## Institute for Mathematical Sciences

## Preprint ims96-10

** A. Epstein and M. Yampolsky**
* 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.

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