相关论文: Substitution Tilings and Separated Nets with Simil…
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…
We study separated nets that correspond to substitution tilings of the Euclidean space. We give a simple condition, in terms of the eigenvalues and eigenspaces of the substitution matrix, to know whether the separated net is a bounded…
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…
We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include…
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…
In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…
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…
We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation…
Cao & Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that every Archimedean tiling is the union of translates of a fixed lattice, we take a more general…
We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove…
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…
We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with…
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…
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a…
In this paper we generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We show that every such group $G$ contains separated nets that are not biLipschitz…
Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles,…
We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an…
Linearly repetitive Delone sets are shown to be rectifiable by a bi-Lipschitz homeomorphisms of the Euclidean space that sends the Delone set to the set of points with integer coordinates.