Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-29T06:18:29.868Z Has data issue: false hasContentIssue false

Describing quadratic Cremer point polynomials by parabolic perturbations

Published online by Cambridge University Press:  01 June 1998

DAN ERIK KRARUP SØRENSEN
Affiliation:
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.

Type
Research Article
Copyright
© 1998 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)