Filled Julia set
The filled-in Julia set of a polynomial is :
is the set of complex numbers
Relation to the Fatou set
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
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.
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 . This makes sense when is connected and full
- spine is invariant under 180 degree rotation,
- spine is a finite topological tree,
- Critical point always belongs to the spine.
- -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
- Simplified version of algorithm:
divides dynamical plane into two components.
Filled Julia set for fc, 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.
- Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester
- 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
- 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.