Here Julia sets J(co) associated with the preperiodic points co near the point z = co are shown. These pictures are compared with the corresponding areas of the Mandelbrot set.
This local similaity between the Mandelbrot set near a preperiodic point co and the Julia set J(co) near z = co shown above is the subject a theorem of Tan Lei.
Here is a partial explanation [1] for it in the case when period of
co is 1. For small e
fCo+eo(n+1)(0) =
fCo+eon(co +
e)
= fCoon(co) + (
d/dc fCon(0) |C=Co +
d/dz fCoon(z) |z=Co)
e + O(e2)
= fCoon(co +
kne) +
O(e2) ,
where
kn = (
d/dc fCon(co) |C=Co +
d/dz fCoon(z) |z=Co)
/ d/dz fCoon(z) |z=Co .
As since
d/dc
fCo(n+1)(co) |C=Co =
2 hn d/dc
fCon(co) |C=Co + 1 ,
d/dz
fCoo(n+1)(z) |z=Co =
2 hn d/dz
fCoon(z) |z=Co ,
hn = fCoon(co)
and hn go to the fixed point h of the critical
orbit of preperiodic point co for large enough n,
then it can be shown, that kn converge to a finite k
[Ravenel].
Equation
fCo+eo(n+1)(0) =
fCoon(co + ke) +
O(e2)
means that for small e the (n+1)th
point in critical orbit of c = co+e
can be approximated by the nth point in the Julia orbit of z* =
co+ke .
I.e. the critical orbit is bounded if and only if the z* orbit is
bounded. This accounts for the local similarity
between the Mandelbrot set near co and the Julia set
J(co) near z = co .
[1] Douglas C. Ravenel Fractals and computer graphics