Julia set

Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given an iterated map of the complex plane to itself, or a collection of such maps, the Julia set for this system can be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Generalisations

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The Julia set can be defined for any iterated map of the complex plane to itself, or collection of maps. The formal mathematical definition is that the Julia set is the smallest closed, completely invariant point set for such a map or collection of maps which contains at least 3 points. It can also be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

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Julia sets can also be defined for any n-dimensional space, not just the complex plane.

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Julia sets typically (though not always) have a fractal structure, and Julia sets can be associated with fractals such as the Sierpinski triangle and the Cantor set.

Related Topics:
Fractal - Sierpinski triangle - Cantor set

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The complement of the Julia set (i.e. the set of points for which nearby points do exhibit similar behaviour) is sometimes called the Fatou set, although some authors use the term Fatou set or Fatou dust for a disconnected Julia set.

Related Topics:
Complement - Fatou set - Disconnected

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Some common generalisations include:

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• The cubic map, $z\_\{n+1\}\; =\; z\_n^3\; +\; b$, or (with greater generality), $z\_\{n+1\}\; =\; z\_n^3\; -\; 3a^2z\; +\; b$, where $a$ and $b$ are complex-valued parameters (as $c$ is in the quadratic mapping). The resulting Mandelbrot set is 4-dimensional. The map has critical points (infinity, zero, $+a$, $-a$). As with the quadratic map, infinity is always an attractor. However, zero is always attracted to infinity. It thus remains to check the orbits of $a$ and $-a$.
• Newton's method of approximation can be used to solve various equations in the complex plane. Specifically, to solve some equation $f(z)=0$, we iterate $z\_\{n+1\}\; =\; z\_n\; -\; f(z)/f\text{'}(z)$ (where $f\text{'}(z)$ is the first derivative of $f$ with respect to $z$.) The method is intended to work starting from a fairly good approximation of the solution. Graphing with arbitrary start points, we typically find parts of the Julia set at points equidistant between roots (so the algorithm "can't decide" which one to go far). The most common example used is the cube roots of unity (i.e., $f(z)\; =\; z^3\; +\; 1$). To form a Mandelbrot set, a parametrised equation is needed.
• In a similar way, Halley's method can be applied, generally resulting in more complex images.
• Certain of the complex transcendental functions (such as sin, cos, exp, log, etc.) can be iterated to produce Julia sets. For example, $z\_\{n+1\}\; =\; lambda\; sin(z\_n)$ where $lambda$ is an adjustable parameter. (Note that transcendental functions are not rational maps - so Julia's result about critical points does not necessarily hold.)