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

Abstract

In the paper, we focus on the connectedness of planar self-affine sets $T(A,\mathcal{D})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set $\mathcal{D}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v,Av\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,\mathcal{D})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\det (A)|>3$ are obtained as well.

15 pages, 10 figures

References in corpus (1)

• Digit sets for connected tiles via similar matrices I: Dilation matrices with rational eigenvalues