Iterated function systems, Ruelle operators, and invariant projective measures
Abstract
We introduce a Fourierbased 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 finitetoone endomorphism r: X→ X which is onto but not onetoone. (2) In the case of affine Iterated Function Systems (IFSs) in {R}^d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B, L in {R}^d 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 R_W acting on functions on X , and a corresponding class H of continuous R_W harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L^2(μ) . For affine IFSs we establish orthogonal bases in L^2(μ) . 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 .
 Publication:

Mathematics of Computation
 Pub Date:
 December 2006
 DOI:
 10.1090/S0025571806018618
 arXiv:
 arXiv:math/0501077
 Bibcode:
 2006MaCom..75.1931D
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Spectral Theory;
 42C10;
 42C40;
 47D07
 EPrint:
 39 pages, LaTeX "amsart" class. v2, a little clarification added. v3: new introductory section, pages 24