Filled Julia set
The filled-in Julia set of a polynomial is a Julia set and its interior, non-escaping set.
Formal definition
The filled-in Julia set of a polynomial is defined as the set of all points of the dynamical plane that have
- is the set of complex numbers
- is the -fold composition of with itself = iteration of function
Relation to the Fatou set
The filled-in Julia set is the
The
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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
The
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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Rabbit Julia set with spine
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Basilica Julia set with spine
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 ,
- 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 :
Images
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Filled Julia set for fc, c=1−φ=−0.618033988749…, where φ is the Golden ratio
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Filled Julia with no interior = Julia set. It is for c=i.
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Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
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Filled Julia set for c = −0.8 + 0.156i.
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Filled Julia set for c = 0.285 + 0.01i.
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Filled Julia set for c = −1.476.
Names
- airplane[6]
- Douady rabbit
- dragon
- basilica or San Marco fractal or San Marco dragon
- cauliflower
- dendrite
- Siegel disc
Notes
- ^ 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
- ^ John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
- ^ Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
- ^ 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.
- ^ K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257
- ^ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher
References
- Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
- Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.