^{1} Department of Mathematics, Technical University of Denmark^{2} Department of Applied Mathematics and Computer Science, Technical University of Denmark

Abstract:

We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given external ray accumulating the Cremer point, but having in itsimpression both the Cremer point and its other preimage. We think of the Julia setas containing a "topologists single comb".By tuning, similar results hold on the boundary of any hyperbolic component of theMandelbrot set.