  We introduce a duality for Affine Iterated Function Systems (AIFS) which is
naturally motivated by group duality in the context of traditional harmonic
analysis. Our affine systems yield fractals defined by iteration of contractive
affine mappings. We build a duality for such systems by scaling in two
directions: fractals in the small by contractive iterations, and fractals in
the large by recursion involving iteration of an expansive matrix. By a fractal
in the small we mean a compact attractor X supporting Hutchinson's canonical
measure mu, and we ask when mu is a spectral measure, i.e., when the Hilbert
space L^2(mu) has an orthonormal basis (ONB) of exponentials. We further
introduce a Fourier duality using a matched pair of such affine systems. Using
next certain extreme cycles, and positive powers of the expansive matrix we
build fractals in the large which are modeled on lacunary Fourier series and
which serve as spectra for X. Our two main results offer simple geometric
conditions allowing us to decide when the fractal in the large is a spectrum
for X. Our results in turn are illustrated with concrete Sierpinski like
fractals in dimensions 2 and 3.
