Invariant measures whose supports possess the strong open set property

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 \cup V \ne 0 $.In that case, the core of $ V, \check{V} = \bigcap_{n=0}^\infty $ 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(\delta V) = 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.