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.

Full text (pdf)

• Gerald S. Goodman
• via Dazzi, 11, 50141 Firenze, Italy
• Received: 2008-01-07.
• Revised: 2008-04-07.
• Accepted: 2008-04-21.

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.

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.