Quasiperiodic Spectra and Orthogonality for Iterated Function System Measures

TL;DR

This paper extends Fourier basis concepts to fractal measures from affine iterated function systems, exploring spectral properties, geometric conditions for spectra, and tensor product decompositions related to complex Hadamard matrices.

Contribution

It introduces geometric criteria for the existence of Fourier spectra on IFS attractors and analyzes their spectral pair compositions and decompositions.

Findings

Identified conditions for spectral duality in IFS measures

Characterized spectral pair decompositions via tensor products

Connected spectral properties with complex Hadamard matrices

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.