Abstract

We let double-struck L sign be the completion of the field of formal Puiseux series and study polynomials with coefficients in double-struck L sign as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in double-struck L sign[ζ]. We show that cubic polynomial dynamics over double-struck L sign and ℂ are intimately related. More precisely, we establish that some elements of double-struck L sign 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.