Ergodic Theory and Dynamical Systems



Describing quadratic Cremer point polynomials by parabolic perturbations


DAN ERIK KRARUP SØRENSEN  1 a1
a1 Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark (e-mail: dan@mat.dtu.dk)

Abstract

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.

Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the 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 explicitly given external ray accumulating 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 one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.

By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

(Received March 8 1996)
(Revised October 1 1996)



Footnotes

1 The research presented in this paper was partly supported by grant number ERBCHBG CT 930324, Leibnitz postdoctoral Fellowship, Human Capital and Mobility, EU, and grant number 9600411, Danish Natural Science Research Council.