# 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 ${\displaystyle f}$

${\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}$
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.

${\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}$

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

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

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

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

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 ${\displaystyle -\beta }$,

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

${\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}}$
divides dynamical plane into two components.

## Notes

1. ^
2. ^ John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
3. ^ Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
4. ^ A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.
5. ^
6. ^ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher

## 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.