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

We study a class of planar self-affine sets T (A, D) generated by the integer expanding matrices A with | det A| = 3 and the non-collinear digit sets D = {0, v, k Av} where k ∈ Z {0} and v ∈ R² such that {v, Av} is linearly independent. By examining the characteristic polynomials of A carefully, we prove that T (A, D) is connected if and only if the parameter k = ±1. Copyright © 2014 Springer Science+Business Media Dordrecht.