Stretching rays and their accumulations, following Pia Willumsen

Abstract

Pia Willumsen did her graduate work with Bodil Branner. Willumsen has written a very beautiful Ph.D. thesis, [W], containing many interesting results. They have, however, remained unpublished up to today. We present here a short account of some of Willumsen’s results, sketch some of the proofs, as well as some immediate extensions. The main topic is stretching rays, which is the analogue in higher dimensional parameter space of external rays of the Mandelbrot set. In the space of cubic polynomials, the interaction of the two critical points create new and interesting phenomena. A typical case deals with maps with a parabolic basin containing both critical points. The results we present here provide necessary conditions for accumulation and landing of stretching rays to these maps. Also, discontinuity of the wring operator on the cubic (in contrast to quadratic) polynomials, is proven.