University of Minnesota
School of Mathematics

A Proof of the Mandelbrot N2 Conjecture

John Guckenheimer
Richard McGehee

Institut Mittag-Leffler Report No. 15 (1984)

Abstract.   This technical report discusses the geometry of the Mandlebrot set for iterations of quadratic functions  fc(z) = z2+c  of one complex variable. The specific focus is upon the bifurcations of period  N  periodic orbits from a fixed point when the multiplier of the fixed point is  ei/N . Bifurcation points of this type accumulate at a parabolic fixed point; i.e., a fixed point with multiplier  1 . Mandelbrot observed that the size of the stability domains of the period  N  orbits scale like  1/N2 . We give an explanation for this phenomenon in terms of the normal forms of resonant bifurcations with multiplier  ei/N . More extensive results establishing that these stability domains have a limiting shape following rescaling are corollaries of the theory of analytic normal forms for parabolic points. See, for example, the paper "Bifurcation of parabolic fixed points'' by M. Shishikura in the volume The Mandelbrot Set, Theme and Variations published as volume 274 of the London Mathematical Society Lecture Notes Series.

Electronic Copies

I scanned one of the original typewritten copies as an image file. It is available here in pdf format.

image file (pdf)


Last update: March 23, 2008 ©2008 Richard McGehee