  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 $(X, \mu)$ where $X$ is a compact subset of $\br^d$,
(the support of $\mu$) and the measure $\mu$ 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 $L^2(X,\mu)$; and (2) the
interplay between the geometry of $(X, \mu)$ on the one side, and the spectral
data entailed by possible Fourier bases.
