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 fc, 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.
• • Filled Julia set for c = −0.4+0.6i.
• 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

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
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.