 Filled Julia set

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 $K(P)={\bigcap }_{n\ge 0}(P^n{)}^{-1}(\{|z|\le B\})$ 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 $\partial K(P)=J(P)$.

The filled Julia set is a mathematical concept in the context of complex dynamics, particularly related to the behavior of iterating complex functions. More specifically, it is derived from the iteration of a complex function, typically of the form \( f(z) = z^2 + c \), where \( z \) is a complex variable and \( c \) is a complex parameter.