Effective medium approximations

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 $\ f$

$\ K(f)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty \ as\ k\to \infty \}$
where :

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

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

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

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

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

$J(f)\,=\partial K(f)=\partial A_{f}(\infty )$

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

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

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

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 $-\beta \,$,

$S_{c}=\left[-\beta ,\beta \right]\,$

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

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 $R\,$ :

$R\ {\overset {\underset {\mathrm {def} }{}}{=}}\ R_{1/2}\ \cup \ S_{c}\ \cup \ R_{0}\,$

divides dynamical plane into two components.

Images

• 
  Filled Julia set for f_c, c=φ−2=-0.38..., where φ means Golden ratio
• 
  Filled Julia with no interior = Julia set. It is for c=i.
• 
  Filled Julia set for c=-1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
• 
  Douady rabbit

Notes

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References

1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.