Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

In the paper, we focus on the connectedness of planar self-affine sets T (A, D) generated by an integer expanding matrix A with | det(A)| = 3 and a collinear digit set D = {0, 1, b}v, where b > 1 and v ∈ R2 such that {v, Av} is linearly independent. We discuss the domain of the digit b to determine the connectedness of T (A, D). Especially, a complete characterization is obtained when we restrict b to be an integer. Some results on the general case of | det(A)| > 3 are obtained as well. Copyright © 2012 Elsevier Inc. All rights reserved.