1 1 Fe b 20 04 Iterated function systems , representations , and Hilbert space

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1, . . . , N , in some Hausdorff space Y . There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y . Typically, some kind of contractivity property for the maps τi is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τi to operators on the Hilbert space L 2 (μ). Instead, we show that it is possible to realize the maps τi quite generally in Hilbert spaces H (X) of squaredensities on X. The elements in H (X) are equivalence classes of pairs (φ, μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫ X |φ| 2 dμ <∞. We say that (φ, μ) ∼ (ψ, ν) if there is a positive Borel measure λ such that μ << λ, ν << λ, and φ √ dμ dλ = ψ √ dν dλ , λ a.e. on X. We prove that, under general conditions on the system (X, τi), there are isometries Si : (φ, μ) 7−→ ( φ ◦ σ, μ ◦ τ i ) in H (X) satisfying ∑i=1 SiS i = I = the identity operator in H (X). For the construction we assume that some mapping σ : X → X satisfies the conditions σ ◦ τi = idX , i = 1, . . . , N . We further prove that this representation in the Hilbert space H (X) has several universal properties. Preprint submitted to Elsevier Science 13 April 2008