M & J-sets similarity for preperiodic points. Lei's theorem
Douglas C. Ravenel

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.
Zoom=3.087

Zoom=2.34
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 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 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)
,
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+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 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 .

[1] Douglas C. Ravenel Fractals and computer graphics


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updated 30 April 2002