Iterated function systems, Ruelle operators, and invariant projective measures

Abstract

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 X comes with a finite-to-one endomorphism r : X → X r\colon X\rightarrow X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d \mathbb {R}^d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B , L B, L in R d \mathbb {R}^d of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) ( X , μ ) (X, \mu ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W W on X X . From W W we define a transition operator R W R_W acting on functions on X X , and a corresponding class H H of continuous R W R_W -harmonic functions. The properties of the functions in H H are analyzed, and they determine the spectral theory of L 2 ( μ ) L^2(\mu ) . For affine IFSs we establish orthogonal bases in L 2 ( μ ) L^2(\mu ) . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d \mathbb {R}^d .