Filled Julia set

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 ${\displaystyle f}$

where:

• ${\displaystyle \mathbb {C} }$ 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.

The attractive basin of infinity is one of the components of the Fatou set.

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:

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

where: ${\displaystyle A_{f}(\infty )}$ denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for ${\displaystyle f}$

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

• 
  Rabbit Julia set with spine
• 
  Basilica Julia set with 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 ${\displaystyle -\beta }$,

with such properties:

• 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 ${\displaystyle R}$:

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

Notes

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.