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.

Topologically mixing tiling of generated by a generalized substitution

This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.

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.

FRACTAL TILINGS FROM SUBSTITUTION TILINGS

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

Cut-and-Project Schemes for Pisot Family Substitution Tilings

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

Cut-and-Project Schemes for Pisot Family Substitution Tilings

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].