• DOI:10.1142/S0129167X04002569
• Corpus ID: 119164953

ITERATED FUNCTION SYSTEMS, REPRESENTATIONS, AND HILBERT SPACE

@article{Jorgensen2004ITERATEDFS,
  title={ITERATED FUNCTION SYSTEMS, REPRESENTATIONS, AND HILBERT SPACE},
  author={Palle E. T. Jorgensen},
  journal={International Journal of Mathematics},
  year={2004},
  volume={15},
  pages={813-832}
}

• P. Jorgensen
• Published 11 February 2004
• Mathematics
• International Journal of Mathematics

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 L2(μ). Instead, we show that it is possible to realize the maps τi 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 We prove that, under general conditions on the system (X,τi), there are isometries in ℋ(X) satisfying 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.