A characterization of connected self-affine fractals arising from collinear digits

TL;DR

This paper provides a complete characterization of connected self-affine fractals generated by collinear digit sets from expanding integer matrices with quadratic characteristic polynomials, extending previous results for specific cases.

Contribution

It generalizes the characterization of connected self-affine fractals to broader classes of matrices and digit sets, expanding on prior work limited to specific parameters.

Findings

Complete characterization of connected self-affine fractals

Extension of previous results from |q|=3 to general cases

Conditions for connectedness based on matrix and digit set parameters

Abstract

Let $A$ be an expanding integer matrix with characteristic polynomial $f(x)=x^{2}+px+q$, and let $\mathcal{D}=\{0,1,\dots,|q|-2,|q|+m\}\mathbf{v}$ be a collinear digit set where $m\geqslant 0, {\mathbf v}\in {\mathbb Z}^2$. It is well known that there exists a unique self-affine fractal $T$ satisfying $AT=T+\mathcal{D}$. In this paper, we give a complete characterization on the connected $T$. That generalizes the previous result of $|q|=3$.