Filled Julia set

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

Formal definition

The filled-in Julia set K ( f ) {\displaystyle K(f)} of a polynomial f {\displaystyle f} is defined as the set of all points z {\displaystyle z} of the dynamical plane that have bounded orbit with respect to f {\displaystyle f}

K ( f ) = d e f { z ∈ C : f ( k ) ( z ) ↛ ∞   as   k → ∞ } {\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}
where:

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

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.

K ( f ) = C ∖ A f ( ∞ ) {\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}

The attractive basin of infinity is one of the components of the Fatou set.

A f ( ∞ ) = F ∞ {\displaystyle A_{f}(\infty )=F_{\infty }}

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:

K ( f ) = F ∞ C . {\displaystyle K(f)=F_{\infty }^{C}.}

Relation between Julia, filled-in Julia set and attractive basin of infinity

Wikibooks has a book on the topic of: Fractals

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

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

A f ( ∞ )   = d e f   { z ∈ C : f ( k ) ( z ) → ∞   a s   k → ∞ } . {\displaystyle 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 {\displaystyle f} are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

  • Rabbit Julia set with spine
    Rabbit Julia set with spine
  • Basilica Julia set with spine
    Basilica Julia set with spine

The most studied polynomials are probably those of the form f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , which are often denoted by f c {\displaystyle f_{c}} , where c {\displaystyle c} is any complex number. In this case, the spine S c {\displaystyle S_{c}} of the filled Julia set K {\displaystyle K} is defined as arc between β {\displaystyle \beta } -fixed point and − β {\displaystyle -\beta } ,

S c = [ − β , β ] {\displaystyle S_{c}=\left[-\beta ,\beta \right]}
with such properties:

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

Algorithms for constructing the spine:

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

Curve R {\displaystyle R} :

R = d e f R 1 / 2 ∪ S c ∪ R 0 {\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}}
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 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 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=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
  • Douady rabbit
  • Filled Julia set for c = −0.8 + 0.156i.
    Filled Julia set for c = −0.8 + 0.156i.
  • Filled Julia set for c = 0.285 + 0.01i.
    Filled Julia set for c = 0.285 + 0.01i.
  • Filled Julia set for c = −1.476.
    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 Archived 2012-02-08 at the Wayback Machine
  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.
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