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Iterated function systems, Ruelle operators, and invariant projective measures

Authors:	Dorin Ervin Dutkay Palle E T Jorgensen

Institution:	Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019 ; Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419

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 comes with a finite-to-one endomorphism $r\colon X\rightarrow X$ which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in $\mathbb{R}^d$ , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets in $\mathbb{R}^d$ of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) $(X, \mu)$ is based on a Markov process on certain paths. The probabilities are determined by a weight function on . From we define a transition operator acting on functions on , and a corresponding class of continuous -harmonic functions. The properties of the functions in are analyzed, and they 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 $\mathbb{R}^d$ .

Keywords:	Measures projective limits transfer operator martingale fixed-point wavelet multiresolution fractal Hausdorff dimension Perron-Frobenius Julia set subshift orthogonal functions Fourier series Hadamard matrix tiling lattice harmonic function

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