1996/08/15 by Adam Epstein, Epstein, Adam L., Michael Yampolsky +1
Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.math/9608213
openalex publication_date 1996/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.