  We study polynomials with coefficients in a field L as dynamical systems
where L is any algebraically closed and complete ultrametric field with dense
valuation group and characteristic zero residual field. We give a complete
description of the dynamical and parameter space of cubic polynomials. In
particular we characterize cubic polynomials with compact Julia sets. Also, we
prove that any infraconnected connected component of a filled Julia set (of a
cubic polynomial) is either a point or eventually periodic. A smallest field S
with the above properties is, up to isomorphism, the completion of the field of
formal Puiseux series with coefficients in an algebraic closure of Q. We show
that some elements of S naturally correspond to the Fourier series of analytic
almost periodic functions (in the sense of Bohr) which parametrize (near
infinity) the quasiconformal classes of non-renormalizable complex cubic
polynomials. Our techniques are based on the ideas introduced by Branner and
Hubbard to study complex cubic polynomials.
