Opuscula Mathematica

Opuscula Math.
, no. 4
 (), 463-470
Opuscula Mathematica

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.
Keywords: address, Baire category, fractal, scaling function, scaling operator, strong open set condition.
Mathematics Subject Classification: 28A80, 54E40, 54E52.
Cite this article as:
Gerald S. Goodman, Fractal sets satisfying the strong open set condition in complete metric spaces, Opuscula Math. 28, no. 4 (2008), 463-470
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.