On the connectedness of planar self-affine sets

In this paper, we consider the connectedness of planar self-affine set T(A,D)T(A,D) arising from an integral expanding matrix A   with characteristic polynomial f(x)=x2+bx+cf(x)=x2+bx+c and a consecutive collinear digit set D={0,1,…,m}vD={0,1,…,m}v. The necessary and sufficient conditions only depending on b,c,mb,c,m are given for the T(A,D)T(A,D) to be connected. Moreover, we also consider the case that DD is non-consecutively collinear.