# 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

* Rabbit Julia set with spine
* Basilica Julia set with 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 ${\mathcal {R}}_{0}^{K}$,
* $-\beta$ is landing point of external ray ${\mathcal {R}}_{1/2}^{K}$.
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.

## Images

* Filled Julia set for f$_c$, c=1−φ=−0.618033988749…, where φ is the Golden ratio
* Filled Julia with no interior = Julia set. It is for c=i.
* Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
* Douady rabbit
* Filled Julia set for c = −0.8 + 0.156i.
* Filled Julia set for c = 0.285 + 0.01i.
* Filled Julia set for c = −1.476.

## Names

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