Filled Julia set

The filled-in Julia set $K(f\_c)$ of a polynomial $f\; \_c$ is defined as the set of all points $z,$ of dynamical plane that have bounded orbit with respect to $f\; \_c$

$K(f\_c)\; overset\{underset\{mathrm\{def\{\{=\}\; \{\; z\; in\; mathbb\{C\}\; :\; f^\{(k)\}\; \_c\; (z)\; ot\; o\; infty\; as\; k\; o\; infty\; \}$
where :

$mathbb\{C\}$ is set of complex numbers

$z,$ is complex variable of function $f\; \_c\; (z)$
$c,$ is complex parameter of function $f\; \_c\; (z)$

:$f\_c:mathbb\; C\; omathbb\; C$
$f\_c$ may be various functions. In typical case $f\_c$ is complex quadratic polynomial.

$f^\{(k)\}\; \_c\; (z)$ is the $k$ -fold compositions of $f\; \_c,$ with itself = iteration of function $f\; \_c,$

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of attractive basin of infinity.
$K(f\_c)\; =\; mathbb\{C\}\; setminus\; A\_\{f\_c\}(infty)$

Attractive basin of infinity is one of components of the Fatou set.
$A\_\{f\_c\}(infty)\; =\; F\_infty$

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
$K(f\_c)\; =\; F\_infty^C$

Relation between Julia, filled-in Julia set and attractive basin of infinity

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

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

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

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

If filled-in Julia set has no interior then Julia set coincides with filled-in Julia set. It happens when $c,$ is Misiurewicz point.

pine

Spine $S\_c,$ of the filled Julia set $K\; ,$ is defined as arc between -fixed point and ,

with such properities:
*spine lays 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] ] . 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. [ [http://arxiv.org/abs/math/9801148 Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case] ]
* -fixed point is a landing point of external ray of angle zero $mathcal\{R\}^K\; \_0$,
* is landing point of external ray $mathcal\{R\}^K\; \_\{1/2\}$.

Algorithms for constructiong the spine:
* is described by A. Douady [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, Ga, USA, 1986.]

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

Curve $R,$ :

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

divides dynamical plane into 2 components.

=

References

# Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0387158518.
# Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathemathics Technical University of Denmark , [http://www2.mat.dtu.dk/publications/uk?id=122 MAT-Report no. 1996-42] .