Equicontractive weak separation property on the line does not imply convex finite type condition

Abstract

Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does not satisfy the convex finite type condition.