On the Convex Hulls of Self-Affine Fractals

Abstract

Suppose that the set ${\mathcal{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, \cdots, d_q\} \subset {\Bbb R}^n$, there exists a unique nonempty compact set $F=F({\mathcal{T}},D)$ satisfying $ F = \bigcup _{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=\cdots =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.