Boundary Points of Self-Affine Sets in R


Abstract: Let A be an n \times n expanding matrix with integer entries and D = \{0, d_1, ... , d_{N-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.

Keywords: Self-affine sets, boundary points, strict product-form digits

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