@Article{Goodman2008,
author = "Goodman, Gerald S.",
title = "Fractal sets satisfying the strong open set condition in complete metric spaces",
journal = "Opuscula Math.",
year = "2008",
volume = "28",
number = "4",
pages = "463--470",
doi = "http://dx.doi.org/10.7494/OpMath.2008.28.4.463",
url = "http://dx.doi.org/10.7494/OpMath.2008.28.4.463",
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",
issn = "1232-9274"
}