Related papers: Pure discrete spectrum for a class of one-dimensio…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
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…
For any $\lambda>2$, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor $\lambda$. In particular, we obtain the first class of examples of substitutive systems with…
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…
The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the…
A one-dimensional discrete Stark Hamiltonian with a continuous electric field is constructed by extension theory methods. In absence of the impurities the model is proved to be exactly solvable, the spectrum is shown to be simple,…
This paper is concerned with the study of diffraction intensities of a relevant class of binary Pisot substitutions via exponential sums. Arithmetic properties of algebraic integers are used to give a new and constructive proof of the fact…
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…
We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on…
The diffraction spectrum of the dart-rhombus random tiling of the plane is derived in rigorous terms. Using the theory of dimer models, it is shown that it consists of Bragg peaks and an absolutely continuous diffuse background, but no…
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…
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…
A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes…
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…
By generalising Rudin's construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We…
In this paper, we generalize and develop results of Queffelec allowing us to characterize the spectrum of an aperiodic substitution in $\mathbb{Z}^d$ by describing the Fourier coefficients of mutually singular measures of pure type giving…
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $\beta$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a…
Suppose a set of prototiles allows $N$ different substitution rules. In this paper we study tilings of $\mathbb{R}^d$ constructed from random application of the substitution rules. The space of all possible tilings obtained from all…
In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from…
We show equivalence of pure point diffraction and pure point dynamical spectrum for measurable dynamical systems build from locally finite measures on locally compact Abelian groups. This generalizes all earlier results of this type. Our…