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

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.

Updated to published version

References in corpus (3)

• Tiling groupoids and Bratteli diagrams
• Fusion tilings with infinite local complexity
• Meyer sets, topological eigenvalues, and Cantor fiber bundles

Cited by in corpus (8)

• Recognizability for sequences of morphisms
• Geometry, dynamics, and arithmetic of $S$-adic shifts
• Dynamical properties of almost repetitive Delone sets
• An uncountable set of tiling spaces with distinct cohomology
• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions
• Tilings and traces
• Pattern-Equivariant Homology of Finite Local Complexity Patterns
• Meyer sets, topological eigenvalues, and Cantor fiber bundles