The Mandelbrot Set and Julia Sets

Combinatorics in the Mandelbrot Set - The 1/n² rule, and deviations from it

The most obvious relation between the length of a cycle and the size of a disc having that cycle is

the longer the period, the smaller the disc

Mandelbrot conjectured that the disc associated with an n-cycle has radius approximately 1/n².

Even a slightly closer inspecton reveals a departure from this rule: the left 5-cycle disc is larger than the right 5-cycle disc. To give a more precise formulation of the rule, we introduce the internal angles of the main cardioid. Then the N^-2 rule is

the disc attached at internal angle m/n has radius approximately rad(m,n) = sin(pi*m/n)/n².

The approximation was proved by Guckenheimer and McGehee. Mandelbrot commented that the deviations from this rule appear to quite intricate. Several years ago, L. Kerry Mitchell and I did some numerical investigations of these departures.

First, we must answer the question, "What do we mean by the radius of these discs?" because only the 2-cycle disc is a true disc. The multiplier provides an answer. By the center of a disc we mean the point c where the multiplier is 0; that is, z = 0 belongs to an n-cycle of z² + c. By the root of a disc the mean the point c where the multiplier is 1; that is, the point where the disc is tangent to the cardioid. For the disc at internal angle n/m, we say the root to center distance, rtc(m,n), is the distance between the center and the root. This is what we shall mean by the radius of disc.

For comparison we considered also ang(m,n), the angle between the root to center line and the line perpendicular to the cardioid at the root point.

Here is a plot of the ratio rtc(m,n)/rad(m,n) for the first 10000 discs attached to the upper half of the main cardioid. (Here "first 10000" means the 10000 having the lowest values of n.) The most obvious aspec of this plot is that the structure between m/n = 1/2 and m/n = 1/3 is repeated between m/n = 1/3 and m/n = 1/4, and between m/n = 1/4 and m/n = 1/5, and so on. For very small values of m/n, the structure seems to disappear, being replaced by a collection of horizontal asymptotes. Presumably, plotting a larger collection of discs would remove this problem.

Note each of these regions appears to be divided into subregions similar to further distorted versions of the whole plot.

Here is a plot of the angle ang(m,n) for the same 10000 discs. As with the plots of rtc(m,n)/rad(m,n), we see structures repeated between 1/2 and 1/3, between 1/3 and 1/4, and so on, and also a similar collection of horizontal asymptotes.

The departures from the N^-2 rule appear to organize themselves into fractal patterns, a sort of secondary fractal structure of the Mandelbrot set. These plots were made on two (different) computers, running programs in different languages (Pascal and Fortran), written by different programmers. So we believe they represent real features, not computational gremlins.

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