Iterated function systems, representations, and Hilbert space

In this paper, we axe 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 tau(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 subset of Y. Typically, some kind of contractivity property for the maps tau(i) is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure p available which allows us to pass the point-maps tau(i) to operators on the Hilbert space L-2(mu). Instead, we show that it is possible to realize the maps tau(i) quite generally in Hilbert spaces H(X) of square-densities on X. The elements in H(X) axe equivalence classes of pairs (phi, mu), where phi is a Borel function on X, mu is a positive Borel measure on X, and integral(X) \phi\(2) dmu < infinity. We say that (phi, mu) similar to (phi, nu) if there is a positive Borel measure lambda such that mu much less than lambda, nu much less than lambda, and phirootdmu/dlambda = psirootdnu/dlambda, lambda a.e. on X. We prove that, under general conditions on the system (X, tau(i)), there are isometries S-i: (phi, mu) --> (phi (.) sigma, mu (.) tau(i)(-1)) in H(X) satisfying Sigma(i=1)(N) SiSi* = I = the identity operator in H(X). For the construction we assume that some mapping sigma: X --> X satisfies the conditions sigma (.) tau(i) = id(X), i = 1,...N. We further prove that this representation in the Hilbert space H(X) has several universal properties.