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 $\ f$

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

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 $\ K \,$.^[1] This makes sense when $K\,$ is connected and full ^[2]
• spine is invariant under 180 degree rotation,
• spine is a finite topological tree,
• Critical point $z_{cr} = 0 \,$ always belongs to the spine.^[3]
• $\beta\,$-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^[4]
• 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 $0$.^[5]

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

1. Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester
2. John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
3. Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
4. A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.
5. K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257

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.