Here Julia sets J(c_o) associated with the preperiodic points c_o near the point z = c_o are shown. These pictures are compared with the corresponding areas of the Mandelbrot set.

J pictures differ from the corresponding M pictures in that there are no black regions in them. On the other hand if we zoom in to a preperiodic point in M, the nearby miniature Ms shrink faster than the view window (if we shrink the picture by a factor of m the miniature Ms shrink by a factor of m²), so they eventually disappear.

This local similarity between the Mandelbrot set near a preperiodic point c_o and the Julia set J(c_o ) near z = c_o shown above is the subject a theorem of Tan Lei.

Here is a partial explanation [1] for it in the case when period of c_o is 1. For small ε
    f_Co+ε^o(n+1)(0) = f_Co+ε^on(c_o + ε)
    = f_Co^on(c_o ) + ( ^d/_dc f_C^on(0) |_C=Co + ^d/_dz f_Co^on(z) |_z=Co ) ε + O(ε²)
    = f_Co^on(c_o + k_nε) + O(ε²) ,
where
    k_n = ( ^d/_dc f_C^on(c_o ) |_C=Co + ^d/_dz f_Co^on(z) |_z=Co ) / ^d/_dz f_Co^on(z) |_z=Co .
As since
    ^d/_dc f_C^o(n+1)(c_o ) |_C=Co = 2 h_n ^d/_dc f_C^on(c_o ) |_C=Co + 1 ,
    ^d/_dz f_Co^o(n+1)(z) |_z=Co = 2 h_n ^d/_dz f_Co^on(z) |_z=Co ,     h_n = f_Co^on(c_o )
and h_n go to the fixed point h of the critical orbit of preperiodic point c_o for large enough n, then it can be shown, that k_n converge to a finite k [Ravenel].

Equation
    f_Co+ε^o(n+1)(0) = f_Co^on(c_o + kε) + O(ε²)
means that for small ε the (n+1)th point in critical orbit of c = c_o+ε can be approximated by the nth point in the Julia orbit of z_∗ = c_o+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 c_o and the Julia set J(c_o ) near z = c_o .

[1] Douglas C. Ravenel Fractals and computer graphics