Related papers: Determining pure discrete spectrum for some self-a…
We introduce a procedure for establishing pure discrete spectrum for substitution tiling systems of Pisot family type and illustrate with several examples.
We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the R-action on the corresponding tiling space has pure discrete spectrum. As a…
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 consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out…
Overlap coincidence in a self-affine tiling in $\R^d$ is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in $\R^d$ and find an…
We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a…
A substitution $\vp$ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot $\vp$ that satisfies a no cycle condition and for…
It is proved that every pseudo-self-affine tiling in R^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoi tessellations is a corollary. Previously, these…
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…
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…
This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete…
As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the…
In this paper, we study 2-port networks and introduce new concepts of voltage drop and $\Pi$-equivalence. The main result is that each planar network is $\Pi$-equivalent to a network with no more than 5 edges. This implies that if an…
A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and…
We define a generic algorithmic framework to prove pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of…
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 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…
The {\alpha}-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - {\alpha}, for a fixed {\alpha} in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their…
This paper shows that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. In other words, the Poisson relation, which asserts that the singular…
We prove the Pisot Conjecture for beta-substitutions: If beta is a Pisot number, the tiling dynamical system associated with the beta-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic…