# Connectedness of a Class of Two-dimensional Self-affine Tiles Associated With Triangular Matrices

#### Abstract

We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A = [(p)(-a) (0)(q)] together with nonconsecutive and noncollinear digit sets of the form D = {l(0), l(1), ... , l(vertical bar p vertical bar-1)} x {m(0) m(1), ... , m(vertical bar q vertical bar-1)}, where {l(0), l(1), ... , l(vertical bar p vertical bar-1)} and {m(0) m(1), ... , m(vertical bar q vertical bar-1)} are residue systems for vertical bar p vertical bar and vertical bar q vertical bar respectively. We give a necessary and sufficient condition for T(A, D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets

*This paper has been withdrawn.*