**M & J-sets similarity for preperiodic points.
Lei's theorem**

*Douglas C. Ravenel*
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 **M**s shrink faster
than the view window (if we shrink the picture by a factor of *m*
the miniature **M**s shrink by a factor of *m*^{2}),
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(ε^{2})

= f_{Co}^{on}(c_{o} +
k_{n}ε) + O(ε^{2}) ,

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(ε^{2})

means that for small *ε* the *(n+1)*th
point in critical orbit of *c = c*_{o}+ε
can be approximated by the *n*th 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

Contents
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*updated* 14 Sep 2013