Quasiperiodic spectra and orthogonality for iterated function system measures

    Authors

    D. Dutkay;P. Jorgensen

    Comments

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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.

    Journal Title

    Mathematische Zeitschrift

    Volume

    261

    Issue/Number

    2

    Publication Date

    1-1-2009

    Document Type

    Article

    First Page

    373

    Last Page

    397

    WOS Identifier

    WOS:000261036500008

    ISSN

    0025-5874

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