Nucleus

Robert P. Munafo, 2003 Sep 22.

The unique point within any mu-atom which has the property of belonging to its own limit cycle. This point is called the superstable point.

This use of the word 'nucleus' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature.

If you set the polynomial formula for a lemniscate ZN equal to zero and solve for C (to get the roots of the polynomial), the roots are the nuclei of the mu-atoms of period N, plus any mu-atoms of periods that divide evenly into N. This procedure has been used numerically by Jay Hill to find all mu-atoms for periods up to about 16.

The nucleus of R2a is the origin, 0 + 0i. The nucleus of R2.1/2a is -1 + 0i.

I also sometimes use "nucleus" as a colloquial term for the feature with 4-fold rotational symmetry at the center of an embedded Julia set. See paramecia.


From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo.     Mu-ency index

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