On the Connectedness of Self-Affine Tiles

Abstract

Let T be a self-affine tile in ℝ n defined by an integral expanding matrix A and a digit set D . The paper gives a necessary and sufficient condition for the connectedness of T . The condition can be checked algebraically via the characteristic polynomial of A . Through the use of this, it is shown that in ℝ 2 , for any integral expanding matrix A , there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.