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.

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