The connectedness of some two-dimensional self-affine sets

In the paper, we mainly discuss the connectedness of two kinds of self-affine sets. One is generated by matrix A=(p0−aq) and digit set D={(is,jt)T:i=0,1,…,|q|−1,j=0,1,…,|p|−1}, where s,t≠0 and p,q∈Z with 3≤|p|+1<|q|<2|p|−1. The other is generated by matrix A=(p0−aq) and digit set D={(is,(di+j)t)T:i=0,1,…,|p|−1,j=0,1,…,|q|−1}, where s,t≠0, and p,q,d∈Z with |p|,|q|≥2. The sufficient or necessary conditions for their connectedness are revealed.