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 $K$ .^[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.

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

^ Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester Archived 2012-02-08 at the Wayback Machine
^ John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
^ Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
^ 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.
^ K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257
^ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher

References

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

[1] Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester Archived 2012-02-08 at the Wayback Machine

[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] K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257

[6] The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher

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[2]

[3]

[4]

[5]

[6]

Fractals
Characteristics	Fractal dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity
Iterated function system	Barnsley fern Cantor set Koch snowflake Menger sponge Sierpiński carpet Sierpiński triangle Apollonian gasket Fibonacci word Space-filling curve Blancmange curve De Rham curve Minkowski Dragon curve Hilbert curve Koch curve Lévy C curve Moore curve Peano curve Sierpiński curve Z-order curve String T-square n-flake Vicsek fractal Gosper curve Pythagoras tree Weierstrass function
Strange attractor	Multifractal system
L-system	Fractal canopy Space-filling curve H tree
Escape-time fractals	Burning Ship fractal Julia set Filled Newton fractal Douady rabbit Lyapunov fractal Mandelbrot set Misiurewicz point Multibrot set Newton fractal Tricorn Mandelbox Mandelbulb
Rendering techniques	Buddhabrot Orbit trap Pickover stalk
Random fractals	Brownian motion Brownian tree Brownian motor Fractal landscape Lévy flight Percolation theory Self-avoiding walk
People	Michael Barnsley Georg Cantor Bill Gosper Felix Hausdorff Desmond Paul Henry Gaston Julia Niels Fabian Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Hamid Naderi Yeganeh Lewis Fry Richardson Wacław Sierpiński
Other	Coastline paradox Fractal art List of fractals by Hausdorff dimension The Fractal Geometry of Nature (1982 book) The Beauty of Fractals (1986 book) Chaos: Making a New Science (1987 book) Kaleidoscope Chaos theory

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