Fractal sets satisfying the strong open set condition in complete metric spaces

Abstract

Let $K$ be a Hutchinson fractal in a complete metric space $X$, invariant under the action $S$ of the union of a finite number of Lipschitz contractions. The Open Set Condition states that $X$ has a non-empty subinvariant bounded open subset $V$, whose images under the maps are disjoint. It is said to be strong if $V$ meets $K$. We show by a category argument that when $K \not\subset V$ and the restrictions of the contractions to $V$ are open, the strong condition implies that $\check{V}=\bigcap_{n=0}^{\infty} S^n(V)$, termed the core of $V$, is non-empty. In this case, it is an invariant, proper, dense, subset of $K$, made up of points whose addresses are unique. Conversely, $\check{V}\neq \emptyset$ implies the SOSC, without any openness assumption.