Analysis of orthogonality and of orbits in affine iterated function systems

    Authors

    D. E. Dutkay;P. E. T. Jorgensen

    Comments

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    Abbreviated Journal Title

    Math. Z.

    Keywords

    Fourier series; affine fractal; spectrum; spectral measure; Hilbert; space; attractor; SELF-SIMILAR FRACTALS; SPECTRAL THEORY; TILES; RN; SIMILARITY; CONJECTURE; SETS; Mathematics

    Abstract

    We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson's canonical measure mu, and we ask when mu is a spectral measure, i.e., when the Hilbert space L-2(mu) has an orthonormal basis (ONB) of exponentials {e(lambda) vertical bar lambda is an element of Lambda}. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3.

    Journal Title

    Mathematische Zeitschrift

    Volume

    256

    Issue/Number

    4

    Publication Date

    1-1-2007

    Document Type

    Article

    Language

    English

    First Page

    801

    Last Page

    823

    WOS Identifier

    WOS:000246614400010

    ISSN

    0025-5874

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