Dergiler / Turkish Journal of Mathematics / 2003 / Cilt: 27 - Sayı: 2
Boundary Points of Self-Affine Sets in R
- Sayfa
- 273–281
- DOI
- —
Özet
Let A be an n \times n expanding matrix with integer entries and D = \{0, d1, ... , dN-1 \} \subseteq {\Bbb{Z}}n be a set of N distinct vectors, called an N-digit set. The unique non-empty compact set T = T(A,D) satisfying AT = T + D is called a self-affine set. If T has positive Lebesgue measure, it is called a self-affine region. In general, it is not clear how to determine a point to be on the boundary of a self-affine region. In this note, we consider one-dimensional self-affine regions T and present a simple approach to get increasing subsets of the boundary of T. This approach also gives a characterization of strict product-form digit sets introduced by Odlyzko.
Abstract
Let A be an n \times n expanding matrix with integer entries and D = \{0, d1, ... , dN-1 \} \subseteq {\Bbb{Z}}n be a set of N distinct vectors, called an N-digit set. The unique non-empty compact set T = T(A,D) satisfying AT = T + D is called a self-affine set. If T has positive Lebesgue measure, it is called a self-affine region. In general, it is not clear how to determine a point to be on the boundary of a self-affine region. In this note, we consider one-dimensional self-affine regions T and present a simple approach to get increasing subsets of the boundary of T. This approach also gives a characterization of strict product-form digit sets introduced by Odlyzko.