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

Abstract

We study a class of planar self-affine sets \(T(A,{\mathcal {D}})\) generated by the integer expanding matrices \(A\) with \(|\det A|=3\) and the non-collinear digit sets \({\mathcal {D}}=\{0, v, kAv\}\) where \(k\in {\mathbb {Z}}\setminus \{0\}\) and \(v\in {\mathbb {R}}^{2}\) such that \(\{v, Av\}\) is linearly independent. By examining the characteristic polynomials of \(A\) carefully, we prove that \(T(A,{\mathcal {D}})\) is connected if and only if the parameter \(k=\pm 1\).