Conformal dynamics of fractal growth patterns without randomness

Many models of fractal growth patterns (such as diffusion limited aggregati on and dielectric breakdown models) combine complex geometry with randomnes s; this double difficulty is a stumbling block to their elucidation. In thi s paper we introduce a wide class of fractal growth models with highly comp lex geometry but without any randomness in their growth rules. The models a re defined in terms of deterministic itineraries of iterated conformal maps , generating the function Phi((n))(omega) which maps the exterior of the un it circle to the exterior of an n-particle growing aggregate. The complexit y of the evolving interfaces is fully contained in the deterministic dynami cs of the conformal map Phi((n))(omega). We focus attention on a class of g rowth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gai ned is used to introduce a scaling theory of the fractal growth patterns an d to identify the exponent that determines the fractal dimension.