Boundaries of Disk-Like Self-affine Tiles

On Disk-like Self-affine Tiles Arising from Polyominoes

In this paper we study a class of plane self-affine lattice tiles that are defined using polyominoes. In particular, we characterize which of these tiles are homeomorphic to a closed disk. It turns out that their topological structure depends very sensitively on their defining parameters. In order to achieve our results we use an algorithm of Scheicher and the second author which allows to dete...

Generating Self-Affine Tiles and Their Boundaries

A tile is a bounded subset of the plane, copies of which may be used to cover the whole plane without gaps or overlap. There are many sources (such as [1]) of beautiful images involving tiling, from medieval Islamic art, through Escher, to more modern work. Perhaps the simplest example of a tile, though, is a solid square, which may tile the plane in a familiar checkerboard pattern. The square ...

Hausdorff Dimension of Boundaries of Self-affine Tiles in R

We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upperand and lower-bound for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the ...

Hausdorff Dimension of Boundaries of Self-affine Tiles in Rn

Rational Self-affine Tiles

An integral self-affine tile is the solution of a set equation AT = ⋃d∈D(T +d), where A is an n× n integer matrix and D is a finite subset of Z. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Qn×n. We define rational self-affine tiles as compact subsets of the open subring R ×∏pKp of the adèle ring AK , where ...