Filled Julia set
The filled-in Julia set K ( f ) {\displaystyle K(f)} of a polynomial f {\displaystyle f} is the union of a Julia set and its interior, non-escaping set. == Formal definition == The filled-in Julia set K ( f ) {\displaystyle K(f)} of a polynomial f {\displaystyle f} is defined as the set of all points z {\displaystyle z} of the dynamical plane that have bounded orbit with respect to f {\displaystyle f} K ( f ) = d e f { z ∈ C : f ( k ) ( z ) ↛ ∞ as k → ∞ } {\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}} where: C {\displaystyle \mathbb {C} } is the set of complex numbers f ( k ) ( z ) {\displaystyle f^{(k)}(z)} is the k {\displaystyle k} -fold composition of f {\displaystyle f} with itself = iteration of function f {\displaystyle f} == Relation to the Fatou set == The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.