It is evident that for large enough z any quadratic map
f is conjugate ("very similar", see [1,2] for rigorous
math :) to the squaring function _{c}(z)f.
We have seen that _{o}(z) = z^{2}f
and its reverse function map circles into circles, therefore conjugate function
_{o}(z)f maps circle
_{c}^{-1}(z) = +-(z - c)^{1/2}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
f 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.
_{c}^{-1}(z) |

If the critical point z (and therefore the
_{c} = 0z = c point) goes to infinity, a very different topological picture
emerges. We take a circle l', which goes through the c .
Each point of l' has two preimages +-(z - c)
(e.g. for ^{1/2}c = 4 the point z = -4 has z) with the exeption of _{0,1} =
+-8^{1/2}iz = c ,
which has the only z preimage. So the preimage
of _{c}l' is the figure eight curve l. Two discks D are mapped by _{0} ,
D_{1}f in one-to-one fashion
onto the _{c}V disk (it containes both D).
The Julia set _{0} , D_{1}J(4) is contained inside
D
and has infinitely many connected components.
_{0} U D_{1} |

Now let

for any

f

f

As a consequence, if two sequences disagree somewhere, say in the

2. Otherwise, the filled Julia set of

[1] *Robert L. Devaney*
Complex Complex
Saddle-Node Bifurcations § 2

[2] *John W. Milnor* "Dynamics in One Complex Variable" § 9

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