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

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 $m⩾0,\mathbf{v}\in {\mathbb{Z}}^2$. 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 $|q|=3$.

17 pages, 3 figures