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G. Julia [[#References|[a1]]] and P. Fatou
 
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:
+
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|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|Perfect set]]); 2) $  J ( f  ) $
 +
equals the closure of the set of repelling periodic points (cf. [[Periodic point|Periodic point]]); 3) $  J ( f  ) $
 +
is either totally disconnected (cf. [[Totally-disconnected space|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|Strange attractor]]) of the inverse iterated mapping $  f ^ { - n } $.  
 +
In almost-all cases $  J( f  ) $
 +
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 $  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) <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" />.
+
a) $  D $
 +
is an attracting domain; $  D $
 +
contains an attracting periodic point $  p $
 +
with  $  0 < | g  ^  \prime  ( p) | < 1 $.
  
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" />.
+
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) <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" />.
+
c) $  D $
 +
is a parabolic domain; its boundary contains a periodic point $  p $
 +
with $  g  ^  \prime  ( p) = 1 $.
  
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.
+
d) $  D $
 +
is a Siegel disc (cf. [[Siegel domain|Siegel domain]]); $  D $
 +
is simply connected and $  g \mid  _ {D} $
 +
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.
+
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 <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" />.)
+
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 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]].
+
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|Chaos]]; [[Routes to chaos|Routes to chaos]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:14, 5 June 2020


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)
How to Cite This Entry:
Julia set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Julia_set&oldid=14959