Question: How are the rotation numbers associated to the primary bulbs arranged around the boundary of the main cardioid?Answer: Amazingly, the order of the rotation numbers around the boundary of the main cardioid is exactly the same as the rational numbers. That is, if you call the "cusp" of the cardioid 0, and then proceed in the counterclockwise direction, you visit each bulb corresponding to rotation number p/q in exact order, ending at 1 when you arrive back at the cusp. Equally amazingly, all you need to prove this is a little (complex) calculus! |
Example: To see a (QuickTime) animation of the filled Julia sets as
c winds around the main cardioid (more precisely, just outside
the main cardioid, sometimes in, sometimes just outside the primary
bulbs), click here.
Warning: This movie takes a LONG time to load---38 Mb!!! Have
a lot of patience.
Exploration: Use the Mandelbrot
Set Iterator
together with the order of the bulbs around the main cardioid to find
the following bulbs associated to the following fractions: (note that
you will have to increase the number of iterations performed to around
200-300 to view magnifications of these bulbs)
For Further Information: See the papers:
Dynamical Systems and Technology Project
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