Abstract.
This technical report discusses the geometry of the Mandlebrot set for
iterations of quadratic functions f_{c}(z) = z^{2}+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 e^{2πi/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/N^{2} . We give an explanation for this
phenomenon in terms of the normal forms of resonant bifurcations with
multiplier e^{2πi/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.