On the connectedness of self-affine attractors

Abstract

Let T = T(A, D) be a self-affine attractor in \( \mathbb{R}^n \) defined by an integral expanding matrix A and a digit set D. In the first part of this paper, in connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \( \{0, 1,\ldots, |\det(A)| - 1\} \) . It is shown that in \( \mathbb{R}^3 \) and \( \mathbb{R}^4 \) , for any integral expanding matrix A, T(A, D) is connected.