The filled-in Julia set $K(f)$ of a polynomial $f$ is a Julia set and its interior, non-escaping set.
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
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
$J(f)=\partial K(f)=\partial A_{f}(\infty )$
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 $K$.^{[1]} This makes sense when $K$ is connected and full^{[2]}
- spine is invariant under 180 degree rotation,
- spine is a finite topological tree,
- Critical point $z_{cr}=0$ always belongs to the spine.^{[3]}
- $\beta$-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^{[4]}
- 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 $0$.^{[5]}
Curve $R$:
$R{\overset {\mathrm {def} }((}={))}R_{1/2}\cup S_{c}\cup R_{0))$
divides dynamical plane into two components.