Fourier frequencies in affine iterated function systems

Abstract

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in $\br^d$, and the ``IFS'' refers to such a finite system of transformations, or functions. The iteration limits are pairs $(X, μ)$ where $X$ is a compact subset of $\br^d$, (the support of $μ$) and the measure $μ$ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space $L^2(X,μ)$; and (2) the interplay between the geometry of $(X, μ)$ on the one side, and the spectral data entailed by possible Fourier bases.