The parabolic implosion for f0(z) = z + zv+1 + O(zv+2)
In this thesis we examine the bifurcation in behaviour (for the dynamics) which occurs when we perturb the holomorphic germ fo(z) =z + zv+1 + O(zv+2) defined in a neigh- bourhood of 0, so that the multiple fixed point at 0 splits into v+1 fixed points (counted with multiplicity). The phenomenon observed is called the parabolic implosion, since the perturbation will typically cause the filled Julia set (if it is defined) to "implode. " The main tool used for studying this bifurcation is the Fatou coordinates and the associated Ecalle cylinders. We show the existence of these for a family of well behaved f's close to fo, and that these depend continuously upon f. Each well behaved f will have a gate structure which gives a qualitative description of the "egg-beater dynamic" for f. Each gate between the fixed points of f will have an associated complex number called the lifted phase. (Minus the real part of the lifted phase is a rough measure of how many iterations it takes for an orbit to pass through the gate. ) The existence of maps with any desired gate structure and any (sensible) lifted phases is shown. This leads to a simple parameterisation of the well behaved maps. We are particularly interested in sequences fk → fo where all the lifted phases of the fk converge to some limits, modulo Z. We show that there is a non-trivial Lavaurs reap g associated with these limits, which commutes with fo. The dynamics of fk are shown to (in some sense) converge to the dynamics of the system (fo, g). We also prove that for any Lavaurs map g there is a sequence fk → fo so that fk k → g as k → +oo, uniformly on compact sets.