Title

Analysis of orthogonality and of orbits in affine iterated function systems

Authors

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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