Parabolic Explosions in Families of Complex Polynomials

Abstract

We present a new algebro-algebraic approach to the parabolic explosion of orbits for polynomials of a fixed degree $d \geq 2$, $P(z) = {z}^{d} + {a}_{d-1}{z}^{d-1} + \ldots + \lambda z,\ {a}_{d-1},\ldots ,{a}_{2},\lambda \in \mathbb{C}$ where 0 is a multiple fixed point of ${P}_{\mathbf{a}}^{\circ q}$ for some $\mathbf{a} = ({a}_{d-1},\ldots ,{a}_{2},{\lambda }_{0})$ with ${\lambda }_{0}^{q} = 1,\ {\lambda }_{0}^{k}\neq 1$ for $k = 1,\ldots ,q - 1$. We show using methods based on Puiseux series that for an open dense set of perturbed maps with $\lambda = {\lambda }_{0}\exp (2\pi iu)$, 0 becomes a simple fixed point and a number of periodic orbits of period q appear which are holomorphic in u ^q. We also prove that the unwrapping coordinates for perturbations of an analytic map with a parabolic periodic point converge uniformly to the unwrapping coordinate for the map itself.