TOPOLOGICAL STRUCTURE OF SELF-SIMILAR SETS

Abstract

We consider the attractor T of injective contractions f 1 , …, f m on R 2 which satisfy the Open Set Condition. If T is connected, then T's interior T° is either empty or has no holes, and T's boundary ∂T is connected; if further T° is non-empty and connected, then ∂T is a simple closed curve, thus T is homeomorphic to the unit disk {x∈R 2 : |x|≤1}.