Constructive Approximation

, Volume 19, Issue 3, pp 373–397

Self-Affine Sets and Graph-Directed Systems


DOI: 10.1007/s00365-002-0515-0

Cite this article as:
He, Lau & Rao Constr. Approx. (2003) 19: 373. doi:10.1007/s00365-002-0515-0


Abstract. A self-affine set in Rn is a compact set T with A(T)= ∪ d∈ D(T+d) where A is an expanding n× n matrix with integer entries and D={d1, d2,···, dN} ⊂Zn is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if To). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.

Key words

Attractor Boundary Contraction Iterated function system Hausdorff dimension Graph-directed construction Overlapping Tiles Self-affine sets Self-similar sets 

AMS Classification

Primary, 52C20, 52C22 Secondary, 28A80. 

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • He
    • 1
  • Lau
    • 2
  • Rao
    • 3
  1. 1.Department of Mathematics The Chinese University of Hong Kong Shatin Hong Kong xghe@sina.comHK
  2. 2.Department of Mathematics The Chinese University of Hong Kong Shatin Hong Kong
  3. 3.Department of Mathematics Wuhan University Wuhan P. R. China

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