Opuscula Math. 28, no. 4 (2008), 463-470

Opuscula Mathematica

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

Gerald S. Goodman

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.

Keywords: address, Baire category, fractal, scaling function, scaling operator, strong open set condition.

Mathematics Subject Classification: 28A80, 54E40, 54E52.

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• Gerald S. Goodman
• via Dazzi, 11, 50141 Firenze, Italy
• Revised: 2008-04-07.
• Accepted: 2008-04-21.