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| The filled-in Julia set <math>\ K(f) </math> of a polynomial <math>\ f </math> is :
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| * a [[Julia set]] and its [[Interior (topology)|interior]],
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| * [[escaping set|non-escaping set]]
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| ==Formal definition==
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| The filled-in [[Julia set]] <math>\ K(f) </math> of a polynomial <math>\ f </math> is defined as the set of all points <math>z\,</math> of the dynamical plane that have [[Bounded sequence|bounded]] [[Orbit (dynamics)|orbit]] with respect to <math>\ f </math>
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| <math> \ K(f) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \not\to \infty\ as\ k \to \infty \} </math>
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| <br>
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| where :
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| <math>\mathbb{C}</math> is the [[Complex number|set of complex numbers]]
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| <math> \ f^{(k)} (z) </math> is the <math>\ k</math> -fold [[Function composition|composition]] of <math>f \,</math> with itself = [[Iterated function|iteration of function]] <math>f \,</math>
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| ==Relation to the Fatou set==
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| The filled-in Julia set is the [[Complement (set theory)|(absolute) complement]] of the [[Basin of attraction|attractive basin]] of [[Point at infinity|infinity]]. <br />
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| <math>K(f) = \mathbb{C} \setminus A_{f}(\infty)</math>
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| The [[Basin of attraction|attractive basin]] of [[Point at infinity|infinity]] is one of the [[Classification of Fatou components|components of the Fatou set]].<br />
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| <math>A_{f}(\infty) = F_\infty </math>
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| In other words, the filled-in Julia set is the [[Complement (set theory)|complement]] of the unbounded [[Classification of Fatou components|Fatou component]]: <br />
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| <math>K(f) = F_\infty^C.</math>
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| ==Relation between Julia, filled-in Julia set and attractive basin of infinity==
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| {{Wikibooks|Fractals }}
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| The [[Julia set]] is the common [[Boundary (topology)|boundary]] of the filled-in Julia set and the [[Basin of attraction|attractive basin]] of [[Point at infinity|infinity]] <br>
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| <br />
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| <math>J(f)\, = \partial K(f) =\partial A_{f}(\infty)</math><br />
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| <br />
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| where : <br />
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| <math>A_{f}(\infty)</math> denotes the [[Basin of attraction|attractive basin]] of [[Point at infinity|infinity]] = exterior of filled-in Julia set = set of escaping points for <math>f</math><br />
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| <br />
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| <math>A_{f}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \to \infty\ as\ k \to \infty \}. </math>
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| If the filled-in Julia set has no [[Interior (topology)|interior]] then the [[Julia set]] coincides with the filled-in Julia set. This happens when all the critical points of <math>f</math> are pre-periodic. Such critical points are often called [[Misiurewicz point]]s.
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| ==Spine ==
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| The most studied polynomials are probably those of the form <math>f(z)=z^2 + c</math>, which are often denoted by <math>f_c</math>, where <math>c</math> is any complex number. In this case, the spine <math>S_c\,</math> of the filled Julia set <math>\ K \,</math> is defined as [[Arc (projective geometry)|arc]] between <math>\beta\,</math>[[Periodic points of complex quadratic mappings|-fixed point]] and <!-- its preimage --> <math>-\beta\,</math>,
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| <math>S_c = \left [ - \beta , \beta \right ]\,</math>
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| with such properties:
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| *spine lies inside <math>\ K \,</math>.<ref>[http://www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester]</ref> This makes sense when <math>K\,</math> is connected and full <ref>[http://www.emis.de/journals/EM/expmath/volumes/13/13.1/Milnor.pdf John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)]</ref>
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| * spine is invariant under 180 degree rotation,
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| * spine is a finite topological tree,
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| *[[Complex quadratic polynomial|Critical point]] <math> z_{cr} = 0 \,</math> always belongs to the spine.<ref>[http://arxiv.org/abs/math/9801148 Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case]</ref>
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| *<math>\beta\,</math>[[Periodic points of complex quadratic mappings|-fixed point]] is a landing point of [[external ray]] of angle zero <math>\mathcal{R}^K _0</math>,
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| *<math>-\beta\,</math> is landing point of [[external ray]] <math>\mathcal{R}^K _{1/2}</math>.
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| Algorithms for constructing the spine:
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| *[[b:Fractals/Iterations in the complex plane/Julia set|detailed version]] is described by A. Douady<ref>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.</ref>
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| *Simplified version of algorithm:
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| **connect <math>- \beta\,</math> and <math> \beta\,</math> within <math>K\,</math> by an arc,
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| **when <math>K\,</math> has empty interior then arc is unique,
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| **otherwise take the shortest way that contains <math>0</math>.<ref>[http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521547666 K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257]</ref>
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| Curve <math>R\,</math> :
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| <math>R\ \overset{\underset{\mathrm{def}}{}}{=} \ R_{1/2}\ \cup\ S_c\ \cup \ R_0 \,</math>
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| divides dynamical plane into two components.
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| ==Images==
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| <gallery>
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| Image:Time escape Julia set from coordinate (phi-2, 0).jpg|Filled Julia set for f<sub>c</sub>, c=φ−2=-0.38..., where φ means [[Golden ratio]]
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| Image:Julia_IIM_1.jpg| Filled Julia with no interior = Julia set. It is for c=i.
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| Image:Filled.jpg| Filled Julia set for c=-1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
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| Image:ColorDouadyRabbit1.jpg|[[Douady rabbit]]
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| </gallery>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| # Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
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| # Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, [http://www2.mat.dtu.dk/publications/uk?id=122 MAT-Report no. 1996-42].
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| {{DEFAULTSORT:Filled Julia Set}}
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| [[Category:Fractals]]
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| [[Category:Limit sets]]
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| [[Category:Complex dynamics]]
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