Filled Julia set#Spine

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 $f$

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

where:

• $\backslash mathbb\{C\}$ is the set of complex numbers
• $f^\{(k)\}\; (z)$ is the $k$ -fold composition of $f$ with itself = iteration of function $f$

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.

$$K(f)\; =\; \backslash mathbb\{C\}\; \backslash setminus\; A\_\{f\}(\backslash infty)$$

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

$$A\_\{f\}(\backslash infty)\; =\; F\_\backslash infty$$

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

$$K(f)\; =\; F\_\backslash infty^C.$$

{{Wikibooks|Fractals}}

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

$$J(f)\; =\; \backslash partial\; K(f)\; =\; \backslash partial\; A\_\{f\}(\backslash infty)$$

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

$$A\_\{f\}(\backslash infty)\; \backslash \; \backslash overset\{\backslash underset\{\backslash mathrm\{def\}\}\{\}\}\{=\}\; \backslash \; \backslash \{\; z\; \backslash in\; \backslash mathbb\{C\}\; :\; f^\{(k)\}\; (z)\; \backslash to\; \backslash infty\backslash \; as\backslash \; k\; \backslash to\; \backslash infty\; \backslash \}.$$

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.

Rabbit Julia set with spine.svg|Rabbit Julia set with spine

Basilica Julia set with spine.svg|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 $\backslash beta$-fixed point and $-\backslash beta$,

$$S\_c\; =\; \backslash left\; [\; -\; \backslash beta\; ,\; \backslash beta\; \backslash right\; ]$$

with such properties:

• spine lies inside $K$.[http://www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester] {{webarchive|url=https://web.archive.org/web/20120208022215/http://www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html |date=2012-02-08 }} This makes sense when $K$ is connected and full[http://www.emis.de/journals/EM/expmath/volumes/13/13.1/Milnor.pdf John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)]
• spine is invariant under 180 degree rotation,
• spine is a finite topological tree,
• Critical point $z\_\{cr\}\; =\; 0$ always belongs to the spine.[https://arxiv.org/abs/math/9801148 Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case]
• $\backslash beta$-fixed point is a landing point of external ray of angle zero $\backslash mathcal\{R\}^K\; \_0$,
• $-\backslash beta$ is landing point of external ray $\backslash mathcal\{R\}^K\; \_\{1/2\}$.

Algorithms for constructing the spine:

• detailed version is described by A. DouadyA. 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.
• Simplified version of algorithm:
• connect $-\; \backslash beta$ and $\backslash beta$ within $K$ by an arc,
• when $K$ has empty interior then arc is unique,
• otherwise take the shortest way that contains $0$.K M. Brucks, H Bruin : [http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521547666 Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257]

Curve $R$:

$$R\; \backslash overset\{\backslash mathrm\{def\}\}\{\{\}=\{\}\}\; R\_\{1/2\}\; \backslash cup\; S\_c\; \backslash cup\; R\_0$$

divides dynamical plane into two components.

Julia-Menge.png|Filled Julia set for f_c, c=1−φ=−0.618033988749…, where φ is the Golden ratio

Julia IIM 1.jpg|Filled Julia with no interior = Julia set. It is for c=i.

Filled.jpg|Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.

ColorDouadyRabbit1.jpg|Douady rabbit

Julia-Menge -0.8 0.156i.png|Filled Julia set for c = −0.8 + 0.156i.

Julia-Menge 0.285 0.01i Julia 002.png|Filled Julia set for c = 0.285 + 0.01i.

Julia-Menge -1.476 0i Julia.png|Filled Julia set for c = −1.476.

• airplane[http://www.math.uni-bonn.de/people/karcher/Julia_Sets.pdf The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher]
• Douady rabbit
• dragon
• basilica or [http://mathworld.wolfram.com/SanMarcoFractal.html San Marco fractal] or [https://planetmath.org/sanmarcodragon San Marco dragon]
• cauliflower
• [http://mathworld.wolfram.com/DendriteFractal.html dendrite]
• Siegel disc

{{Reflist}}

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, [https://web.archive.org/web/20070625193229/http://www2.mat.dtu.dk/publications/uk?id=122 MAT-Report no. 1996-42].

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Category:Fractals

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