Iterated function systems, Ruelle operators, and invariant projective measures

Dutkay, Dorin; Jorgensen, Palle

doi:10.1090/S0025-5718-06-01861-8

Iterated function systems, Ruelle operators, and invariant projective measures
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by Dorin Ervin Dutkay and Palle E. T. Jorgensen;

Math. Comp. 75 (2006), 1931-1970

DOI: https://doi.org/10.1090/S0025-5718-06-01861-8

Published electronically: May 31, 2006

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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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