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# Filled Julia set

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

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

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

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

$f_c:mathbb Ctomathbb C$
$f_c$ may be various functions. In typical case $f_c$ is complex quadratic polynomial.

$f^\left\{\left(k\right)\right\} _c \left(z\right)$ 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.
$K\left(f_c\right) = mathbb\left\{C\right\} setminus A_\left\{f_c\right\}\left(infty\right)$

Attractive basin of infinity is one of components of the Fatou set.
$A_\left\{f_c\right\}\left(infty\right) = F_infty$

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
$K\left(f_c\right) = F_infty^C$

## 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

$J\left(f_c\right), = partial K\left(f_c\right) =partial A_\left\{f_c\right\}\left(infty\right)$

where :
$A_\left\{f_c\right\}\left(infty\right)$ denotes attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for $f_c$

$A_\left\{f_c\right\}\left(infty\right) overset\left\{underset\left\{mathrm\left\{def\right\}\right\}\left\{\right\}\right\}\left\{=\right\} \left\{ z in mathbb\left\{C\right\} : f^\left\{\left(k\right)\right\} _c \left(z\right) to infty as k to infty \right\}$

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 $-beta,$,

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

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 $z_\left\{cr\right\} = 0 ,$ always belongs to the spine.
• $beta,$ -fixed point is a landing point of external ray of angle zero $mathcal\left\{R\right\}^K _0$,
• $-beta,$ is landing point of external ray $mathcal\left\{R\right\}^K _\left\{1/2\right\}$.

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 $R,$ :

$R overset\left\{underset\left\{mathrm\left\{def\right\}\right\}\left\{\right\}\right\}\left\{=\right\} R_\left\{1/2\right\} cup S_c cup R_0 ,$

divides dynamical plane into 2 components.

## 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

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