Filled Julia set

The filled-in Julia set $\ K(f)$ of a polynomial $\ f$ is a Julia set and its interior.

Formal definition

The filled-in Julia set $\ K(f)$ of a polynomial $\ f$ is defined as the set of all points $z\,$ of 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 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 attractive basin of infinity.
$K(f) = \mathbb{C} \setminus A_{f}(\infty)$

Attractive basin of infinity is one of 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

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

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

where :
$A_{f}(\infty)$ denotes 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 Julia set coincides with 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 $\mathcal{R}^K _0$,
• $-\beta\,$ is landing point of external ray $\mathcal{R}^K _{1/2}$.

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

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.