Filled Julia set

From HandWiki

The filled-in Julia set [math]\displaystyle{ K(f) }[/math] of a polynomial [math]\displaystyle{ f }[/math] is a Julia set and its interior, non-escaping set

Formal definition

The filled-in Julia set [math]\displaystyle{ K(f) }[/math] of a polynomial [math]\displaystyle{ f }[/math] is defined as the set of all points [math]\displaystyle{ z }[/math] of the dynamical plane that have bounded orbit with respect to [math]\displaystyle{ f }[/math] [math]\displaystyle{ K(f) \overset{\mathrm{def}}{{}={}} \left \{ z \in \mathbb{C} : f^{(k)} (z) \not\to \infty ~ \text{as} ~ k \to \infty \right\} }[/math] where:

  • [math]\displaystyle{ \mathbb{C} }[/math] is the set of complex numbers
  • [math]\displaystyle{ f^{(k)} (z) }[/math] is the [math]\displaystyle{ k }[/math] -fold composition of [math]\displaystyle{ f }[/math] with itself = iteration of function [math]\displaystyle{ f }[/math]

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. [math]\displaystyle{ K(f) = \mathbb{C} \setminus A_{f}(\infty) }[/math]

The attractive basin of infinity is one of the components of the Fatou set. [math]\displaystyle{ A_{f}(\infty) = F_\infty }[/math]

In other words, the filled-in Julia set is the complement of the unbounded Fatou component: [math]\displaystyle{ K(f) = F_\infty^C. }[/math]

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 [math]\displaystyle{ J(f) = \partial K(f) = \partial A_{f}(\infty) }[/math] where: [math]\displaystyle{ A_{f}(\infty) }[/math] denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for [math]\displaystyle{ f }[/math]

[math]\displaystyle{ A_{f}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \to \infty\ as\ k \to \infty \}. }[/math]

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 [math]\displaystyle{ f }[/math] are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

  • Rabbit Julia set with spine

  • Basilica Julia set with spine

The most studied polynomials are probably those of the form [math]\displaystyle{ f(z) = z^2 + c }[/math], which are often denoted by [math]\displaystyle{ f_c }[/math], where [math]\displaystyle{ c }[/math] is any complex number. In this case, the spine [math]\displaystyle{ S_c }[/math] of the filled Julia set [math]\displaystyle{ K }[/math] is defined as arc between [math]\displaystyle{ \beta }[/math]-fixed point and [math]\displaystyle{ -\beta }[/math], [math]\displaystyle{ S_c = \left [ - \beta , \beta \right ] }[/math] with such properties:

  • spine lies inside [math]\displaystyle{ K }[/math].[1] This makes sense when [math]\displaystyle{ K }[/math] is connected and full[2]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point [math]\displaystyle{ z_{cr} = 0 }[/math] always belongs to the spine.[3]
  • [math]\displaystyle{ \beta }[/math]-fixed point is a landing point of external ray of angle zero [math]\displaystyle{ \mathcal{R}^K _0 }[/math],
  • [math]\displaystyle{ -\beta }[/math] is landing point of external ray [math]\displaystyle{ \mathcal{R}^K _{1/2} }[/math].

Algorithms for constructing the spine:

  • detailed version is described by A. Douady[4]
  • Simplified version of algorithm:
    • connect [math]\displaystyle{ - \beta }[/math] and [math]\displaystyle{ \beta }[/math] within [math]\displaystyle{ K }[/math] by an arc,
    • when [math]\displaystyle{ K }[/math] has empty interior then arc is unique,
    • otherwise take the shortest way that contains [math]\displaystyle{ 0 }[/math].[5]

Curve [math]\displaystyle{ R }[/math]: [math]\displaystyle{ R \overset{\mathrm{def}}{{}={}} R_{1/2} \cup S_c \cup R_0 }[/math] divides dynamical plane into two components.

Images

  • Filled Julia set for fc, c=1−φ=−0.618033988749…, where φ is the 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.

  • Filled Julia set for c = −0.8 + 0.156i.

  • Filled Julia set for c = 0.285 + 0.01i.

  • Filled Julia set for c = −1.476.

Names

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
  6. ↑ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher

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.



Retrieved from "https://handwiki.org/wiki/index.php?title=Filled_Julia_set&oldid=3375389"