= Filled Julia set
{title2=$K(P)$}
For a polynomial $P$ of degree at least two, the filled Julia set $K(P)$ consists of the points with bounded forward orbit. Choose an escape radius $B$ so that $|P(z)|>2|z|$ for $|z|>B$. Then $K(P)=\bigcap_{n\geq0}(P^n)^{-1}(\{|z|\leq B\})$ is compact, and its complement is the basin of infinity. On the interior all iterates are bounded by $B$, so they form a <normal family>; on the complement they converge locally uniformly to infinity. At a boundary point, normality would give a subsequential limit equal to infinity on an open escaping part and hence identically infinity, contradicting the uniform bound at a nearby point of $K(P)$. Therefore $\partial K(P)=J(P)$.
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