Editing Julia set (section)

==Quadratic polynomials==

A very popular complex dynamical system is given by the family of [[complex quadratic polynomial]]s, a special case of  [[Rational function|rational maps]]. Such quadratic polynomials can be expressed as 
: <math>f_c(z) = z^2 + c ~,</math>
where ''c'' is a complex parameter. Fix some <math>R > 0</math> large enough that <math>R^2 - R \ge |c|.</math> (For example, if {{mvar|c}} is in the [[Mandelbrot set]], then <math>|c| \le 2,</math> so we may simply let <math>R = 2~.</math>) Then the filled Julia set for this system is the subset of the complex plane given by
: <math>K(f_c) = \left\{z \in \mathbb C : \forall n \in \mathbb N, |f_c^n(z)| \le R \right\}~,</math>
where <math>f_c^n(z)</math> is the ''n''th [[iterated function|iterate]] of <math>f_c(z).</math> The Julia set <math>J(f_c)</math> of this function is the boundary of <math>K(f_c)</math>.

<gallery widths="180" heights="180">
File:JSr07885.gif|alt=Julia sets for                               z                      2                          +        0.7885                          e                      i            a                          ,              {\displaystyle z^{2}+0.7885\,e^{ia},}   where a ranges from 0 to                     2        π              {\displaystyle 2\pi }|Julia sets for <math> z^2 + 0.7885\,e^{ia} ,</math> where ''a'' ranges from 0 to <math>2\pi</math>
File:Julia circling.ogg|A video of the Julia sets as left
File:Julia-Menge.png|Filled Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = 1 − ''φ''}}, where ''φ'' is the [[golden ratio]]
File:Julia 0.4 0.6.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|''c'' {{=}} (''φ'' − 2) + (''φ'' − 1)''i'' {{=}} −0.4 + 0.6''i''}}
File:Julia 0.285 0.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = 0.285 + 0''i''}}
File:Julia 0.285 0.01.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = 0.285 + 0.01''i''}}
File:Julia 0.45 0.1428.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = 0.45 + 0.1428''i''}}
File:Julia -0.70176 -0.3842.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = −0.70176 − 0.3842''i''}}
File:Julia -0.835 -0.2321.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = −0.835 − 0.2321''i''}}
File:Julia -0.8 0.156.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = −0.8 + 0.156''i''}}
File:Julia -0.7269 0.1889.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=v''c'' = −0.7269 + 0.1889''i''}}
File:Julia 0 0.8.png|Julia set for ''f<sub>c</sub>'', {{math|size=100%|1=''c'' = -0.8''i''}}
File:JuliaSet.35.png|Julia set for ''f<sub>c</sub>'', {{math|1=''c'' = 0.35 + 0.35''i''|size=100%}}
File:JuliaSet.4.png|Julia set for ''f<sub>c</sub>'', {{math|1=''c'' = 0.4 + 0.4''i''|size=100%}}
File:Julia Mandelbrot Relationship.png|Collection of Julia sets laid out in a 100 × 100 grid such that the center of each image corresponds to the same position in the complex plane as the value of the set. When laid out like this, the overall image resembles a [[Photographic mosaic]] depicting a [[Mandelbrot set]]. 
</gallery>
The parameter plane of quadratic polynomials&nbsp;– that is, the plane of possible ''c'' values&nbsp;– gives rise to the famous [[Mandelbrot set]]. Indeed, the Mandelbrot set is defined as the set of all ''c'' such that <math>J(f_c)</math> is [[connected set|connected]]. For parameters outside the Mandelbrot set, the Julia set is a [[Cantor space]]: in this case it is sometimes referred to as '''Fatou dust'''.

In many cases, the Julia set of ''c'' looks like the Mandelbrot set in sufficiently small neighborhoods of ''c''. This is true, in particular, for so-called [[Misiurewicz point|Misiurewicz parameters]], i.e. parameters ''c'' for which the critical point is pre-periodic. For instance:
* At ''c'' = ''i'', the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
* At ''c'' = −2, the tip of the long spiky tail, the Julia set is a straight line segment.

In other words, the Julia sets <math>J(f_c)</math> are locally similar around [[Misiurewicz point]]s.<ref>[[Tan Lei]], [http://projecteuclid.org/euclid.cmp/1104201823 "Similarity between the Mandelbrot set and Julia Sets"], Communications in Mathematical Physics 134 (1990), pp. 587&ndash;617.</ref>