Self-similar Tilings Research Articles

Tiling iterated function systems

This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be fractals and the tiled set maybe a complicated unbounded subset of RM.

Product-form Hadamard triples and its spectral self-similar measures

In a previous work by Łaba and Wang, it was proved that whenever there is a Hadamard triple (N,D,L), then the associated one-dimensional self-similar measure μN,D generated by maps N−1(x+d) with d∈D, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples (N,Ak,Lk) by putting each triple into different scales of N. Our main result is to prove that the associated self-similar measure μN,D is a spectral measure. This result allows us to show that product-form self-similar tiles are spectral sets as long as the tiles in the group ZN obey the Coven-Meyerowitz (T1), (T2) tiling condition. Moreover, we show that all self-similar tiles with N=pαq are spectral sets, answering a question by Fu, He and Lau in 2015. Finally, our results allow us to offer new singular spectral measures not generated by a single Hadamard triple. Such new examples allow us to classify all spectral self-similar measures generated by four equi-contraction maps, which will appear in a forthcoming paper.

On arithmetic progressions in non-periodic self-affine tilings

AbstractWe study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$ .

FRACTAL REP TILES OF ℝ2 AND ℝ3 USING INTEGER MATRICES

This paper is devoted to the study of self-similar fractal rep tiles of the Euclidean planes [Formula: see text] and [Formula: see text] using integer matrices. We construct many examples appearing in the literature, as well as new examples by integer matrices. Several exotic and radially symmetric self-similar rep tiles and their variations are obtained with change of bases extending the work by Bandt and others. We also present a few examples of polyhedral self-similar rep tiles including variations.

Self-Similar Polygonal Tiling

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting and to provide a novel method for the construction of such polygonal tilings.

Topological properties of self-similar fractals with one parameter

In this paper, we study two classes of planar self-similar fractals Tε with a shifting parameter ε. The first one is a class of self-similar tiles by shifting x-coordinates of some digits. We give a detailed discussion on the disk-likeness (i.e., the property of being a topological disk) in terms of ε. We also prove that Tε determines a quasi-periodic tiling if and only if ε is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for Tε to be connected.

Spectral Measure at Zero for Self-Similar Tilings

The goal of this paper is to study the action of the group of translations over self-similar tilings in the euclidian space $\mathbb{R}^d$. It investigates the behaviour near zero for spectral measures for such dynamical systems. Namely the paper gives a Holder asymptotic expansion near zero for these spectral measures. It is a generalization to higher dimension of a result by Bufetov and Solomyak who studied self similar-suspension flows for substitutions. The study of such asymptotics mostly involves the understanding of the deviations of some ergodic averages.

Topological substitution for the aperiodic Rauzy fractal tiling

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.

Pinwheel tiling fractal graph- a notion to edge cordial and cordial labeling

<p>A fractal is a complex geometric figure that continues to display self-similarity when viewed on all scales. Tile substitution is the process of repeatedly subdividing shapes according to certain rules. These rules are also sometimes referred to as inflation and deflation rule. One notable example of a substitution tiling is the so-called Pinwheel tiling of the plane. Many examples of self-similar tiling are made of fractiles: tiles with fractal boundaries. . The pinwheel tiling was the first example of this sort. There are many as such as family of tiling fractal curves, but for my study, I have considered this Pinwheel and its two intriguing Pinwheel properties of tiling fractals. These fractals have been considered as a graph and the same has been viewed under the scope of cordial and edge cordial labeling to apply this concept for further study in Engineering and science applications.</p>

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and Lt are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.

Spectrality of Self-Similar Tiles

We call a set $$K \subset {\mathbb {R}}^s$$ with positive Lebesgue measure a spectral set if $$L^2(K)$$ admits an exponential orthonormal basis. It was conjectured that K is a spectral set if and only if K is a tile (Fuglede’s conjecture). Although the conjecture was proved to be false in $${\mathbb {R}}^s$$ , $$s\ge 3$$ (Kolountzakis in Forum Math 18:519–528, 2006; Tao in Math Res Lett 11:251–258, 2004), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties of a class of self-similar tiles K in $${\mathbb {R}}$$ that has a product structure on the associated digit sets. We show that strict product-form tiles and the associated modulo product-form tiles are spectral sets. For the converse question, we give a pilot study for the self-similar set K generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality and verify Fuglede’s conjecture for this special case. We also make use of this case to illustrate the theorems and discuss some questions that arise in the general situation.

Digit sets for self-similar tiles

A new class of digit sets, which we called very weak product-form digit sets, is introduced and it is shown that they are tile digit set for self-similar tiles. This extends previous results about product-form and weak product-form digit sets.

Self-similar aperiodic patterns with 9-fold symmetry

We show there exists one planar self-similar tiling with D9 symmetry. The tiles are the regular hexagon and the rhombi with 20, 40, or 80 degree angles. It has two almost isometric tilings with D3 symmetry.

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.

Fractal tiles and quasidisks

We show that the interiors of a large class of planar disklike self-similar tiles are quasidisks. We also show that in general, the interior of a disklike self-affine tile may not be a John domain and thus may not be a quasidisk.

Fusion: a General Framework for Hierarchical Tilings

One well studied way to construct quasicrystalline tilings is via in ate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings the Penrose, octagonal, and pinwheel tilings are famous examples. We present a di erent model for generating hierarchical tilings we call fusion rules . In ate-and-subdivide rules are a special case of fusion rules, but general fusion rules are more exible and allow for defects, changes in geometry, and even constrained randomness. A condition that produces homogeneous structures and a method for computing frequency for fusion tiling spaces are discussed.

Second-order ergodic theorem for self-similar tiling systems

AbstractWe consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space ${ \mathbb{R} }^{d} $. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule.

Spectral structure of digit sets of self-similar tiles on ${\mathbb R}^1$

We study the structure of the digit sets D for the integral selfsimilar tiles T (b,D) (we call such a D a tile digit set with respect to b). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial PD , i.e., the zeros of PD on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomials, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order.

Limit Theorems for Self-Similar Tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.

On Self-Tiling Tile Sets

SummaryA novel species of self-similar tilings is introduced. Results divide naturally into two categories, involving either polyforms or non-polyforms. Some eye-catching findings are presented in what promises to be a rich field of research. The discovery of a class of triangle pairs showing arresting properties will undoubtedly surprise and appeal to many, not least those readers already conversant with rep-tiles.