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

TL;DR

This paper investigates the connectedness of certain planar self-affine sets generated by specific matrices and digit sets, providing a criterion based solely on the parameter k for determining connectedness.

Contribution

It introduces a new criterion for connectedness of self-affine sets with non-collinear digit sets, based on the characteristic polynomials of the generating matrix.

Findings

Connectedness depends only on the parameter k.

A case-by-case analysis of characteristic polynomials yields the criterion.

The criterion simplifies the determination of connectedness for these 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 $T(A,{\mathcal{D}})$.