The Herman Ring Formula

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Ever since I was a small child I knew the Sullivan Classification by heart. It describes all of the possible kinds of attractor in a chaotic rational mapping of complex numbers. Those possibilities are:

1. An attracting fixed point or cycle.
2. A parabolic fixed point or cycle.
3. A Siegel disk or a cycle of these.
4. A Herman ring or a cycle of these.

Attracting point	Parabolic point on the verge of becoming 7	Siegel disk
Attracting 3-cycle	Parabolic 3-cycle on the verge of becoming 9	Cycle of 2 Siegel disks

Fractint PAR file for replicating these images

The Herman ring does not appear in the Mandelbrot set, or indeed in any polynomial. I had experimented for years with numerous rational functions and had never seen anything I didn't recognize as one of the three cases outlined above.

One day last fall (1997) I decided I just had to have some Herman rings, and so I logged onto the university library computer and input a search for "Herman ring". Nada. I tried "Complex mappings" and got a few titles... including an obscure Springer Verlag published title whose name and description looked promising. I tracked it down to a dusty corner of the basement on a shelf full of arcane mathematical texts. An illustration of a quadratic Julia set on the front cover told me I was on the right track. I popped open the book at the back and found the index, then the "h", and then "Herman ring". Two pages referenced. I checked the second, and found nothing of interest except a recap of the Sullivan classification, which I already knew. I checked the first of the pages, and I was damned if I didn't see near the bottom of the page an orbit plot and near the top a mathematical equation. That equation was, in fact, a rational mapping with Herman rings. I pulled out my note binder and a pencil and wrote down the formula. I plugged it into Fractint that very night, and the rest, as they say, is history.

The equation: z ->

The formula proved to be a real treat. It has a fascinating and complex four-dimensional Mandelbrot set (owing to the two parameters), lovely Julia sets, and provides many amazing vistas the likes of which I'd never seen before. About the only thing it won't do is produce a cycle of Herman rings. The Julia sets always have an attractor at infinity and another at zero, and most of my Julia images show distinct color schemes for the two basins.
There are two critical points in this formula, which produce two congruent, mirror image quadratic Mandelbrots...usually. If one fixes c = 3 or c = -3 and looks at the a-plane, one sees a cubic Mandelbrot set, because the critical points, which depend on c, coincide and thus there is a single critical point with multiplicity two. (This also happens for c = 1 or c = -1, but for these and c = 0, the formula reduces to raising z to some power and multiplying by a, and the set reduces to the unit circle, which is rather boring.)

Fractint Formula File

Fractint PAR File

Herman Ring

Rubies

Herman Mandelbrot

Herman Mandelbrot 2