On the frequency module of the hull of a primitive substitution tiling

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

Primitive Tilings and Coherent Frames

This paper characterizes primitive substitution tiling systems for Lie groups for which every element of the associated tiling space generates coherent frames for corresponding representation spaces.

Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Substitutive number systems

In this paper, we associate a primitive substitution with a family of non-integer positional number systems with respect to the same base but with different sets of digits. In this way, we generalize the classical Dumont–Thomas numeration which corresponds to one specific case. Therefore, our concept also covers beta-expansions induced by Parry numbers. But we establish links to variants of beta-expansions such as symmetric beta-expansions, too. In other words, we unify several well-known notions of non-integer representations within one general framework. A focus in our research is set on finiteness and periodicity properties. It turns out that these characteristics mainly depend on the substitution. As a consequence we are able to relate known finiteness properties that are viewed independently yet.

Primitive substitution tilings with rotational symmetries

This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.

A Class of Hyperspace Systems and Distributional Chaos

Research on dynamical system has penetrated into many problems of agricultural production, such as prediction of corn yield, analysis on operational situation of irrigation district and research on ecological difference equation. In this paper, we investigated dynamical properties for non-primitive substitution and the set-valued maps induced by the substitution. We proved In two cases that the hyperspace systems induced by non-primitive substitution are not distributional chaotic.

Integer Semigroups Associated with Dumont-Thomas Numeration Systems

Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers.

C*-algebras arising from substitutions

In this paper, we introduce a C*-algebra associated with a primitive substitution. We show that when σ is proper, the C*-algebra is simple and purely infinite and contains the associated Cuntz–Krieger algebra and the crossed product C*-algebra of the corresponding Cantor minimal system. We calculate the K-groups.

Kolakoski-(3, 1) Is a (Deformed) Model Set

Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding biin finite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.