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 $f\; \_c$

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

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

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

$f\_c:mathbb\; Ctomathbb\; C$

$f\_c$ 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 $f\_c$

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

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

Spine

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 properities:

• spine lays 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 constructiong the spine:

• Fractals/Iterations in the complex plane/Julia set 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 shorest way that contains $0$.

Curve $R,$ :

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

divides dynamical plane into 2 components.

Images

References

1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0387158518.
2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathemathics Technical University of Denmark , MAT-Report no. 1996-42