We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations of the satellite type. The basic tool at each step is to control the behaviour of certain external rays. We distinguish between perturbations to left and right satellites and investigate the differences between one-sided and two-sided (ie. alternating) parabolic perturbations. In the one-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: ``the topologists spiral''. In the two-sided case, we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at a particular point, but having in its impression also the symmetric point. The limit Julia set resembles in a certain way the classical non-locally connected set: ``the topologists sine''. Furthermore we obtain, by using a more general strategy, that all of these Julia sets fail to be arc-wise connected.
|Journal||Journal of Geometric Analysis|
|Publication status||Published - 1997|