“Julia Set”的概念、定义、习题解题方法及翻译-数学辞典

单词

Julia Set

释义

Julia Set

Let be a rational function

(1)

where

is the Riemann Sphere

, and

and

arePolynomials 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.

Quadratic Julia sets are generated by the quadratic mapping

(3)

for fixed

. The special case

is called Douady's Rabbit Fractal,

is called the SanMarco Fractal, and

is the Siegel Disk Fractal. For every

, this transformation generatesa 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

. Thetransformation 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'sMethod, 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.

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