Iterated function systems, Ruelle operators, and invariant projective measures
Abstract
We introduce an harmonic analysis for iterated function systems (IFS) (X, mu) which 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 of R_W-harmonic functions. The properties of these functions determine the spectral theory of L^2(mu). For affine IFSs we establish orthogonal bases in 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.
Cite
@article{arxiv.math/0501077,
title = {Iterated function systems, Ruelle operators, and invariant projective measures},
author = {Dorin Ervin Dutkay and Palle E. T. Jorgensen},
journal= {arXiv preprint arXiv:math/0501077},
year = {2015}
}
Comments
39 pages, LaTeX "amsart" class. v2, a little clarification added. v3: new introductory section, pages 2-4; explanation of notation, below (6.13) on page 26; correction in the statement of Theorem 8.4 on page 31: S\Lambda + L contained in \Lambda; typographic, notational, and linguistic refinements, and improved cross-references, throughout the paper, and both formal and substantive corrections in the bibliography