NationMaster - Encyclopedia: Julia set

In complex dynamics, the Julia set $J(f),$ of a holomorphic function $f,$ informally consists of those points whose long-time behavior under repeated iteration of $f,$ can change drastically under arbitrarily small perturbations. Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...

The Fatou set $F(f),$ of $f,$ is the complement of the Julia set: that is, the set of points which exhibit 'stable' behavior. The word complement (with an e in the second syllable, not to be confused with a different word, compliment with an i) has a number of uses. ...

Thus on $F(f),$ , the behavior of $f,$ is 'regular', while on $J(f),$ , it is 'chaotic'. A plot of the trajectory Lorenz system for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...

These sets are named in honor of the French mathematicians Gaston Julia and Pierre Fatou, who initiated the theory of complex dynamics in the early 20th century. Gaston Maurice Julia (February 3, 1893 – March 19, 1978) was a French mathematician who devised the formula for the Julia set. ... Pierre Fatou was the first to define the Mandelbrot set. ... Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ...

A Julia set

3D slice of a 4D Quaternion Julia set

1 Formal definition
2 Equivalent descriptions of the Julia set
3 Quadratic polynomials
4 Generalisations
5 Plotting the Julia set using backwards iteration
6 See also
7 References

Download high resolution version (1024x768, 436 KB)A julia set with seed coordinates (-0. ... Download high resolution version (1024x768, 436 KB)A julia set with seed coordinates (-0. ... Image File history File links Download high-resolution version (4677x3307, 5922 KB) Save the following content as julia. ... Image File history File links Download high-resolution version (4677x3307, 5922 KB) Save the following content as julia. ...

Formal definition

Let

$f:Xto X,$

be an analytic self-map of a Riemann surface $X,$ . We will assume that $X,$ is either the Riemann sphere, the complex plane, or the once-punctured complex plane, as the other cases do not give rise to interesting dynamics. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

We will be considering $f,$ as a discrete dynamical system on the phase space $X,$ , so we are interested in the behavior of the iterates $f^n,$ of $f,$ (that is, the $n,$ -fold compositions of $f,$ with itself). A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... For other senses of this term, see phase space (disambiguation). ... In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...

The Fatou set of $f,$ consists of all points $zin X,$ such that the family of iterates

$(f^n)_{ninmathbb{N}}$

forms a normal family in the sense of Montel when restricted to some open neighborhood of $z,$ . In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. ... Paul Antoine Aristide Montel (born 29 April 1876, died 22 January 1975) was a French mathematician. ...

The Julia set of $f,$ is the complement of the Fatou set in $X,$ .

Equivalent descriptions of the Julia set

$J(f),$ is the smallest closed set containing at least three points which is completely invariant under $f,$ .
$J(f),$ is the closure of the set of repelling periodic points.
For all but at most two points $zin X,$ , the Julia set is the set of limit points of the full backwards orbit $bigcup_n f^{-n}(z)$ . (This suggests a simple algorithm for plotting Julia sets, see below.)
If $f,$ is an entire function - in particular, when $f,$ is a polynomial, then $J(f),$ is the boundary of the set of points which converge to infinity under iteration.
If $f,$ is a polynomial, then $J(f),$ is the boundary of the filled Julia set; that is, those points whose orbits under $f,$ remain bounded.

In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, a periodic point x is a point for which , where is the nth iterate of some function. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...

Quadratic polynomials

A very popular complex dynamical system is given by the family of quadratic polynomials,

$f_c(z) = z^2 + c,$

(where $c,$ is a complex parameter).

Filled Julia set for f_c, c=φ−2 Download high resolution version (640x640, 48 KB)Example of a julia set, a fractal. ...	Julia set for f_c, c=(φ−2)+(phi−1)i Download high resolution version (640x640, 82 KB)Example of a julia set, a fractal. ...	Julia set for f_c, c=0.285 Download high resolution version (640x640, 47 KB)Example of a julia set, a fractal. ...	Julia set for f_c, c = 0.285 + 0.01i Image File history File links Download high-resolution version (2048x2048, 1116 KB) Julia set, a fractal. ...
Julia set for c=0.45 - 0.1428i Image File history File links Julia_set_camp3. ...	Julia set for f_c, c=-0.70176 -0.3842i Image File history File links Julia_set_camp1. ...	Julia set for f_c, c=-0.835-0.2321i Image File history File links Julia_set_camp2. ...

The parameter plane of quadratic polynomials - that is, the plane of possible $c$ -values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all $c$ such that $J(f_c),$ is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor set: in this case it is sometimes referred to as Fatou dust. Initial image of a Mandelbrot set zoom sequence with continuously colored environment. ... In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

Map of 121 Julia sets in position over the Mandelbrot set

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' parameters, i.e. parameters $c$ for which the critical point is pre-periodic. For instance: File links The following pages link to this file: Julia set Categories: Public domain images | Images of fractals ... File links The following pages link to this file: Julia set Categories: Public domain images | Images of fractals ...

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.

Generalisations

The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Epstein's 'finite-type maps'. In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...

Julia sets are also commonly defined in the study of dynamics in several complex variables.

Plotting the Julia set using backwards iteration

A Julia set plot, generated using backwards iteration

As mentioned above, the Julia set can be found as the set of limit points of the set of preimages of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point $z,$ we know to be in the Julia set, such as a repelling periodic point, and compute all preimages of $z,$ under some high iterate $f^n,$ of $f,$ . Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ...

Unfortunately, as the number of iterated preimages grows exponentially, this is not computationally feasible. However, we can adjust this method, in a similar way as the "random game" method for iterated function systems. That is, in each step, we choose at random one of the inverse images of $f,$ . Menger sponge, created by using IFS. Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ...

For example, for the quadratic polynomial $f_c,$ , the backwards iteration is described by

$z_{n+1}^2 = z_n - c.$

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite hard to reach with the reverse Julia algorithm. For this reason, other methods usually produce better images.

References

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
Adrien Douady and John H. Hubbard, "Etude dynamique des polynômes complexes", Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006 (First appeared in 1990 as a Stony Brook IMS Preprint, available as arXiV:math.DS/9201272.)
Alexander Bogomolny, "Mandelbrot Set and Indexing of Julia Sets" at cut-the-knot.
Evgeny Demidov, "The Mandelbrot and Julia sets Anatomy" (2003)

Categories: Fractals | Limit sets Lennart Carleson (b. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

Results from FactBites:

Julia set - Wikipedia, the free encyclopedia (1472 words)
	Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane.
	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.
	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.

Encyclopedia4U - Julia set - Encyclopedia Article (554 words)
	The julia set is the smallest fixed point set for such a map or collection of maps, not counting the empty set.
	For example, the Sierpinski triangle is a fixed point set of three maps, each of which maps the triangle to one of the corners, shrinking by a factor of 1/2.
	The cantor set is the julia set of this pair of maps.

More results at FactBites »

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