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Julia Set


Let R(z) be a rational function

 R(z)=(P(z))/(Q(z)),
(1)

where z in C^*, C^* is the Riemann sphere C union {infty}, and P and Q are polynomials without common divisors. The "filled-in" Julia set J_R is the set of points z which do not approach infinity after R(z) is repeatedly applied (corresponding to a strange attractor). The true Julia set J is the boundary of the filled-in set (the set of "exceptional points"). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).

JuliaSets

Quadratic Julia sets are generated by the quadratic mapping

 z_(n+1)=z_n^2+c
(2)

for fixed c. For almost every c, this transformation generates a fractal. Examples are shown above for various values of c. The resulting object is not a fractal for c=-2 (Dufner et al. 1998, pp. 224-226) and c=0 (Dufner et al. 1998, pp. 125-126), although it does not seem to be known if these two are the only such exceptional values.

DendriteFractalDouadysRabbitFractal
SanMarcoFractalSiegelDisk

The special case of c=i on the boundary of the Mandelbrot set is called a dendrite fractal (top left figure), c=-0.123+0.745i is called Douady's rabbit fractal (top right figure), c=-0.75 is called the San Marco fractal (bottom left figure), and c=-0.391-0.587i is the Siegel disk fractal (bottom right figure).

The equation for the quadratic Julia set is a conformal mapping, so angles are preserved. Let J be the Julia set, then x^'|->x leaves J invariant. If a point P is on J, then all its iterations are on J. The transformation has a two-valued inverse. If b=0 and y is started at 0, then the map is equivalent to the logistic map. The set of all points for which J is connected is known as the Mandelbrot set.

For a Julia set J_c with c<<1, the capacity dimension is

 d_(capacity)=1+(|c|^2)/(4ln2)+O(|c|^3).
(3)

For small c, J_c is also a Jordan curve, although its points are not computable.


See also

Dendrite Fractal, Douady's Rabbit Fractal, Fatou Dust, Fatou Set, Fractal, Mandelbrot Set, Newton's Method, San Marco Fractal, Siegel Disk Fractal, Strange Attractor

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References

Dickau, R. M. "Julia Sets." http://mathforum.org/advanced/robertd/julias.html.Dickau, R. M. "Another Method for Calculating Julia Sets." http://mathforum.org/advanced/robertd/inversejulia.html.Douady, A. "Julia Sets and the Mandelbrot Set." In The Beauty of Fractals: Images of Complex Dynamical Systems (Ed. H.-O. Peitgen and D. H. Richter). Berlin: Springer-Verlag, p. 161, 1986.Dufner, J.; Roser, A.; and Unseld, F. Fraktale und Julia-Mengen. Harri Deutsch, 1998.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 124-126, 138-148, and 177-179, 1991.Mendes-France, M. "Nevertheless." Math. Intell. 10, 35, 1988.Peitgen, H.-O. and Saupe, D. (Eds.). "The Julia Set," "Julia Sets as Basin Boundaries," "Other Julia Sets," and "Exploring Julia Sets." §3.3.2 to 3.3.5 in The Science of Fractal Images. New York: Springer-Verlag, pp. 152-163, 1988.Schroeder, M. Fractals, Chaos, Power Laws. New York: W. H. Freeman, p. 39, 1991.Wagon, S. "Julia Sets." §5.4 in Mathematica in Action. New York: W. H. Freeman, pp. 163-178, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 126-127, 1991.

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Julia Set

Cite this as:

Weisstein, Eric W. "Julia Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JuliaSet.html

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