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Mathematics of Computation

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Iterated function systems, Ruelle operators, and invariant projective measures

Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Math. Comp. 75 (2006), 1931-1970
MSC (2000): Primary 28A80, 31C20, 37F20, 39B12, 41A63, 42C40, 47D07, 60G42, 60J45
Published electronically: May 31, 2006
MathSciNet review: 2240643
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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$ 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 $ B, L$ 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 $ 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(\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$.

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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019

Palle E. T. Jorgensen
Affiliation: Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419

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
Received by editor(s): January 5, 2005
Received by editor(s) in revised form: June 16, 2005
Published electronically: May 31, 2006
Additional Notes: This research was supported in part by the National Science Foundation DMS-0139473 (FRG)
Article copyright: © Copyright 2006 American Mathematical Society

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