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,..., vertical bar q vertical bar - 1, j = 0, 1,..., vertical bar p vertical bar -1} where s,t not equal 0 and p,q is an element of Z with 3 <= vertical bar p vertical bar + 1 < vertical bar q vertical bar < 2 vertical bar p vertical bar - 1. The other is generated by matrix A = ((p0)(-aq)) and digit set D = {(is, (di + j)t)(T) : i = 0, 1, ...,vertical bar p vertical bar - 1 ,j = 0, 1,..., vertical bar q vertical bar - 1}, where s,t not equal 0, and p, q, d is an element of Z with vertical bar q vertical bar, vertical bar q vertical bar >= 2. The sufficient or necessary conditions for their connectedness are revealed. (C) 2014 Elsevier Inc. All rights reserved.