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

Formal definition

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\))$$

• ${\displaystyle \mathbb {C} }$ is the set of complex numbers
• ${\displaystyle f^{(k)}(z)}$ is the ${\displaystyle k}$ -fold composition of ${\displaystyle f}$ with itself = iteration of function ${\displaystyle f}$

Relation to the Fatou set

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:

$${\displaystyle K(f)=F_{\infty }^{C}.}$$

Relation between Julia, filled-in Julia set and attractive basin of infinity

Wikibooks has a book on the topic of: Fractals

The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity

$${\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )}$$

${\displaystyle A_{f}(\infty )}$

${\displaystyle f}$

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

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.

Spine

The most studied polynomials are probably those of the form ${\displaystyle f(z)=z^{2}+c}$, which are often denoted by ${\displaystyle f_{c))$, where ${\displaystyle c}$ is any complex number. In this case, the spine ${\displaystyle S_{c))$ of the filled Julia set ${\displaystyle K}$ is defined as arc between ${\displaystyle \beta }$-fixed point and ${\displaystyle -\beta }$,

$${\displaystyle S_{c}=\left[-\beta ,\beta \right]}$$

• spine lies inside ${\displaystyle K}$.^[1] This makes sense when ${\displaystyle K}$ is connected and full^[2]
• spine is invariant under 180 degree rotation,
• spine is a finite topological tree,
• Critical point ${\displaystyle z_{cr}=0}$ always belongs to the spine.^[3]
• ${\displaystyle \beta }$-fixed point is a landing point of external ray of angle zero ${\displaystyle {\mathcal {R))_{0}^{K))$,
• ${\displaystyle -\beta }$ is landing point of external ray ${\displaystyle {\mathcal {R))_{1/2}^{K))$.

Algorithms for constructing the spine:

• detailed version is described by A. Douady^[4]
• Simplified version of algorithm:
  • connect ${\displaystyle -\beta }$ and ${\displaystyle \beta }$ within ${\displaystyle K}$ by an arc,
  • when ${\displaystyle K}$ has empty interior then arc is unique,
  • otherwise take the shortest way that contains ${\displaystyle 0}$.^[5]

Curve ${\displaystyle R}$:

$${\displaystyle R{\overset {\mathrm {def} }((}={))}R_{1/2}\cup S_{c}\cup R_{0))$$

Images

Names

• airplane^[6]
• Douady rabbit
• dragon
• basilica or San Marco fractal or San Marco dragon
• cauliflower
• dendrite
• Siegel disc