Puiseux series polynomial dynamics and iteration of complex cubic polynomials

Abstract

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.