The Mandelbrot set and Julia Sets

The boundary of the Mandelbrot set

For a point c in an n-cycle component, the iterates of z₀ = 0 converge to an n-cycle. In each n-cycle component, there is exactly one c value for which z₀ = 0 belongs to an n-cycle.

That is, taking z₀ = 0, we have z_n = 0. This c is called the center of the n-cycle component.

For example, the center of the big cardioid (1-cycle component) is c = 0. Observe

z0 = 0,

z1 = z02 + c = 02 + 0 = 0,

and so all zn = 0.

For another example, the center of the 2-cycle component is c = -1. Observe

z0 = 0,

z1 = z02 + c = 02 - 1 = -1,

z2 = z12 + c = (-1)2 - 1 = 0,

so z3 = -1,

z4 = 0,

and so on.

A few centers (indicated by red crosses) are pictured above.

Return to Some features of the Mandelbrot set boundary.