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

References

1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.