We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct bord…
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 t…
We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $θ>1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $θ$ being a Pisot number.
The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of $\mathbb{Z}^{2}$. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalisation to bijective block substitutions of trivial height on $\mathbb{Z}^{d}$.
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.
Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all substitution tilings showing tiles in infinitely many orientations.
We consider substitution tilings in R^d that give rise to point sets that are not bounded displacement (BD) equivalent to a lattice and study the cardinality of BD(X), the set of distinct BD class representatives in the corresponding tiling space X. We prove a sufficient condition under which the tiling space contains continuously many distinct BD classes and present such an example in the plane. In particular, we show here for the first time that this cardinality can be greater than one.
In this paper we show that the function defined by a primitive, star shaped substitu- tion tiling of the plane, can be realized as a Jacobian of a biLipschitz homeomorphism of R^2. In particular it holds for any Penrose tiling.
In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use [n/2] rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain parallelograms.
A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with 11-fold symmetry appearing in the literature.
Two results about equidistribution of tile orientations in primitive substitution tilings are stated, one for finitely many, one for infinitely many orientations. Furthermore, consequences for the associated diffraction spectra and the dynamical systems are discussed.
There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map $Q$. Our first result gives a partial answer to a question of Lagarias and Wang: we char…
The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.
General substitution rules for non-periodic rhomb tilings are derived. From the requirement that all substitution tiles consist of a discrete number of prototiles, it follows that a substitution tile with angle s*pi/n must be built out of pairs of prototiles with angles (s+t)*pi/n and (s-t)*pi/n, except if t=0. In addition, we require that a discrete number of prototile edges must fit between the beginning and endpoint of the substitution tile edge. By comparing the total area of the discrete number of prototiles constituting the substitution tile and the total area derived from the assumed ed…
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates onl…
We describe a computer algorithm that searches for substitution rules on a set of triangles, the angles of which are all integer multiples of π/n. We find new substitution rules admitting 7-fold rotational symmetry at many different inflation factors.
A separated net is a set of points which is relatively dense and uniformly discrete (another name for a Delone set). We are dealing with tilings and separated nets in Euclidean spaces and with the question whether a given separated net is biLipschitz to the integer lattice. In this paper we show, as an answer to a question of Burago and Kleiner, that the net that is obtained form the Penrose tiling is biLipschitz to the integer lattice.
Overlap coincidence is an equivalent criterion to pure discrete spectrum of the dynamics of self affine tilings. In the case of one dimension, strong coincidence on m letter irreducible substitution has been introduced in Dekking (1978) and Arnoux and Ito (2001) which implies that the system is metrically conjugate to a domain exchange. However being a domain exchange does not imply the property of pure discrete spectrum of the tiling dynamics. The relation between two coincidences has not been established completely. In this paper we generalize strong coincidence to higher dimensions and show…
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) whose vertices are supported by the $2n$-th cyclotomic field $\mathbb{Q}\left(ζ_{2n}\right)$ is extended to cases with Dense Tile Orientations (DTO). It is shown that every CAST with DTO has an inflation multiplier $η$ with irrational argument so that $\frac{kπ}{2n}\neq\arg\left(η\right)\notin\mathbb{πQ}$. The minimal inflation multiplier $η_{min.irr.}$ is discussed for $n\geqq2$. Examples of CASTs with DTO, minimal inflation multiplier $η_{min.irr.}$ and individual dihedral symmetry $D_{2n}$ are introduced for $n\in\left\{ 2,3,4,5,…
We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.
By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of planar tilings generated by boundary substitutions due to the paper [17] by Kenyon.
We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap distributions, and slope gap distributions for the Ammann Chair tiling, the recently discovered fifteenth pentagonal tiling, and a few pertinent tilings related to these famous examples. We also consider the spatial statistics of generalized Ulam sets in two dimensions. Additionally, we carefully prove a tight asymptotic formula for the time steps in which Ulam …
We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile rearrangements in the substitution rules associated to edge flips.
In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $λ\ZZ^d$ for some positive $λ$. This condition includes primitive Pisot substitution tilings but also concern…
The paper studies ways in which the sets of a partition of a lattice in $\RR^n$ become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in $\RR^n$ gives rise to regular model sets (based on $p$-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the $n-$dimensional chair tiling and the sphinx tiling are pure point diffractive.
The nature of this paper is twofold: On one hand, we will give a short introduction and overview of the theory of model sets in connection with nonperiodic substitution tilings and generalized Rauzy fractals. On the other hand, we will construct certain Rauzy fractals and a certain substitution tiling with interesting properties, and we will use a new approach to prove rigorously that the latter one arises from a model set. The proof will use a duality principle which will be described in detail for this example. This duality is mentioned as early as 1997 by Gelbrich in the context of iterated…
We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.
Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot--Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier matrix cocycle in internal space. This cocycle leads to a transfer matrix equation and thus to a closed expression of matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a c…
