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 (corresponding to a strange
attractor). 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 (Fatou
set) and Cantor sets (Fatou
dust).
Quadratic Julia sets are generated by the quadratic mapping
 |
(2)
|
for fixed 
.
For almost every 
,
this transformation generates a fractal. Examples are
shown above for various values of 
. The resulting object is not a fractal for 
(Dufner et al. 1998, pp. 224-226) and 
(Dufner et al. 1998, pp. 125-126),
although it does not seem to be known if these two are the only such exceptional
values.
|
|
|
|
The special case of 
on the boundary of the Mandelbrot set is called
a dendrite fractal (top left figure), 
is called Douady's
rabbit fractal (top right figure), 
is called the San Marco
fractal (bottom left figure), and 
is the Siegel
disk fractal (bottom right figure).
The equation for the quadratic Julia set is a conformal mapping, 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.
For a Julia set 
with 
, the capacity
dimension is
 |
(3)
|
For small 
,
is also a Jordan
curve, although its points are not computable.
