Julia set
G. Julia [a1] and P. Fatou
studied the iteration of rational mappings . Let
denote the
-fold composite of the function
with itself. A point
is an element of the so-called Fatou set
of
if there exists a neighbourhood
of
in
such that the family of iterates
is a normal family. The Julia set
is the complement of the Fatou set.
has the following properties: 1)
is non-empty and perfect (cf. Perfect set); 2)
equals the closure of the set of repelling periodic points (cf. Periodic point); 3)
is either totally disconnected (cf. Totally-disconnected space) or connected by Jordan arcs or coincides with
; 4)
is invariant with respect to
and
; and 5)
is an attractor (cf. Strange attractor) of the inverse iterated mapping
. In almost-all cases
has a fractal dimension and may be termed a fractal (cf. Fractals). D. Sullivan has given an exhaustive classification of the elements of the Fatou set with respect to their dynamics. Every component of
is either periodic or pre-periodic. Let
be such a periodic domain and let
be its period. Writing
one has the following five kinds of dynamics:
a) is an attracting domain;
contains an attracting periodic point
with
.
b) is a super-attractive domain;
contains a periodic point
which is also a critical point, i.e.
.
c) is a parabolic domain; its boundary contains a periodic point
with
.
d) is a Siegel disc (cf. Siegel domain);
is simply connected and
is analytically equivalent to a rotation.
e) is a Herman ring:
is conformally equivalent to an annulus and
is analytically conjugate to a rigid rotation of an annulus.
Here a pre-periodic point is a point some iterate of which is periodic. A fixed point of
is super-attractive if
. (Recall that if
is a fixed point, then
is attractive if
and repelling if
.)
The existence of Herman rings has been proved, but they have never yet (1989) been observed.
The best studied case is the quadratic mapping . All phenomena are present there, with the exception of a Herman ring. All
for which
is connected form the Mandelbrot set, the bifurcation diagram of
in the parameter space of
. See also Chaos; Routes to chaos.
References
[a1] | G. Julia, "Mémoire sur l'iteration des fonctions rationnelles" J. de Math. , 8 (1918) pp. 47–245 |
[a2a] | P. Fatou, "Sur les équations fonctionnelles" Bull. Soc. Math. France , 47 (1919) pp. 161–271 |
[a2b] | P. Fatou, "Sur les équations fonctionnelles" Bull. Soc. Math. France , 48 (1920) pp. 33–94 |
[a2c] | P. Fatou, "Sur les équations fonctionnelles" Bull. Soc. Math. France , 48 (1920) pp. 208–314 |
[a3] | H. Brolin, "Invariant sets under iteration of rational functions" Ark. Mat. , 6 (1965) pp. 103–144 |
[a4] | P. Blanchard, "Complex analytic dynamics" Bull. Amer. Math. Soc. , 11 (1984) pp. 84–141 |
[a5] | R.L. Devaney, "An introduction to chaotic dynamical systems" , Benjamin/Cummings (1986) |
[a6] | H.-O. Peitgen, P.H. Richter, "The beauty of fractals" , Springer (1986) |
Julia set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Julia_set&oldid=14959