<s>
Siegel	B-Algorithm
disc	I-Algorithm
is	O
a	O
connected	O
component	B-Algorithm
in	I-Algorithm
the	I-Algorithm
Fatou	I-Algorithm
set	I-Algorithm
where	O
the	O
dynamics	O
is	O
analytically	O
conjugate	O
to	O
an	O
irrational	O
rotation	O
.	O
</s>
<s>
Given	O
a	O
holomorphic	O
endomorphism	O
on	O
a	O
Riemann	O
surface	O
we	O
consider	O
the	O
dynamical	O
system	O
generated	O
by	O
the	O
iterates	B-Algorithm
of	O
denoted	O
by	O
.	O
</s>
<s>
We	O
then	O
call	O
the	O
orbit	B-Algorithm
of	O
as	O
the	O
set	O
of	O
forward	O
iterates	B-Algorithm
of	O
.	O
</s>
<s>
We	O
are	O
interested	O
in	O
the	O
asymptotic	O
behavior	O
of	O
the	O
orbits	B-Algorithm
in	O
(	O
which	O
will	O
usually	O
be	O
,	O
the	O
complex	O
plane	O
or	O
,	O
the	O
Riemann	O
sphere	O
)	O
,	O
and	O
we	O
call	O
the	O
phase	O
plane	O
or	O
dynamical	O
plane	O
.	O
</s>
<s>
In	O
this	O
last	O
case	O
where	O
is	O
the	O
period	B-Algorithm
and	O
means	O
is	O
a	O
fixed	O
point	O
.	O
</s>
<s>
We	O
can	O
then	O
define	O
the	O
multiplier	O
of	O
the	O
orbit	B-Algorithm
as	O
and	O
this	O
enables	O
us	O
to	O
classify	O
periodic	B-Algorithm
orbits	I-Algorithm
as	O
attracting	O
if	O
superattracting	O
if	O
)	O
,	O
repelling	O
if	O
and	O
indifferent	O
if	O
.	O
</s>
<s>
Indifferent	O
periodic	B-Algorithm
orbits	I-Algorithm
can	O
be	O
either	O
rationally	O
indifferent	O
or	O
irrationally	O
indifferent	O
,	O
depending	O
on	O
whether	O
for	O
some	O
or	O
for	O
all	O
,	O
respectively	O
.	O
</s>
<s>
Siegel	B-Algorithm
discs	I-Algorithm
are	O
one	O
of	O
the	O
possible	O
cases	O
of	O
connected	O
components	O
in	O
the	O
Fatou	B-Language
set	I-Language
(	O
the	O
complementary	O
set	O
of	O
the	O
Julia	B-Language
set	I-Language
)	O
,	O
according	O
to	O
Classification	B-Algorithm
of	I-Algorithm
Fatou	I-Algorithm
components	I-Algorithm
,	O
and	O
can	O
occur	O
around	O
irrationally	O
indifferent	O
periodic	O
points	O
.	O
</s>
<s>
The	O
Fatou	B-Language
set	I-Language
is	O
,	O
roughly	O
,	O
the	O
set	O
of	O
points	O
where	O
the	O
iterates	B-Algorithm
behave	O
similarly	O
to	O
their	O
neighbours	O
(	O
they	O
form	O
a	O
normal	O
family	O
)	O
.	O
</s>
<s>
Let	O
be	O
a	O
holomorphic	O
endomorphism	O
where	O
is	O
a	O
Riemann	O
surface	O
,	O
and	O
let	O
U	O
be	O
a	O
connected	O
component	O
of	O
the	O
Fatou	B-Language
set	I-Language
.	O
</s>
<s>
We	O
say	O
U	O
is	O
a	O
Siegel	B-Algorithm
disc	I-Algorithm
of	O
f	O
around	O
the	O
point	O
if	O
there	O
exists	O
a	O
biholomorphism	O
where	O
is	O
the	O
unit	O
disc	O
and	O
such	O
that	O
for	O
some	O
and	O
.	O
</s>
<s>
Siegel	O
's	O
theorem	O
proves	O
the	O
existence	O
of	O
Siegel	B-Algorithm
discs	I-Algorithm
for	O
irrational	O
numbers	O
satisfying	O
a	O
strong	O
irrationality	O
condition	O
(	O
a	O
Diophantine	O
condition	O
)	O
,	O
thus	O
solving	O
an	O
open	O
problem	O
since	O
Fatou	O
conjectured	O
his	O
theorem	O
on	O
the	O
Classification	B-Algorithm
of	I-Algorithm
Fatou	I-Algorithm
components	I-Algorithm
.	O
</s>
<s>
This	O
is	O
part	O
of	O
the	O
result	O
from	O
the	O
Classification	B-Algorithm
of	I-Algorithm
Fatou	I-Algorithm
components	I-Algorithm
.	O
</s>
