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.

Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings

“…Full details can be found in [KSS] for the case that the interval A is an entire edge, and the required background concerning the Perron-Frobenius and the Wiener-Ikehara theorems can be found in [Ga, Chapter XIII] and [MV,Chapter 8.3], respectively. The refinement to general intervals is then simple, and can be found in the proof of [SS,Theorem 7.2]. We note that though the result is stated for a half-closed half-open interval A, it is the same whether the boundaries of A are included or not.…”

“…In the non-random case, if σ is a substitution scheme in R d then an important observation is the following correspondence between tiles in a patch of the form F σ t pT i q and walks on the associated graph: for every t ą 0, the tiles of F σ t pT i q are in one-to-one correspondence with walks of length d ¨t originating at the vertex i in G σ . Moreover, if T is a tile of type r and volume volT , then the corresponding walk γ T terminates at a point on an edge that terminates at vertex r, and the termination point is at distance log 1 volT from the vertex r. See [SS,§2] for additional details, examples and illustrations, and notice the slight difference in the definition of the associated graph pointed out in Remark 2.2.…”

“…As noted above, general formulas for tile frequencies in the non-random case appeared in [SS,§7], see also [Sm,Theorem 1.13], using a similar strategy to the one applied in the current contribution. While allowing for computations in specific cases and examples, the formulas remained somewhat opaque, and were not given explicitly in terms of the associated matrices or in terms of density functions, making them challenging to decipher and giving little additional insight to the way the frequencies depend on the generating substitution scheme.…”

“…In fact, they are at least of order 1{ log k volpe t Sq for some fixed k ě 1, which depends only on the underlying system. See [SS,§8] for more details, and for a discussion about how these "bad" error terms are related to questions on bounded displacement and bilipschitz equivalence classes of the associated Delone sets.…”

A dichotomy for bounded displacement equivalence of Delone sets

“…This important geometric consequence of minimality is called almost repetitivity, and a precise definition and additional details are given in §3. For a proof of this equivalence, see [FR,Theorem 3.11] and [SS,Theorem 6.5], and see also the discussion included in [KL].…”