Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in Rd, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B,L in Rd of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) (X,μ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator RW acting on functions on X, and a corresponding class H of continuous RW harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L2(μ). For affine IFSs we establish orthogonal bases in L2(μ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in Rd.