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)