The connectedness of some two-dimensional self-affine sets

Abstract

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. © 2014 Elsevier Inc.