View source for Julia set

G. Julia [[#References|[a1]]] and P. Fatou

studied the iteration of rational mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543901.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543902.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543903.png" />-fold composite of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543904.png" /> with itself. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543905.png" /> is an element of the so-called Fatou set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543906.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543907.png" /> if there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j0543909.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439010.png" /> such that the family of iterates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439011.png" /> is a [[Normal family|normal family]]. The Julia set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439012.png" /> is the complement of the Fatou set. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439013.png" /> has the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439014.png" /> is non-empty and perfect (cf. [[Perfect set|Perfect set]]); 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439015.png" /> equals the closure of the set of repelling periodic points (cf. [[Periodic point|Periodic point]]); 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439016.png" /> is either totally disconnected (cf. [[Totally-disconnected space|Totally-disconnected space]]) or connected by Jordan arcs or coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439017.png" />; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439018.png" /> is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439020.png" />; and 5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439021.png" /> is an attractor (cf. [[Strange attractor|Strange attractor]]) of the inverse iterated mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439022.png" />. In almost-all cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439023.png" /> has a [[Fractal dimension|fractal dimension]] and may be termed a fractal (cf. [[Fractals|Fractals]]). D. Sullivan has given an exhaustive classification of the elements of the Fatou set with respect to their dynamics. Every component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439024.png" /> is either periodic or pre-periodic. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439025.png" /> be such a periodic domain and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439026.png" /> be its period. Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439027.png" /> one has the following five kinds of dynamics:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439028.png" /> is an attracting domain; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439029.png" /> contains an attracting periodic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439030.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439031.png" />.

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439032.png" /> is a super-attractive domain; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439033.png" /> contains a periodic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439034.png" /> which is also a critical point, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439035.png" />.

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439036.png" /> is a parabolic domain; its boundary contains a periodic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439038.png" />.

d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439039.png" /> is a Siegel disc (cf. [[Siegel domain|Siegel domain]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439040.png" /> is simply connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439041.png" /> is analytically equivalent to a rotation.

e) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439042.png" /> is a Herman ring: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439043.png" /> is conformally equivalent to an annulus and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439044.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439046.png" /> is super-attractive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439047.png" />. (Recall that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439048.png" /> is a fixed point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439049.png" /> is attractive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439050.png" /> and repelling if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439051.png" />.)

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

The best studied case is the quadratic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439052.png" />. All phenomena are present there, with the exception of a Herman ring. All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439053.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439054.png" /> is connected form the Mandelbrot set, the bifurcation diagram of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439055.png" /> in the parameter space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054390/j05439056.png" />. See also [[Chaos|Chaos]]; [[Routes to chaos|Routes to chaos]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Julia,   "Mémoire sur l'iteration des fonctions rationnelles"  ''J. de Math.'' , '''8'''  (1918)  pp. 47–245</TD></TR><TR><TD valign="top">[a2a]</TD> <TD valign="top">  P. Fatou,   "Sur les équations fonctionnelles"  ''Bull. Soc. Math. France'' , '''47'''  (1919)  pp. 161–271</TD></TR><TR><TD valign="top">[a2b]</TD> <TD valign="top">  P. Fatou,   "Sur les équations fonctionnelles"  ''Bull. Soc. Math. France'' , '''48'''  (1920)  pp. 33–94</TD></TR><TR><TD valign="top">[a2c]</TD> <TD valign="top">  P. Fatou,   "Sur les équations fonctionnelles"  ''Bull. Soc. Math. France'' , '''48'''  (1920)  pp. 208–314</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Brolin,   "Invariant sets under iteration of rational functions"  ''Ark. Mat.'' , '''6'''  (1965)  pp. 103–144</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Blanchard,   "Complex analytic dynamics"  ''Bull. Amer. Math. Soc.'' , '''11'''  (1984)  pp. 84–141</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.L. Devaney,   "An introduction to chaotic dynamical systems" , Benjamin/Cummings  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.-O. Peitgen,   P.H. Richter,   "The beauty of fractals" , Springer  (1986)</TD></TR></table>