Opuscula Math. 28, no. 4 (2008), 471-480

 
Opuscula Mathematica

Invariant measures whose supports possess the strong open set property

Gerald S. Goodman

Abstract. Let \(X\) be a complete metric space, and \(S\) the union of a finite number of strict contractions on it. If \(P\) is a probability distribution on the maps, and \(K\) is the fractal determined by \(S\), there is a unique Borel probability measure \(\mu _P\) on \(X\) which is invariant under the associated Markov operator, and its support is \(K\). The Open Set Condition (OSC) requires that a non-empty, subinvariant, bounded open set \(V \subset X\) exists whose images under the maps are disjoint; it is strong if \(K \cap V \neq \emptyset\). In that case, the core of \(V\), \(\check{V}=\bigcap_{n=0}^{\infty} S^n (V)\), is non-empty and dense in \(K\). Moreover, when \(X\) is separable, \(\check{V}\) has full \(\mu _P\)-measure for every \(P\). We show that the strong condition holds for \(V\) satisfying the OSC iff \(\mu_P (\partial V) =0\), and we prove a zero-one law for it. We characterize the complement of \(\check{V}\) relative to \(K\), and we establish that the values taken by invariant measures on cylinder sets defined by \(K\), or by the closure of \(V\), form multiplicative cascades.

Keywords: core, fractal, fractal measure, invariant measure, scaling function, scaling operator, strong open set condition, zero-one law.

Mathematics Subject Classification: 28A80, 60F20.

Full text (pdf)

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

Cite this article as:
Gerald S. Goodman, Invariant measures whose supports possess the strong open set property, Opuscula Math. 28, no. 4 (2008), 471-480

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.