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 similarity 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  for it in the case when period of
co is 1. For small ε
fCo+εo(n+1)(0) = fCo+εon(co + ε)
= fCoon(co ) + ( d/dc fCon(0) |C=Co + d/dz fCoon(z) |z=Co ) ε + O(ε2)
= fCoon(co + knε) + O(ε2) ,
kn = ( d/dc fCon(co ) |C=Co + d/dz fCoon(z) |z=Co ) / d/dz fCoon(z) |z=Co .
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].
fCo+εo(n+1)(0) = fCoon(co + kε) + O(ε2)
means that for small ε the (n+1)th point in critical orbit of c = co+ε can be approximated by the nth point in the Julia orbit of z∗ = co+kε . 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 .
 Douglas C. Ravenel Fractals and computer graphics