
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Abstract: | 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 minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing. |
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