Quasiperiodic spectra and orthogonality for iterated function system measures

Abstract

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.

Families of Spectral Sets for Bernoulli Convolutions

“…The quantity in square brackets is even, perhaps divisible by 4, but not divisible by 8. Therefore, when t − γ 1 is written in the form of (22), the power of 2 is not congruent to 1 modulo 3 and hence t − γ is not in Z 3 8 .…”

“…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].…”

Spectrum is periodic for n-intervals

“…Research on these problems has led to the spectral theory for fractal measures and has received a lot of attention in recent years [10,7,8,12].…”

Non-spectral problem for a class of planar self-affine measures

“…More recently, Li [29] proved that for the self-affine measure μ M,D corresponding to 5) there are at most 3 mutually orthogonal exponential functions in L 2 (μ M,D ), and the number 3 is the best. In the plane, the above set D (usually called the digit set) which consists of the canonical vectors in R 2 is fundamental, many digit sets can be obtained from this set.…”

“…[6,24,25]). The spectral self-affine measure problem at the present day consists in determining conditions under which μ M,D is a spectral measure, and has been studied in the papers [2][3][4][5][6]18,23,25,27,28,32] (see also [33,34] for the main goal). In the opposite direction, the non-spectral Lebesgue measure problem has been studied in the papers [10,11,[15][16][17]22,26] and [13,14] where the conjecture that the disk has no more than 3 orthogonal exponentials is still unsolved.…”