The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Cycle Components

One way the sequence zn can remain bounded is by converging to a fixed point or a cycle.

To every disc and cardioid component of the Mandelbrot set there corresponds a cycle {z1, ..., zn}. (A cycle with n = 1 is a fixed point.)

That is,

Fc(z1) = z2, Fc(z2) = z3, ... , Fc(zn) = z1

The cycle lengths (n) of some components are shown in the diagram above.

Note each zi is a fixed point for Fcn(z):

Fcn(z1) = Fcn-1(Fc(z1)) = Fcn-1(z2) = ... = Fc(zn) = z1.

Return to Hurwitz-Robucci scaling.