Condition for Connectedness of Julia Set of z^2 + c

Theorem

Let $J$ be the Julia set of the rational function on $\overline C$ defined as:

$\forall z \in \overline C: z \mapsto z^2 + c$

for some constant $c \in \overline C$.

Then $J$ is connected in $\overline C$ if and only if $c$ is an element of the Mandelbrot set.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Julia set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Julia set