A family of measures associated with iterated function systems

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Let (X,d) be a compact metric space, and let an iterated function system (IFS) be given on X, i.e., a finite set of continuous maps σᵢ: X→ X, i=0,1,..., N-1. The maps σᵢ transform the measures μ on X into new measures μᵢ. If the diameter of σᵢ₁∘ >... ∘ σᵢₖ(X) tends to zero as k→ ∞, and if pᵢ>0 satisfies ∑ᵢpᵢ=1, then it is known that there is a unique Borel probability measure μ on X such that μ=∑ᵢpᵢ μᵢ *. In this paper, we consider the case when the pᵢs are replaced with a certain system of sequilinear functionals. This allows us to study the variable coefficient case of (*), and moreover to understand the analog of (*) which is needed in the theory of wavelets.