Filled Julia Set

The filled-in Julia set $K(f)$ of a polynomial $f$ is a Julia set and its interior, non-escaping 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 \left \$
where:
* $\mathbb$ is the set of complex numbers
* $f^ (z)$ 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.
$K(f) = \mathbb \setminus A_(\infty)$

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: $A_(\infty)$ denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for $f$

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

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 $f$ 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 /math>
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 $z_ = 0$ always belongs to the spine.
*$\beta$ -fixed point is a landing point of external ray of angle zero $\mathcal^K _0$,
*$-\beta$ is landing point of external ray $\mathcal^K _$.

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 $R$:
$R \overset R_ \cup S_c \cup R_0$
divides dynamical plane into two components.

Images

File:Julia-Menge.png, Filled Julia set for f_c, c=1−φ=−0.618033988749…, where φ is the Golden ratio
File:Julia IIM 1.jpg, Filled Julia with no interior = Julia set. It is for c=i.
File:Filled.jpg, Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
File:ColorDouadyRabbit1.jpg, Douady rabbit
File:Julia-Menge -0.4+0.6i.png, Filled Julia set for c = −0.4+0.6i.
File:Julia-Menge -0.8 0.156i.png, Filled Julia set for c = −0.8 + 0.156i.
File:Julia-Menge 0.285 0.01i Julia 002.png, Filled Julia set for c = 0.285 + 0.01i.
File:Julia-Menge -1.476 0i Julia.png, Filled Julia set for c = −1.476.

Names

* airplaneThe Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher
/ref>
* Douady rabbit
* dragon
* basilica o
San Marco fractal
o
San Marco dragon
* cauliflower


* Siegel disc

Notes

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

{{DEFAULTSORT:Filled Julia Set
Fractals
Limit sets
Complex dynamics