Connectedness of self-affine sets with product digit sets

TL;DR

This paper establishes a precise criterion for the connectedness of self-affine sets generated by specific expanding matrices and product digit sets, extending previous results in the mathematical understanding of fractal geometry.

Contribution

It provides a necessary and sufficient condition for the connectedness of self-affine sets with product digit sets, generalizing earlier findings.

Findings

Derived a complete characterization of connectedness

Extended known results to broader class of matrices

Clarified the geometric structure of self-affine sets

Abstract

Let $T(A,\mathcal{D})$ be a self-affine set generated by an expanding matrix $A=\left[\begin{array}{rr} p & 0\cr -a & q \end{array}\right]$ and a product digit set $\mathcal{D}=\{0,1,\dots,m-1\}\times \{0,1,\dots,n-1\}$. We provide a necessary and sufficient condition for the $T(A,\mathcal{D})$ to be connected, which generalizes the known results.