On the connectedness of planar self-affine sets

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The necessary and sufficient conditions only depending on $b,c,m$ are given for the $T(A,\mathcal{D})$ to be connected. Moreover, we also consider the case that ${\mathcal D}$ is non-consecutively collinear.