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Quasiperiodic spectra and orthogonality for iterated function system measures

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Abstract

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL, 32816-1364, USA

    Dorin Ervin Dutkay

  2. Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242-1419, USA

    Palle E. T. Jorgensen

Authors
  1. Dorin Ervin Dutkay
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  2. Palle E. T. Jorgensen
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Correspondence to Dorin Ervin Dutkay.

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Research supported in part by a grant from the National Science Foundation DMS-0704191.

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Dutkay, D.E., Jorgensen, P.E.T. Quasiperiodic spectra and orthogonality for iterated function system measures. Math. Z. 261, 373–397 (2009). https://doi.org/10.1007/s00209-008-0329-2

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  • DOI: https://doi.org/10.1007/s00209-008-0329-2

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