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

TL;DR

This paper investigates the conditions under which planar self-affine sets generated by specific matrices and collinear digit sets are connected, providing a complete characterization for integer digit parameters and extending results to larger determinants.

Contribution

It offers a complete characterization of connectedness for self-affine sets with a particular digit set structure, especially when the digit parameter is an integer.

Findings

Connectedness depends on the digit parameter b.

Complete characterization for integer b.

Extended results for |det(A)| > 3.

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.