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

• Main Author: Gerald S. Goodman
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2008-01-01
• Series: Opuscula Mathematica
• Subjects: address; Baire category; fractal; scaling function; scaling operator; strong open set condition
• Online Access: http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2834.pdf

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.