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
  • Received: 2008-01-07.
  • Revised: 2008-04-07.
  • Accepted: 2008-04-21.
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Gerald S. Goodman, Fractal sets satisfying the strong open set condition in complete metric spaces, Opuscula Math. 28, no. 4 (2008), 463-470

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