Julia Set

Let $R(z)$ be a rational function

$$R(z)\equiv {P(z)\over Q(z)},$$ (1)

where $z\in \Bbb{C}^*$, $\Bbb{C}^*$ is the Riemann Sphere $\Bbb{C} \cup \{\infty\}$, 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. 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 $J_c$ with $c\ll 1$, the Capacity Dimension is

$$d_{\rm capacity} = 1+{\vert c\vert^2\over 4\ln 2} + {\mathcal O}(\vert c\vert^3).$$ (2)

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

Quadratic Julia sets are generated by the quadratic mapping

$$z_{n+1} = {z_n}^2+c$$ (3)

for fixed $c$. The special case $c=-0.123+0.745i$ is called Douady's Rabbit Fractal, $c=-0.75$ is called the San Marco Fractal, and $c=-0.391-0.587i$ is the Siegel Disk Fractal. For every $c$, this transformation generates a Fractal. It is a Conformal Transformation, so angles are preserved. Let $J$ be the Julia Set, then $x'\mapsto x$ 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.

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.