Filled Julia set

Summary

The filled-in Julia set of a polynomial is a Julia set and its interior, non-escaping set.

Formal definition

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The filled-in Julia set   of a polynomial   is defined as the set of all points   of the dynamical plane that have bounded orbit with respect to  

 
where:

Relation to the Fatou set

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

 

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

 

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

 

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

 
where:   denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for  

 

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   are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

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The most studied polynomials are probably those of the form  , which are often denoted by  , where   is any complex number. In this case, the spine   of the filled Julia set   is defined as arc between  -fixed point and  ,

 
with such properties:
  • spine lies inside  .[1] This makes sense when   is connected and full[2]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point   always belongs to the spine.[3]
  •  -fixed point is a landing point of external ray of angle zero  ,
  •   is landing point of external ray  .

Algorithms for constructing the spine:

  • detailed version is described by A. Douady[4]
  • Simplified version of algorithm:
    • connect   and   within   by an arc,
    • when   has empty interior then arc is unique,
    • otherwise take the shortest way that contains  .[5]

Curve  :

 
divides dynamical plane into two components.

Images

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Names

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Notes

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  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

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  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.