Filled Julia set explained

The filled-in Julia set

K(f)

of a polynomial

is a Julia set and its interior, non-escaping set.

Formal definition

K(f)

of a polynomial

is defined as the set of all points

of the dynamical plane that have bounded orbit with respect to

K(f) \overset \left \

where:

is the set of complex numbers

f^(k)(z)

is the

-fold composition of

with itself = iteration of function

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. $K(f) = \mathbb \setminus A_(\infty)$

The attractive basin of infinity is one of the components of the Fatou set. $A_(\infty) = F_\infty$

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 $J(f) = \partial K(f) = \partial A_(\infty)$ where:

A_f(infty)

denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for

$A_(\infty) \ \overset \ \.$

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

are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

f(z)=z²+c

, which are often denoted by

f_c

, where

is any complex number. In this case, the spine

S_c

of the filled Julia set

is defined as arc between

\beta

-fixed point and

-\beta

S_c = \left [- \beta, \beta \right ]

with such properties:

spine lies inside

.^[1] This makes sense when

is connected and full^[2]

spine is invariant under 180 degree rotation,
spine is a finite topological tree,

z_cr=0

always belongs to the spine.^[3]

\beta

-fixed point is a landing point of external ray of angle zero

	K
l{R}
	0

-\beta

is landing point of external ray

	K
l{R}
	1/2

Algorithms for constructing the spine:

detailed version is described by A. Douady^[4]
Simplified version of algorithm:
- connect

-\beta

and

\beta

within

by an arc,

- when

has empty interior then arc is unique,

- otherwise take the shortest way that contains

.^[5]

Curve

R \overset R_ \cup S_c \cup R_0

divides dynamical plane into two components.

Names

airplane^[6]
Douady rabbit
dragon
basilica or San Marco fractal or San Marco dragon
cauliflower
dendrite
Siegel disc

References

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

Notes and References

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
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)
https://arxiv.org/abs/math/9801148 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.
[Karen Brucks|K M. Brucks]
http://www.math.uni-bonn.de/people/karcher/Julia_Sets.pdf The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher