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Fusion: a general framework for hierarchical tilings of \(\mathbb{R }^d\)

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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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Notes

  1. If we wish, we can also add labels to the supertiles, so that the information carried in an \(n\)-supertile is more than just its composition as a patch in a tiling. This generalization is useful for collaring constructions, as in Sect. 5.

  2. Ergodic theorems are often stated not with balls, but in terms of Følner or van Hove sequences that have special properties, such as being “regular” or “tempered”. That generality is useful for computing frequencies using different sampling regions, or when considering more complicated groups than \(\mathbb{R }^d\). For our purposes, however, balls are sufficient.

  3. Note that this condition is translation-invariant, as every point in \(T\) would then lie in a sequence of unexceptional supertiles whose union is the entire line.

  4. This is connected to the height of a substitution or fusion. If a substitution has height one, then all eigenvalues of \(X_\mathcal{R }\) are eigenvalues of \(S_\mathcal{R }\) [47]. One can similarly define a notion of height for fusions.

  5. There is some flexibility with the geometry of the prototiles. They could be parallelograms or rectangles, and there are two vertical and two horizontal degrees of freedom for the lengths of the sides.

  6. The absence of the \(\sqrt{5}\) that is present in Example 4.4 is due to the integer size of the prototiles.

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Acknowledgments

We thank Mike Boyle, Lewis Bowen, Kariane Calta, Amos Nevo, E. Arthur Robinson, Jr. and Boris Solomyak for helpful discussions. The work of L.S. is partially supported by NSF Grants DMS-0701055 and DMS-1101326.

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  1. Department of Mathematics, Vassar College, Poughkeepsie, NY, 12604, USA

    Natalie Priebe Frank

  2. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Lorenzo Sadun

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Frank, N.P., Sadun, L. Fusion: a general framework for hierarchical tilings of \(\mathbb{R }^d\) . Geom Dedicata 171, 149–186 (2014). https://doi.org/10.1007/s10711-013-9893-7

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