  Many models of fractal growth patterns (like Diffusion Limited Aggregation
and Dielectric Breakdown Models) combine complex geometry with randomness; this
double difficulty is a stumbling block to their elucidation. In this paper we
introduce a wide class of fractal growth models with highly complex geometry
but without any randomness in their growth rules. The models are defined in
terms of deterministic itineraries of iterated conformal maps, automatically
generating the global conformal function which maps the exterior of the unit
circle to the exterior of a n-particle growing aggregate. The complexity of the
evolving interfaces is fully contained in the deterministic dynamics of the
conformal map. We focus attention to a class of growth models in which the
itinerary is quasiperiodic. Such itineraries can be approached via a series of
rational approximants. The analytic power gained is used to introduce a scaling
theory of the fractal growth patterns. We explain the mechanism for the
fractality of the clusters and identify the exponent that determines the
fractal dimension.
