On the Convex Hulls of Self-Affine Fractals

Ibrahim Kirat; Ilker Kocyigit

paper

On the Convex Hulls of Self-Affine Fractals

arXiv:1504.07396

Abstract

Suppose that the set $T = {T_{1}, T_{2}, ..., T_{q}}$ of real $n \times n$ matrices has joint spectral radius less than $1$ . Then for any digit set $D = {d_{1}, \dots, d_{q}} \subset R^{n}$ , there exists a unique nonempty compact set $F = F (T, D)$ satisfying $F = ⋃_{j = 1}^{q} T_{j} (F + d_{j})$ , which is called a self-affine fractal. We consider an existing criterion for the convex hull of $F$ to be a polytope, which is due to Kirat and Kocyigit. In this note, we strengthen our criterion for the case $T_{1} = T_{2} = \dots = T_{q}$ . More specifically, we give an upper bound for the number of steps needed for deciding whether the convex hull of $F$ is a polytope or not. This improves our earlier result on the topic.