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.