DESCRIBING QUADRATIC CREMER POINT POLYNOMIALS BY PARABOLIC PERTURBATIONS

We describe two infinite-order parabolic perturbation procedures yield ing quadratic polynomials having a Cremer fixed point. The main idea i s to obtain the polynomial as the limit of repeated parabolic perturba tions. The basic tool at each step is to control the behaviour of cert ain external rays. Polynomials of the Cremer type correspond to parame ters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We invest igate the differences between two-sided (i.e. alternating) and one-sid ed parabolic perturbations. In the two-sided case, we prove the existe nce of polynomials having an explicitly given external ray accumulatin g both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a 'topologist's double comb'. In the on e-sided case we prove a weaker result: the existence of polynomials ha ving an explicitly given external ray accumulating at the Cremer point , but having in the impression of the ray both the Cremer point and it s other preimage. We think of the Julia set as containing a 'topologis t's single comb'. By tuning, similar results hold on the boundary of a ny hyperbolic component of the Mandelbrot set.