Analysis of orthogonality and of orbits in affine iterated function systems

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 µ, and we ask when µ is a spectral measure, i.e., when the Hilbert space L 2 (µ) has an orthonormal basis (ONB) of exponentials {e λ | λ ∈ Λ}. 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.

Affine Systems: Asymptotics at Infinity for Fractal Measures

“…In higher dimensions, the spectral question is also difficult. In [12] it is conjectured that the only measures for which μ λ is a spectral measure is the rational case (see Sect. 2.3).…”

“…[12,27], which explore further conditions on the pair (A, B) which guarantee that L 2 (μ (A,B) ) has an orthogonal basis of complex exponentials. By this we mean that there is a subset of R d such that the set of complex exponential functions…”

“…The papers [12] and [25,26] study such questions for various classes of affine IFSs. In [26], we considered orthogonal complex exponentials in L 2 (X, μ (λ,{±1}) ) for the nonrational case in R. As we noted there, relatively little is known about either the spectral or continuity properties of μ (A,B) .…”

Families of Spectral Sets for Bernoulli Convolutions

“…This general framework includes subjects such as universal tiling sets and the dual spectral set conjecture (see [11,22,25,27,35]). Starting with [16], questions about Fourier duality have received considerable attention with respect to pure harmonic analysis [5,8,9,24,29,30,33,34] and with respect to applications such as wavelets, sampling, algorithms, martingales, and substitution-dynamical systems [2][3][4]26].…”

Spectral self-affine measures on the spatial Sierpinski gasket

“…The present paper will follow the paper [8] (1.4), it has been an interesting topic to examine the spectrality of self-affine measure μ M,D . With the effort of Jorgensen and Pedersen [2], [3,Example 7.1], Strichartz [10], [11,Example 2.9(e)], Dutkay and Jorgensen [1,Theorem 5.1(iii)], and the author [5, Theorem 1], [8], the spectrality or the non-spectrality of μ M,D can be summarized as the following Theorem A.…”