# M and J-sets similarity. Embedded Julia midgets

Tan Lei's theorem says, that the M-set at a preperiodic
point (e.g. at the band merging point *M = -1.543689* in the second
picture) and the Julia set *J(M)* at *z = M* are similar and have the
same scaling constant.
In the vicinity of the periodic point *P = -1.75488* (in the center of
the period-3 midget) the Mandelbrot set is similar again to the Julia set
*J(P)*. As since *f*_{c}^{o3} is renormalizable for
*c = P* , we see the *J(0)* set at *z = P* and its shrinking
copies (instead of sequence of tiny midgets in the M-set).
At last this image corresponds to *A = -1.7465234* at the cusp of
the period-3 midget. The point *A* lays from outside of tiny cardioid,
therefore we see small copies of the cauliflower Cantor dust *J(0.35)*
set.

To make the last step we look at the M-set near *A* . As since at small
scale the M-set is similar to corresponding *J(A)* set, we see embedded
*J(0.35)* midget in the Mandelbrot set (in the right applet).
Note however, that this "cauliflower" is replaced by the M-midget in the
left window. Every embedded cauliflower has a tiny M-midget in its center.
For more details see Robert P. Munafo's
Embedded
Julia Set.

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*updated* 31 Dec 2013