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Referring to the picture of the Mandelbrot set above, we see
attached to the left of the big cardioid (the fixed point, or 1-cycle, component) is a 2-cycle component,
attached to the left of that is a 4-cycle component,
attached to the left of that is an 8-cycle component,
and so on.
This is called the period-doubling cascade.
In the 1970s this cascade was studied (for a process equivalent to looking at the Mandelbrot set for real numbers) by Mitchell Feigenbaum, and Pierre Coulette and Charles Tresser.
Certainly, the 2n-cycle components get smaller as n increases. Is there some pattern to the rate of shrinking of these components?
Return to Scalings in the Mandelbrot set.