Filled Julia set

The filled-in Julia set ${\displaystyle K(f)}$ of a polynomial ${\displaystyle f}$ is a Julia set and its interior, non-escaping set.

The filled-in Julia set ${\displaystyle K(f)}$ of a polynomial ${\displaystyle f}$ is defined as the set of all points ${\displaystyle z}$ of the dynamical plane that have bounded orbit with respect to ${\displaystyle f}$ $${\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}$$ where:

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. $${\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}$$

The attractive basin of infinity is one of the components of the Fatou set. $${\displaystyle A_{f}(\infty )=F_{\infty }}$$

In other words, the filled-in Julia set is the complement of the unbounded Fatou component: $${\displaystyle K(f)=F_{\infty }^{C}.}$$

The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity $${\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )}$$ where: ${\displaystyle A_{f}(\infty )}$ denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for ${\displaystyle f}$

$${\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}$$

If the filled-in Julia set has no interior then the Julia set coincides with the filled-in Julia set. This happens when all the critical points of ${\displaystyle f}$ are pre-periodic. Such critical points are often called Misiurewicz points.