Self-Affine Tiles in Rn

Self-Affine Tiles in Rn

A self-affine tile in R is a set T of positive measure with A(T) = d ∈ $ < (T + d), where A is an expanding n × n real matrix with det (A) = m on integer, and $ = {d 1 ,d 2 , . . . , d m } ⊆ R is a set of m digits. It is known that self-affine tiles always give tilings of R by translation. This paper extends the known characterization of digit sets $ yielding self-affine tiles. It proves seve...

Hausdorff Dimension of Boundaries of Self-affine Tiles in Rn

INTEGRAL SELF-AFFINE TILES IN Rn II. LATTICE TILINGS

Let A be an expanding n n integer matrix with j det(A)j = m. A standard digit set D for A is any complete set of coset representatives for Z n =A(Z n). Associated to a given D is a set T(A; D), which is the attractor of an aane iterated function system, satisfying T = d2D (T + d). It is known that T(A; D) tiles R n by some subset of Z n. This paper proves that every standard digit set D gives a...

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 ...

Geometry of Self { Affine Tiles

For a self{similar or self{aane tile in R n we study the following questions: 1) What is the boundary? 2) What is the convex hull? We show that the boundary is a graph directed self{aane fractal, and in the self{similar case we give an algorithm to compute its dimension. We give necessary and suucient conditions for the convex hull to be a polytope, and we give a description of the Gauss map of...