Julia set

G. Julia [a1] and P. Fatou

studied the iteration of rational mappings $ f: \overline{\mathbf C}\; \rightarrow \overline{\mathbf C}\; $. Let $ f ^ { n } $ denote the $ n $- fold composite of the function $ f $ with itself. A point $ z \in C $ is an element of the so-called Fatou set $ F ( f ) $ of $ f $ if there exists a neighbourhood $ U $ of $ z $ in $ \mathbf C $ such that the family of iterates $ \{ f ^ { n } \mid _ {U} \} $ is a normal family. The Julia set $ J ( f ) $ is the complement of the Fatou set. $ J ( f ) $ has the following properties: 1) $ J ( f ) $ is non-empty and perfect (cf. Perfect set); 2) $ J ( f ) $ equals the closure of the set of repelling periodic points (cf. Periodic point); 3) $ J ( f ) $ is either totally disconnected (cf. Totally-disconnected space) or connected by Jordan arcs or coincides with $ \mathbf C $; 4) $ J ( f ) $ is invariant with respect to $ f $ and $ f ^ { - 1 } $; and 5) $ J ( f ) $ is an attractor (cf. Strange attractor) of the inverse iterated mapping $ f ^ { - n } $. In almost-all cases $ J( f ) $ 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 $ F ( f ) $ is either periodic or pre-periodic. Let $ D $ be such a periodic domain and let $ n $ be its period. Writing $ g = f ^ { n } $ one has the following five kinds of dynamics:

a) $ D $ is an attracting domain; $ D $ contains an attracting periodic point $ p $ with $ 0 < | g ^ \prime ( p) | < 1 $.

b) $ D $ is a super-attractive domain; $ D $ contains a periodic point $ p $ which is also a critical point, i.e. $ g ^ \prime ( p) = 0 $.

c) $ D $ is a parabolic domain; its boundary contains a periodic point $ p $ with $ g ^ \prime ( p) = 1 $.

d) $ D $ is a Siegel disc (cf. Siegel domain); $ D $ is simply connected and $ g \mid _ {D} $ is analytically equivalent to a rotation.

e) $ D $ is a Herman ring: $ D $ is conformally equivalent to an annulus and $ g \mid _ {D} $ 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 $ z _ {0} $ of $ f $ is super-attractive if $ f ^ { \prime } ( z _ {0} ) = 0 $. (Recall that if $ z _ {0} $ is a fixed point, then $ z _ {0} $ is attractive if $ | f ^ { \prime } ( z _ {0} ) | < 1 $ and repelling if $ | f ^ { \prime } ( z _ {0} ) | > 1 $.)

The existence of Herman rings has been proved, but they have never yet (1989) been observed.

The best studied case is the quadratic mapping $ f = z ^ {2} + c $. All phenomena are present there, with the exception of a Herman ring. All $ c $ for which $ J ( f ) $ is connected form the Mandelbrot set, the bifurcation diagram of $ J ( f ) $ in the parameter space of $ c $. 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)