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It is evident that for large enough z any quadratic map
fc(z) is conjugate ("very similar", see [1,2] for rigorous
math :) to the squaring function fo(z) = z2.
We have seen that fo(z)
and its reverse function map circles into circles, therefore conjugate function
fc-1(z) = +-(z - c)1/2 maps circle
l' into oval l .
One can extend this conjugacy up to the z = c point (which has only
one preimage, namely 0 , whereas any other point has two distinct
preimages). So if c lays within Julia set, it follows that we can
continue this process indefinitely to the entire basin of attraction of
infinity (and a simple "similar to a circle" loop is mapped by
fc-1(z) into one more simple loop).
This basin is simply connected, therefore its compliment is a closed,
connected set. But this compliment is just the filled Julia set.
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