Connectedness of planar self-affine sets associated with non-collinear digit sets

Abstract

We study the connectedness of the planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det(A)|=3$ and a non-collinear digit set ${\mathcal D}=\{0, v, kAv\}$ where $k\in {\mathbb Z}\setminus\{0\}$ and $v\in {\mathbb Z}^2$ such that $\{v, Av\}$ is linearly independent. By checking the characteristic polynomials of $A$ case by case, we obtain a criterion concerning only $k$ to determine the connectedness of $T(A,{\mathcal{D}})$.