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

Let A be an expanding integer matrix with characteristic polynomial f(x)=x2+px+q, and let D={0,1,…,|q|−2,|q|+m}v be a collinear digit set where m⩾0, v∈Z2. It is well known that there exists a unique self-affine fractal T satisfying AT=T+D. In this paper, we give a complete characterization of connectedness of T. That generalizes the previous result for |q|=3.