On the connectedness of planar self-affine sets

TL;DR

This paper establishes necessary and sufficient conditions based on matrix parameters for the connectedness of planar self-affine sets generated by integral expanding matrices and digit sets, including non-consecutive collinear cases.

Contribution

It provides a complete characterization of connectedness for planar self-affine sets depending solely on matrix coefficients and digit set parameters.

Findings

Connectedness conditions depend only on $b,c,m$

Criteria established for both consecutive and non-consecutive collinear digit sets

Results applicable to a broad class of self-affine fractals

Abstract

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The necessary and sufficient conditions only depending on $b,c,m$ are given for the $T(A,\mathcal{D})$ to be connected. Moreover, we also consider the case that ${\mathcal D}$ is non-consecutively collinear.