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 ∈ R 2 such that { v , A v } 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.