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Invariant measures whose supports possess the strong open set property

Goodman G. S.

Opuscula Mathematica

2008

Vol. 28, no. 4

471-480

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 µp 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⊂ X exists whose images under the maps are disjoint; it is strong if K ∩ V ≠ 0.In that case, the core of [formula] is non-empty and dense in K. Moreover, when X is separable, V has full µp-measure for every P. We show that the strong condition holds for V satisfying the OSC iff µp(ϑV) = 0, and we prove a zero-one law for it. We characterize the complement of 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.