Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

We study the connectedness of planar self-affine sets T(A,D)T(A,D) generated by a matrix of the form A=[p0−aq] together with nonconsecutive and noncollinear digit sets of the form D={l0,l1,…,l|p|−1}×{m0,m1,…,m|q|−1}D={l0,l1,…,l|p|−1}×{m0,m1,…,m|q|−1}, where {l0,l1,…,l|p|−1}{l0,l1,…,l|p|−1} and {m0,m1,…,m|q|−1}{m0,m1,…,m|q|−1} are residue systems for |p||p| and |q||q| respectively. We give a necessary and sufficient condition for T(A,D)T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.