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).

Attractors and repellers

As stated above, for any given c, the majority of orbits tend toward infinity. For this reason, infinity is described as an attractor of the system. The set of all points z_0 with orbits that are "attracted to" infinity makes up the basin of attraction to infinity.

Related Topics:
Attractor - Basin of attraction

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Depending on the choice of c, there may also exist one other attractor in the system. While infinity is a point attractor, the second attractor may be either a point attractor or a periodic cycle. (A point attractor is essentially a periodic cycle of period 1.) The exact shape of the basin of attraction to this second attractor depends on c.

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If this second attractor does exist for a particular c, then the Julia set is topologically connected, and is in fact the boundary between the basin of attraction to infinity and the basin of attraction to the finite attractor. If there is no second attractor (i.e., infinity is the only attractor) then the Julia set is a disconnected Cantor dust set.

Related Topics:
Topologically connected - Cantor dust

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One of Gaston Julia's important results is that every basin of attraction always contains at least one critical point of the map (provided the map is rational - which the quadratic map is). Since the quadratic map has two critical points (0 and infinity), there can only ever be at most 2 basins of attraction. Since it turns out that infinity is always an attractor, one can determine whether a second attractor exists simply by examining the orbit of 0. This is why the Mandelbrot set can be drawn by examining the behaviour of quadratic maps at 0.

Related Topics:
Critical point - Rational

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Since (in general) the Julia set is the boundary between basins of attraction, the Julia set is sometimes described as being a repeller because all orbits tend away from it.

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