Fourier frequencies in affine iterated function systems

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 Rd, 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 Rd (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 L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.