## Julia Set

Let be a rational function

 (1)

where , is the Riemann Sphere , and and are Polynomials without common divisors. The filled-in'' Julia set is the set of points which do not approach infinity after is repeatedly applied. The true Julia set is the boundary of the filled-in set (the set of exceptional points''). There are two types of Julia sets: connected sets and Cantor Sets.

For a Julia set with , the Capacity Dimension is

 (2)

For small , is also a Jordan Curve, although its points are not Computable.

 (3)

for fixed . The special case is called Douady's Rabbit Fractal, is called the San Marco Fractal, and is the Siegel Disk Fractal. For every , this transformation generates a Fractal. It is a Conformal Transformation, so angles are preserved. Let be the Julia Set, then leaves invariant. If a point is on , then all its iterations are on . The transformation has a two-valued inverse. If and is started at 0, then the map is equivalent to the Logistic Map. The set of all points for which is connected is known as the Mandelbrot Set.

See also Dendrite Fractal, Douady's Rabbit Fractal, Fatou Set, Mandelbrot Set, Newton's Method, San Marco Fractal, Siegel Disk Fractal

References

Dickau, R. M. Julia Sets.'' http://forum.swarthmore.edu/advanced/robertd/julias.html.

Dickau, R. M. Another Method for Calculating Julia Sets.'' http://forum.swarthmore.edu/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.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 124-126, 138-148, and 177-179, 1991.

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.