<s>
In	O
the	O
context	O
of	O
complex	O
dynamics	O
,	O
a	O
branch	O
of	O
mathematics	O
,	O
the	O
Julia	B-Language
set	I-Language
and	O
the	O
Fatou	B-Language
set	I-Language
are	O
two	O
complementary	O
sets	O
(	O
Julia	O
"	O
laces	O
"	O
and	O
Fatou	O
"	O
dusts	O
"	O
)	O
defined	O
from	O
a	O
function	O
.	O
</s>
<s>
Informally	O
,	O
the	O
Fatou	B-Language
set	I-Language
of	O
the	O
function	O
consists	O
of	O
values	O
with	O
the	O
property	O
that	O
all	O
nearby	O
values	O
behave	O
similarly	O
under	O
repeated	B-Algorithm
iteration	I-Algorithm
of	O
the	O
function	O
,	O
and	O
the	O
Julia	B-Language
set	I-Language
consists	O
of	O
values	O
such	O
that	O
an	O
arbitrarily	O
small	O
perturbation	O
can	O
cause	O
drastic	O
changes	O
in	O
the	O
sequence	O
of	O
iterated	B-Algorithm
function	I-Algorithm
values	O
.	O
</s>
<s>
Thus	O
the	O
behavior	O
of	O
the	O
function	O
on	O
the	O
Fatou	B-Language
set	I-Language
is	O
"	O
regular	O
"	O
,	O
while	O
on	O
the	O
Julia	B-Language
set	I-Language
its	O
behavior	O
is	O
"	O
chaotic	O
"	O
.	O
</s>
<s>
The	O
Julia	B-Language
set	I-Language
of	O
a	O
function	O
is	O
commonly	O
denoted	O
and	O
the	O
Fatou	B-Language
set	I-Language
is	O
denoted	O
These	O
sets	O
are	O
named	O
after	O
the	O
French	O
mathematicians	O
Gaston	O
Julia	O
and	O
Pierre	O
Fatou	O
whose	O
work	O
began	O
the	O
study	O
of	O
complex	O
dynamics	O
during	O
the	O
early	O
20th	O
century	O
.	O
</s>
<s>
These	O
sets	O
are	O
the	O
Fatou	O
domains	O
of	O
,	O
and	O
their	O
union	O
is	O
the	O
Fatou	B-Language
set	I-Language
of	O
.	O
</s>
<s>
The	O
complement	O
of	O
is	O
the	O
Julia	B-Language
set	I-Language
of	O
.	O
</s>
<s>
Otherwise	O
,	O
is	O
a	O
nowhere	O
dense	O
set	O
(	O
it	O
is	O
without	O
interior	O
points	O
)	O
and	O
an	O
uncountable	O
set	O
(	O
of	O
the	O
same	O
cardinality	B-Application
as	O
the	O
real	O
numbers	O
)	O
.	O
</s>
<s>
This	O
means	O
that	O
behaves	O
chaotically	O
on	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
Although	O
there	O
are	O
points	O
in	O
the	O
Julia	B-Language
set	I-Language
whose	O
sequence	O
of	O
iterations	O
is	O
finite	O
,	O
there	O
are	O
only	O
a	O
countable	O
number	O
of	O
such	O
points	O
(	O
and	O
they	O
make	O
up	O
an	O
infinitesimal	O
part	O
of	O
the	O
Julia	B-Language
set	I-Language
)	O
.	O
</s>
<s>
There	O
has	O
been	O
extensive	O
research	O
on	O
the	O
Fatou	B-Language
set	I-Language
and	O
Julia	B-Language
set	I-Language
of	O
iterated	O
rational	O
functions	O
,	O
known	O
as	O
rational	O
maps	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
known	O
that	O
the	O
Fatou	B-Language
set	I-Language
of	O
a	O
rational	O
map	O
has	O
either	O
0	O
,	O
1	O
,	O
2	O
or	O
infinitely	O
many	O
components	O
.	O
</s>
<s>
Each	O
component	O
of	O
the	O
Fatou	B-Language
set	I-Language
of	O
a	O
rational	O
map	O
can	O
be	O
classified	O
into	O
one	O
of	O
four	B-Algorithm
different	I-Algorithm
classes	I-Algorithm
.	O
</s>
<s>
For	O
all	O
but	O
at	O
most	O
two	O
points	O
the	O
Julia	B-Language
set	I-Language
is	O
the	O
set	O
of	O
limit	O
points	O
of	O
the	O
full	O
backwards	O
orbit	O
(	O
This	O
suggests	O
a	O
simple	O
algorithm	O
for	O
plotting	O
Julia	B-Language
sets	I-Language
,	O
see	O
below	O
.	O
)	O
</s>
<s>
If	O
f	O
is	O
a	O
polynomial	O
,	O
then	O
is	O
the	O
boundary	O
of	O
the	O
filled	B-Algorithm
Julia	I-Algorithm
set	I-Algorithm
;	O
that	O
is	O
,	O
those	O
points	O
whose	O
orbits	O
under	O
iterations	O
of	O
f	O
remain	O
bounded	O
.	O
</s>
<s>
The	O
Julia	B-Language
set	I-Language
and	O
the	O
Fatou	B-Language
set	I-Language
of	O
f	O
are	O
both	O
completely	O
invariant	O
under	O
iterations	O
of	O
the	O
holomorphic	O
function	O
f	O
:	O
</s>
<s>
For	O
the	O
Julia	B-Language
set	I-Language
is	O
the	O
unit	O
circle	O
and	O
on	O
this	O
the	O
iteration	O
is	O
given	O
by	O
doubling	O
of	O
angles	O
(	O
an	O
operation	O
that	O
is	O
chaotic	O
on	O
the	O
points	O
whose	O
argument	O
is	O
not	O
a	O
rational	O
fraction	O
of	O
)	O
.	O
</s>
<s>
For	O
the	O
Julia	B-Language
set	I-Language
is	O
the	O
line	O
segment	O
between	O
−2	O
and	O
2	O
.	O
</s>
<s>
There	O
is	O
one	O
Fatou	B-Algorithm
domain	I-Algorithm
:	O
the	O
points	O
not	O
on	O
the	O
line	O
segment	O
iterate	B-Algorithm
towards	O
∞	O
.	O
</s>
<s>
For	O
such	O
an	O
iteration	O
the	O
Julia	B-Language
set	I-Language
is	O
not	O
in	O
general	O
a	O
simple	O
curve	O
,	O
but	O
is	O
a	O
fractal	O
,	O
and	O
for	O
some	O
values	O
of	O
c	O
it	O
can	O
take	O
surprising	O
shapes	O
.	O
</s>
<s>
For	O
some	O
functions	O
f(z )	O
we	O
can	O
say	O
beforehand	O
that	O
the	O
Julia	B-Language
set	I-Language
is	O
a	O
fractal	O
and	O
not	O
a	O
simple	O
curve	O
.	O
</s>
<s>
This	O
means	O
that	O
each	O
point	O
of	O
the	O
Julia	B-Language
set	I-Language
is	O
a	O
point	O
of	O
accumulation	O
for	O
each	O
of	O
the	O
Fatou	O
domains	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
there	O
are	O
more	O
than	O
two	O
Fatou	O
domains	O
,	O
each	O
point	O
of	O
the	O
Julia	B-Language
set	I-Language
must	O
have	O
points	O
of	O
more	O
than	O
two	O
different	O
open	O
sets	O
infinitely	O
close	O
,	O
and	O
this	O
means	O
that	O
the	O
Julia	B-Language
set	I-Language
cannot	O
be	O
a	O
simple	O
curve	O
.	O
</s>
<s>
A	O
very	O
popular	O
complex	O
dynamical	O
system	O
is	O
given	O
by	O
the	O
family	O
of	O
complex	B-Algorithm
quadratic	I-Algorithm
polynomials	I-Algorithm
,	O
a	O
special	O
case	O
of	O
rational	O
maps	O
.	O
</s>
<s>
where	O
is	O
the	O
nth	O
iterate	B-Algorithm
of	O
The	O
Julia	B-Language
set	I-Language
of	O
this	O
function	O
is	O
the	O
boundary	O
of	O
.	O
</s>
<s>
The	O
parameter	O
plane	O
of	O
quadratic	O
polynomials	O
–	O
that	O
is	O
,	O
the	O
plane	O
of	O
possible	O
c	O
values	O
–	O
gives	O
rise	O
to	O
the	O
famous	O
Mandelbrot	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
defined	O
as	O
the	O
set	O
of	O
all	O
c	O
such	O
that	O
is	O
connected	O
.	O
</s>
<s>
For	O
parameters	O
outside	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
the	O
Julia	B-Language
set	I-Language
is	O
a	O
Cantor	O
space	O
:	O
in	O
this	O
case	O
it	O
is	O
sometimes	O
referred	O
to	O
as	O
Fatou	B-Language
dust	I-Language
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
the	O
Julia	B-Language
set	I-Language
of	O
c	O
looks	O
like	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
in	O
sufficiently	O
small	O
neighborhoods	O
of	O
c	O
.	O
This	O
is	O
true	O
,	O
in	O
particular	O
,	O
for	O
so-called	O
Misiurewicz	O
parameters	O
,	O
i.e.	O
</s>
<s>
At	O
c	O
=	O
i	O
,	O
the	O
shorter	O
,	O
front	O
toe	O
of	O
the	O
forefoot	O
,	O
the	O
Julia	B-Language
set	I-Language
looks	O
like	O
a	O
branched	O
lightning	O
bolt	O
.	O
</s>
<s>
At	O
c	O
=	O
−2	O
,	O
the	O
tip	O
of	O
the	O
long	O
spiky	O
tail	O
,	O
the	O
Julia	B-Language
set	I-Language
is	O
a	O
straight	O
line	O
segment	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
Julia	B-Language
sets	I-Language
are	O
locally	O
similar	O
around	O
Misiurewicz	O
points	O
.	O
</s>
<s>
The	O
definition	O
of	O
Julia	O
and	O
Fatou	B-Language
sets	I-Language
easily	O
carries	O
over	O
to	O
the	O
case	O
of	O
certain	O
maps	O
whose	O
image	O
contains	O
their	O
domain	O
;	O
most	O
notably	O
transcendental	O
meromorphic	O
functions	O
and	O
Adam	O
Epstein	O
's	O
finite-type	O
maps	O
.	O
</s>
<s>
Julia	B-Language
sets	I-Language
are	O
also	O
commonly	O
defined	O
in	O
the	O
study	O
of	O
dynamics	O
in	O
several	O
complex	O
variables	O
.	O
</s>
<s>
The	O
Julia	B-Language
set	I-Language
for	O
is	O
the	O
unit	O
circle	O
,	O
and	O
on	O
the	O
outer	O
Fatou	B-Algorithm
domain	I-Algorithm
,	O
the	O
potential	O
function	O
φ(z )	O
is	O
defined	O
by	O
φ(z )	O
=	O
log|z|	O
.	O
</s>
<s>
For	O
the	O
more	O
general	O
iteration	O
,	O
it	O
has	O
been	O
proved	O
that	O
if	O
the	O
Julia	B-Language
set	I-Language
is	O
connected	O
(	O
that	O
is	O
,	O
if	O
c	O
belongs	O
to	O
the	O
(	O
usual	O
)	O
Mandelbrot	B-Algorithm
set	I-Algorithm
)	O
,	O
then	O
there	O
exist	O
a	O
biholomorphic	B-Algorithm
map	I-Algorithm
ψ	O
between	O
the	O
outer	O
Fatou	B-Algorithm
domain	I-Algorithm
and	O
the	O
outer	O
of	O
the	O
unit	O
circle	O
such	O
that	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
potential	O
function	O
on	O
the	O
outer	O
Fatou	B-Algorithm
domain	I-Algorithm
defined	O
by	O
this	O
correspondence	O
is	O
given	O
by	O
:	O
</s>
<s>
This	O
formula	O
has	O
meaning	O
also	O
if	O
the	O
Julia	B-Language
set	I-Language
is	O
not	O
connected	O
,	O
so	O
that	O
we	O
for	O
all	O
c	O
can	O
define	O
the	O
potential	O
function	O
on	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
containing	O
∞	O
by	O
this	O
formula	O
.	O
</s>
<s>
For	O
a	O
general	O
rational	O
function	O
f(z )	O
such	O
that	O
∞	O
is	O
a	O
critical	O
point	O
and	O
a	O
fixed	O
point	O
,	O
that	O
is	O
,	O
such	O
that	O
the	O
degree	O
m	O
of	O
the	O
numerator	O
is	O
at	O
least	O
two	O
larger	O
than	O
the	O
degree	O
n	O
of	O
the	O
denominator	O
,	O
we	O
define	O
the	O
potential	O
function	O
on	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
containing	O
∞	O
by	O
:	O
</s>
<s>
Therefore	O
the	O
number	O
is	O
almost	O
independent	O
of	O
k	O
.	O
We	O
define	O
the	O
potential	O
function	O
on	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
by	O
:	O
</s>
<s>
If	O
the	O
cycle	O
is	O
neutral	O
,	O
we	O
cannot	O
colour	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
in	O
a	O
natural	O
way	O
.	O
</s>
<s>
In	O
each	O
Fatou	B-Algorithm
domain	I-Algorithm
(	O
that	O
is	O
not	O
neutral	O
)	O
there	O
are	O
two	O
systems	O
of	O
lines	O
orthogonal	O
to	O
each	O
other	O
:	O
the	O
equipotential	O
lines	O
(	O
for	O
the	O
potential	O
function	O
or	O
the	O
real	O
iteration	O
number	O
)	O
and	O
the	O
field	O
lines	O
.	O
</s>
<s>
If	O
we	O
colour	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
according	O
to	O
the	O
iteration	O
number	O
(	O
and	O
not	O
the	O
real	O
iteration	O
number	O
,	O
as	O
defined	O
in	O
the	O
previous	O
section	O
)	O
,	O
the	O
bands	O
of	O
iteration	O
show	O
the	O
course	O
of	O
the	O
equipotential	O
lines	O
.	O
</s>
<s>
If	O
the	O
iteration	O
is	O
towards	O
∞	O
(	O
as	O
is	O
the	O
case	O
with	O
the	O
outer	O
Fatou	B-Algorithm
domain	I-Algorithm
for	O
the	O
usual	O
iteration	O
)	O
,	O
we	O
can	O
easily	O
show	O
the	O
course	O
of	O
the	O
field	O
lines	O
,	O
namely	O
by	O
altering	O
the	O
colour	O
according	O
as	O
the	O
last	O
point	O
in	O
the	O
sequence	O
of	O
iteration	O
is	O
above	O
or	O
below	O
the	O
x-axis	O
(	O
first	O
picture	O
)	O
,	O
but	O
in	O
this	O
case	O
(	O
more	O
precisely	O
:	O
when	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
is	O
super-attracting	O
)	O
we	O
cannot	O
draw	O
the	O
field	O
lines	O
coherently	O
-	O
at	O
least	O
not	O
by	O
the	O
method	O
we	O
describe	O
here	O
.	O
</s>
<s>
Let	O
z	O
be	O
a	O
point	O
in	O
the	O
attracting	O
Fatou	B-Algorithm
domain	I-Algorithm
.	O
</s>
<s>
If	O
we	O
iterate	B-Algorithm
z	O
a	O
large	O
number	O
of	O
times	O
,	O
the	O
terminus	O
of	O
the	O
sequence	O
of	O
iteration	O
is	O
a	O
finite	O
cycle	O
C	O
,	O
and	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
is	O
(	O
by	O
definition	O
)	O
the	O
set	O
of	O
points	O
whose	O
sequence	O
of	O
iteration	O
converges	O
towards	O
C	O
.	O
The	O
field	O
lines	O
issue	O
from	O
the	O
points	O
of	O
C	O
and	O
from	O
the	O
(	O
infinite	O
number	O
of	O
)	O
points	O
that	O
iterate	B-Algorithm
into	O
a	O
point	O
of	O
C	O
.	O
And	O
they	O
end	O
on	O
the	O
Julia	B-Language
set	I-Language
in	O
points	O
that	O
are	O
non-chaotic	O
(	O
that	O
is	O
,	O
generating	O
a	O
finite	O
cycle	O
)	O
.	O
</s>
<s>
In	O
order	O
to	O
colour	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
,	O
we	O
have	O
chosen	O
a	O
small	O
number	O
ε	O
and	O
set	O
the	O
sequences	O
of	O
iteration	O
to	O
stop	O
when	O
,	O
and	O
we	O
colour	O
the	O
point	O
z	O
according	O
to	O
the	O
number	O
k	O
(	O
or	O
the	O
real	O
iteration	O
number	O
,	O
if	O
we	O
prefer	O
a	O
smooth	O
colouring	O
)	O
.	O
</s>
<s>
A	O
colouring	O
of	O
the	O
field	O
lines	O
of	O
the	O
Fatou	B-Algorithm
domain	I-Algorithm
means	O
that	O
we	O
colour	O
the	O
spaces	O
between	O
pairs	O
of	O
field	O
lines	O
:	O
we	O
choose	O
a	O
number	O
of	O
regularly	O
situated	O
directions	O
issuing	O
from	O
,	O
and	O
in	O
each	O
of	O
these	O
directions	O
we	O
choose	O
two	O
directions	O
around	O
this	O
direction	O
.	O
</s>
<s>
As	O
it	O
can	O
happen	O
that	O
the	O
two	O
field	O
lines	O
of	O
a	O
pair	O
do	O
not	O
end	O
in	O
the	O
same	O
point	O
of	O
the	O
Julia	B-Language
set	I-Language
,	O
our	O
coloured	O
field	O
lines	O
can	O
ramify	O
(	O
endlessly	O
)	O
in	O
their	O
way	O
towards	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
the	O
Julia	B-Language
set	I-Language
can	O
be	O
found	O
as	O
the	O
set	O
of	O
limit	O
points	O
of	O
the	O
set	O
of	O
pre-images	O
of	O
(	O
essentially	O
)	O
any	O
given	O
point	O
.	O
</s>
<s>
So	O
we	O
can	O
try	O
to	O
plot	O
the	O
Julia	B-Language
set	I-Language
of	O
a	O
given	O
function	O
as	O
follows	O
.	O
</s>
<s>
Start	O
with	O
any	O
point	O
z	O
we	O
know	O
to	O
be	O
in	O
the	O
Julia	B-Language
set	I-Language
,	O
such	O
as	O
a	O
repelling	O
periodic	O
point	O
,	O
and	O
compute	O
all	O
pre-images	O
of	O
z	O
under	O
some	O
high	O
iterate	B-Algorithm
of	O
f	O
.	O
</s>
<s>
However	O
,	O
we	O
can	O
adjust	O
this	O
method	O
,	O
in	O
a	O
similar	O
way	O
as	O
the	O
"	O
random	O
game	O
"	O
method	O
for	O
iterated	B-Algorithm
function	I-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
certain	O
parts	O
of	O
the	O
Julia	B-Language
set	I-Language
are	O
quite	O
difficult	O
to	O
access	O
with	O
the	O
reverse	O
Julia	O
algorithm	O
.	O
</s>
<s>
As	O
a	O
Julia	B-Language
set	I-Language
is	O
infinitely	O
thin	O
we	O
cannot	O
draw	O
it	O
effectively	O
by	O
backwards	O
iteration	O
from	O
the	O
pixels	O
.	O
</s>
<s>
Since	O
the	O
iteration	O
count	O
changes	O
vigorously	O
near	O
the	O
Julia	B-Language
set	I-Language
,	O
a	O
partial	O
solution	O
is	O
to	O
imply	O
the	O
outline	O
of	O
the	O
set	O
from	O
the	O
nearest	O
color	O
contours	O
,	O
but	O
the	O
set	O
will	O
tend	O
to	O
look	O
muddy	O
.	O
</s>
<s>
A	O
better	O
way	O
to	O
draw	O
the	O
Julia	B-Language
set	I-Language
in	O
black	O
and	O
white	O
is	O
to	O
estimate	O
the	O
distance	O
of	O
pixels	O
(	O
DEM	O
)	O
from	O
the	O
set	O
and	O
to	O
color	O
every	O
pixel	O
whose	O
center	O
is	O
close	O
to	O
the	O
set	O
.	O
</s>
<s>
It	O
has	O
been	O
proven	O
that	O
the	O
value	O
found	O
by	O
this	O
formula	O
(	O
up	O
to	O
a	O
constant	O
factor	O
)	O
converges	O
towards	O
the	O
true	O
distance	O
for	O
z	O
converging	O
towards	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
