Sofic systems and extreme points of self-affine limit sets

Abstract

Given any iterated function system (IFS) on the plane whose affine transformations share the same contracting matrix $T$, we characterize the set of extreme points of the attractor $A$ (along the convex hull $\operatorname{conv}(A)$) in terms of a directed graph on the edges of $\operatorname{conv}(A)$. This graph corresponds to a sofic system on the coding space associated to the IFS. When $T$ is proper orthogonal or purely real, we give conditions for $\operatorname{conv}(A)$ to be a finite polygon and, under additional symmetry of the transformations, we give conditions for $A$ itself to coincide with $\operatorname{conv}(A)$.