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\n 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\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n comes with a finite-to-one endomorphism\n \n \n \n \n r<\/mml:mi>\n \n :\n \n <\/mml:mo>\n X<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n r\\colon X\\rightarrow X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n \\mathbb {R}^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets\n \n \n \n \n B<\/mml:mi>\n ,<\/mml:mo>\n L<\/mml:mi>\n <\/mml:mrow>\n B, L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n \\mathbb {R}^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the same cardinality which generate complex Hadamard matrices.\n <\/p>\n

\n Our harmonic analysis for these iterated function systems (IFS)\n \n \n \n \n (<\/mml:mo>\n X<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (X, \\mu )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is based on a Markov process on certain paths. The probabilities are determined by a weight function\n \n \n \n W<\/mml:mi>\n W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . From\n \n \n \n W<\/mml:mi>\n W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n we define a transition operator\n \n \n \n \n R<\/mml:mi>\n W<\/mml:mi>\n <\/mml:msub>\n R_W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n acting on functions on\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and a corresponding class\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of continuous\n \n \n \n \n R<\/mml:mi>\n W<\/mml:mi>\n <\/mml:msub>\n R_W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -harmonic functions. The properties of the functions in\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are analyzed, and they determine the spectral theory of\n \n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L^2(\\mu )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For affine IFSs we establish orthogonal bases in\n \n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L^2(\\mu )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n \\mathbb {R}^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-06-01861-8","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1931-1970","source":"Crossref","is-referenced-by-count":85,"title":["Iterated function systems, Ruelle operators, and invariant projective measures"],"prefix":"10.1090","volume":"75","author":[{"given":"Dorin","family":"Dutkay","sequence":"first","affiliation":[]},{"given":"Palle","family":"Jorgensen","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,31]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s00365-003-0539-0","article-title":"Nonuniform average sampling and reconstruction in multiply generated shift-invariant spaces","volume":"20","author":"Aldroubi, Akram","year":"2004","journal-title":"Constr. 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