The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Cycle Components

One way the sequence z_n 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, ..., z_n}. (A cycle with n = 1 is a fixed point.)

That is,

F_c(z₁) = z₂, F_c(z₂) = z₃, ... , F_c(z_n) = z₁

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

Note each z_i is a fixed point for F_cⁿ(z):

F_cⁿ(z₁) = F_c^n-1(F_c(z₁)) = F_c^n-1(z₂) = ... = F_c(z_n) = z₁.

Return to Hurwitz-Robucci scaling.