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To describe some fractal properties of a self–similar set or measure, such as the Hausdorff dimension and the multifractal spectrum, it is useful that it satisfy the strong open set condition, which means there is an open set satisfying the open set condition and, additionally, a part of the self–similar set must meet the open set. It is known that in the non–random case the strong open set condition and the open set condition are equivalent. This paper treats the random case. If the open set condition is assumed, we show that there is a random open set satisfying the strong open set condition. Further, we give an application to multifractal analysis of the random self–similar fractal. 1. Notation In this section we recall the definition of self–similar sets and self–similar measures, and give some properties. A more detailed introduction and the proofs of the properties may be found in [PZ] and in [AP]. Let K ⊂ R be a fixed compact set with K = intK. We are given a positive integer N ≥ 2 and a probability measure μ on Sim × [0, 1] , where Sim is the space of all similarities of R equipped with the usual topology of uniform convergence on compact sets. In this paper we assume the following. Assumption 1. (I) ∫ N ∑ i=1 pi μ(d(S1, . . . , SN ; p1, . . . , pN )) = 1, (II) Si(intK) ⊂ intK for all i = 1, . . . , N and Si(intK) ∩ Sj(intK) = ∅ for all i 6= j for μ–a. a. (S1, . . . , SN ; p1, . . . , pN), (III) there exist pmin > 0 and rmin > 0 such that pi ≥ pmin and LipSi ≥ rmin for all i = 1, . . . , N and μ–a. a. (S1, . . . , SN ; p1, . . . , pN ). Condition (II) is known as the open set condition. In the sequel we often make use of symbolic dynamics. Let Σ = {1, . . . , N}N be the code space over the indices 1, . . . , N , Σn = {1, . . . , N}n the space of all sequences of length n, and Σ∗ = ⋃∞ n=0 Σn. For τ ∈ Σn denote by |τ | = n the length of τ , and by τ |k the truncation of τ to the first k entries, k ≤ n. For τ ∈ Σ∗ and σ ∈ Σ∗ ∪ Σ we write τ ≺ σ if there is a σ′ ∈ Σ∗ ∪ Σ with τ = σσ′. Further, let [τ ] = {σ ∈ Σ : τ ≺ σ} be the cylinder sets in Σ, τ ∈ Σ∗. Received by the editors January 16, 1996 and, in revised form, February 7, 1996. 1991 Mathematics Subject Classification. Primary 28A80; Secondary 60D05, 60G57.

The strong open set condition in the random case