Geography of the cubic connectedness locus I: Intertwining surgery
Abstract
We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs' proof of non local-connectivity for the cubic connectedness locus. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction is an {\it intertwining surgery} on two quadratics. The idea of intertwining first appeared in a collection of problems edited by Bielefeld. Using quasiconformal surgery techniques of Branner and Douady, we show that any two quadratics may be intertwined to obtain a cubic polynomial. The proof of continuity in our two-parameter setting requires further considerations involving ray combinatorics and a pullback argument.
Keywords
Cite
@article{arxiv.math/9608213,
title = {Geography of the cubic connectedness locus I: Intertwining surgery},
author = {Adam L. Epstein and Michael Yampolsky},
journal= {arXiv preprint arXiv:math/9608213},
year = {2016}
}