Filled Julia 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.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 \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}^K _0,
• -\beta is landing point of external ray \mathcal{R}^K _{1/2}.

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

Julia-Menge.png|Filled Julia set for fc, 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.

Names

• airplane
• 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. .
2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.

References

1. [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. link. (2012-02-08)
2. [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)]
3. [https://arxiv.org/abs/math/9801148 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. [[Karen Brucks
6. [http://www.math.uni-bonn.de/people/karcher/Julia_Sets.pdf The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher]