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Iterated function systems, representations, and Hilbert space
Journal article   Open access   Peer reviewed

Iterated function systems, representations, and Hilbert space

Palle E.T Jorgensen
International journal of mathematics, Vol.15(8), pp.813-832
2004
DOI: 10.1142/S0129167X04002569
url
https://arxiv.org/pdf/math/0402175View
Open Access

Abstract

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 τ i , 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 τ i 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 τ i to operators on the Hilbert space L 2 (μ). Instead, we show that it is possible to realize the maps τ i quite generally in Hilbert spaces ℋ(X) of square-densities 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 [Formula: see text] We prove that, under general conditions on the system (X,τ i ), there are isometries [Formula: see text] in ℋ(X) satisfying [Formula: see text] the identity operator in ℋ(X). For the construction we assume that some mapping σ:X→X satisfies the conditions σ◦τ i = id X , i=1,…,N. We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

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