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Geography of the cubic connectedness locus I: Intertwining surgery

Adam L. Epstein·Michael Yampolsky
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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.

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Adam L. Epstein, Michael Yampolsky. 1996-08-15. Geography of the cubic connectedness locus I: Intertwining surgery. https://arxiv.org/abs/math/9608213

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