It may happen that these values stay small or perhaps they don't, that is, repeatedly applying f to a yields arbitrarily large values. So the set of numbers is partitioned into two parts, and the Julia set associated to f is the boundary between these sets. The "filled" Julia set includes those numbers x = a for which the iterates of f applied to a remain bounded.
If you consider complex numbers rather than real numbers, it is the complex plane that is partitioned into two sets, and the resulting picture can be quite striking. For instance, here is a Julia set for the function
f
(
z
) =
z
2
- 0.75.
(Click to download a larger image.)
(Only the square in the complex plane including numbers
z = x + iy
where both
x
and
y
are between ±1.5 is shown.)
That's an example of iterating a quadratic function. Linear functions don't yield interesting partitions of the complex plane, but quadratic and higher degree polynomials do.
A natural source of iterated functions comes from the approximation of roots of functions by Newton's method .
Consider a whole family of functions parameterized by a variable. Although any family of functions can be studied, we'll look at the most studied family, that being the family of quadratic polynomials
f
(
x
) =
x
2
- µ, where µ is a complex parameter. As µ varies, the Julia set will vary on the complex plane. Some of these Julia sets will be connected, and some will be disconnected, and so this character of the Julia sets will partition the µ-parameter plane into two parts. Those values of µ for which the Julia set is connected is called the Mandelbrot set in the parameter plane. The boundary between the Mandelbrot set and its complement is often called the Mandelbrot separator curve. The Mandelbrot set is the black shape in the picture. This is the portion of the plane where
x
varies from -1 to 2 and
y
varies between -1.5 and 1.5. There are some surprising details in this image, and it's well worth
exploring
.
The bulk of the Mandelbrot set is the black cardioid. (A cardioid is a heart-shaped figure). It's studded with circles all around its boundary.
Here's a magnification of the region of the circle on top of the cardioid. Note that this circle as well as the other circles you can see are also studded with circles. There are infinitely many circles on the cardioid, each of those circles has infinitely many circles on them, and on and on
ad infinitum.
That makes for a lot of circles!
The green figure is a Julia set with the parameter µ taken from the center of the circle on top of the cardioid.
If you look close, you'll see the strands of dark blue above the circle under discussion. So, what's going on up there? Here's a blowup of that portion of the figure.
Aha! There's something black up near the top of the picture. What's that?
It's another cardioid with associated circles! Not exactly the same, but close. In fact, there are lots of these tiny little clusters. You can find them along the filaments connecting everything together.
Bibliography on Julia and Mandelbrot sets.
Web references to Julia and Mandelbrot sets.
Don't forget to check out the Mandelbrot and Julia Set Explorer . If you have a particular Mandelbrot or Julia set you want to produce, you can use the Mandelbrot and Julia Set Generator .
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Copyright © 1994, 1995.
David E. Joyce Department of Mathematics and Computer Science Clark University Worcester, MA 01610 Email: djoyce@clarku.edu These pages are located at http://aleph0.clarku.edu/~djoyce/julia/julia.html |
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