Boundary points of self-affine sets in $Bbb{R}$

Abstract

Let -A be an $n x n$ 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 seifaffine 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-affhre 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.