Related papers: Digit sets for connected tiles via similar matrice…

Let $T(A,\mathcal{D})$ be a self-affine set generated by an expanding matrix $A=\left[\begin{array}{rr} p & 0\cr -a & q \end{array}\right]$ and a product digit set $\mathcal{D}=\{0,1,\dots,m-1\}\times \{0,1,\dots,n-1\}$. We provide a…

We consider digit systems $(A,\mathcal{D})$, where $ A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F}…

We study self-similar attractors in the space $\mathbb{R}^d$, i.e., self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix $M$ of the linear part of affine…

The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory…

Let $A$ be a $d \times d$ matrix with rational entries which has no eigenvalue $\lambda \in \mathbb{C}$ of absolute value $|\lambda| < 1$ and let $\mathbb{Z}^d[A]$ be the smallest nontrivial $A$-invariant $\mathbb{Z}$-module. We lay down a…

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…

Let $R$ be an $n\times n$ expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the integral self-affine set…

An iterated function system $\Phi$ consisting of contractive similarity mappings has a unique attractor $F \subseteq \mathbb{R}^d$ which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the…

Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a…

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The…

In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a…

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names:…

We study the connectedness of the planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det(A)|=3$ and a non-collinear digit set ${\mathcal D}=\{0, v, kAv\}$ where $k\in {\mathbb Z}\setminus\{0\}$…

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we…

A set is said to tile the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For…

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then…

In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in…

An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of…

Let $A$ be an $n \times n$ matrix with rational entries and let \[ \mathbb{Z}^n[A] := \bigcup_{k=1}^{\infty} \left( \mathbb{Z}^n + A\mathbb{Z}^n + \dots + A^{k-1}\mathbb{Z}^n\right) \] be the minimal $A$-invariant $\mathbb{Z}$-module…

We introduce rational Dyck tilings, or $(a,b)$-Dyck tilings, and study them by the decomposition into $(1,1)$-Dyck tilings. This decomposition allows us to make use of combinatorial models for $(1,1)$-Dyck tilings such as the Hermite…