Filled Julia set

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

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

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

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

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

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

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

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 $$f$$ are pre-periodic. Such critical points are often called Misiurewicz points.

Spine
The most studied polynomials are probably those of the form $$f(z) = z^2 + c$$, which are often denoted by $$f_c$$, where $$c$$ is any complex number. In this case, the spine $$S_c$$ of the filled Julia set $$K $$ is defined as arc between $$\beta$$-fixed point and $$-\beta$$, $$S_c = \left [ - \beta, \beta \right ]$$ with such properties:
 * spine lies inside $$K$$. This makes sense when $$K$$ is connected and full
 * spine is invariant under 180 degree rotation,
 * spine is a finite topological tree,
 * Critical point $$ z_{cr} = 0 $$ always belongs to the spine.
 * $$\beta$$-fixed point is a landing point of external ray of angle zero $$\mathcal{R}^K _0$$,
 * $$-\beta$$ is landing point of external ray $$\mathcal{R}^K _{1/2}$$.

Algorithms for constructing the spine:
 * detailed version is described by A. Douady
 * Simplified version of algorithm:
 * connect $$- \beta$$ and $$ \beta$$ within $$K$$ by an arc,
 * when $$K$$ has empty interior then arc is unique,
 * otherwise take the shortest way that contains $$0$$.

Curve $$R$$: $$R \overset{\mathrm{def}}{{}={}} R_{1/2} \cup S_c \cup R_0 $$ divides dynamical plane into two components.

Names

 * airplane
 * Douady rabbit
 * dragon
 * basilica or San Marco fractal or San Marco dragon
 * cauliflower
 * dendrite
 * Siegel disc