On the Connectedness of Self-Affine Tiles
نویسندگان
چکیده
منابع مشابه
On the Connectedness of Self-affine Tiles
Let T be a self-affine tile in 2n defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in 2#, for any integral expanding matrix A, there exists a digit set D such that the correspondi...
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Little is known about the connectedness of self-affine tiles inRn . In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the height reducing property, on expanding polynomials (i.e., all the roots have moduli > 1), we show that all such tiles in Rn, n ≤ 3, are connected. The problem is...
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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 ...
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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...
متن کاملSelf-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...
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ژورنال
عنوان ژورنال: Journal of the London Mathematical Society
سال: 2000
ISSN: 0024-6107
DOI: 10.1112/s002461070000106x