Opuscula Mathematica

Opuscula Math.
, no. 4
 (), 471-480
Opuscula Mathematica

Invariant measures whose supports possess the strong open set property

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.
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
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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