<s>
In	O
the	O
mathematical	O
discipline	O
known	O
as	O
complex	O
dynamics	O
,	O
the	O
Herman	B-Algorithm
ring	I-Algorithm
is	O
a	O
Fatou	B-Algorithm
component	I-Algorithm
where	O
the	O
rational	O
function	O
is	O
conformally	O
conjugate	O
to	O
an	O
irrational	O
rotation	O
of	O
the	O
standard	O
annulus	O
.	O
</s>
<s>
So	O
the	O
dynamics	O
on	O
the	O
Herman	B-Algorithm
ring	I-Algorithm
is	O
simple	O
.	O
</s>
<s>
It	O
was	O
introduced	O
by	O
,	O
and	O
later	O
named	O
after	O
,	O
Michael	O
Herman	O
(	O
1979	O
)	O
who	O
first	O
found	O
and	O
constructed	O
this	O
type	O
of	O
Fatou	B-Algorithm
component	I-Algorithm
.	O
</s>
<s>
Polynomials	O
do	O
not	O
have	O
Herman	B-Algorithm
rings	I-Algorithm
.	O
</s>
<s>
Rational	O
functions	O
can	O
have	O
Herman	B-Algorithm
rings	I-Algorithm
.	O
</s>
<s>
According	O
to	O
the	O
result	O
of	O
Shishikura	O
,	O
if	O
a	O
rational	O
function	O
ƒ	O
possesses	O
a	O
Herman	B-Algorithm
ring	I-Algorithm
,	O
then	O
the	O
degree	O
of	O
ƒ	O
is	O
at	O
least3	O
.	O
</s>
<s>
meromorphic	O
functions	O
can	O
possess	O
Herman	B-Algorithm
rings	I-Algorithm
.	O
</s>
<s>
Herman	B-Algorithm
rings	I-Algorithm
for	O
transcendental	O
meromorphic	O
functions	O
have	O
been	O
studied	O
by	O
T	O
.	O
Nayak	O
.	O
</s>
<s>
According	O
to	O
a	O
result	O
of	O
Nayak	O
,	O
if	O
there	O
is	O
an	O
omitted	O
value	O
for	O
such	O
a	O
function	O
then	O
Herman	B-Algorithm
rings	I-Algorithm
of	O
period	O
1	O
or	O
2	O
do	O
not	O
exist	O
.	O
</s>
<s>
Also	O
,	O
it	O
is	O
proved	O
that	O
if	O
there	O
is	O
only	O
a	O
single	O
pole	O
and	O
at	O
least	O
an	O
omitted	O
value	O
,	O
the	O
function	O
has	O
no	O
Herman	B-Algorithm
ring	I-Algorithm
of	O
any	O
period	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
rational	O
function	O
which	O
possesses	O
a	O
Herman	B-Algorithm
ring	I-Algorithm
.	O
</s>
<s>
The	O
picture	O
shown	O
on	O
the	O
right	O
is	O
the	O
Julia	B-Language
set	I-Language
of	O
ƒ	O
:	O
the	O
curves	O
in	O
the	O
white	O
annulus	O
are	O
the	O
orbits	O
of	O
some	O
points	O
under	O
the	O
iterations	O
of	O
ƒ	O
while	O
the	O
dashed	O
line	O
denotes	O
the	O
unit	O
circle	O
.	O
</s>
<s>
There	O
is	O
an	O
example	O
of	O
rational	O
function	O
that	O
possesses	O
a	O
Herman	B-Algorithm
ring	I-Algorithm
,	O
and	O
some	O
periodic	B-Algorithm
parabolic	I-Algorithm
Fatou	I-Algorithm
components	I-Algorithm
at	O
the	O
same	O
time	O
.	O
</s>
<s>
Further	O
,	O
there	O
is	O
a	O
rational	O
function	O
which	O
possesses	O
a	O
Herman	B-Algorithm
ring	I-Algorithm
with	O
period	O
2	O
.	O
</s>
<s>
which	O
possesses	O
a	O
Siegel	B-Algorithm
disk	I-Algorithm
with	O
period2	O
.	O
</s>
<s>
then	O
the	O
period	O
of	O
one	O
of	O
the	O
Herman	B-Algorithm
ring	I-Algorithm
of	O
ga	O
,	O
b	O
,	O
c	O
is3	O
.	O
</s>
<s>
Shishikura	O
also	O
given	O
an	O
example	O
:	O
a	O
rational	O
function	O
which	O
possesses	O
a	O
Herman	B-Algorithm
ring	I-Algorithm
with	O
period	O
2	O
,	O
but	O
the	O
parameters	O
showed	O
above	O
are	O
different	O
from	O
his	O
.	O
</s>
<s>
So	O
there	O
is	O
a	O
question	O
:	O
How	O
to	O
find	O
the	O
formulas	O
of	O
the	O
rational	O
functions	O
which	O
possess	O
Herman	B-Algorithm
rings	I-Algorithm
with	O
higher	O
period	O
?	O
</s>
<s>
This	O
question	O
can	O
be	O
answered	O
(	O
for	O
any	O
period	O
>	O
0	O
)	O
by	O
using	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
for	O
the	O
rational	O
functions	O
ga	O
,	O
b	O
,	O
c	O
.	O
</s>
<s>
The	O
classic	O
Mandelbrot	B-Algorithm
set	I-Algorithm
(	O
for	O
quadratic	O
polynomials	O
)	O
is	O
approximated	O
by	O
iterating	O
the	O
critical	O
point	O
for	O
each	O
such	O
polynomial	O
,	O
and	O
identifying	O
the	O
polynomials	O
for	O
which	O
the	O
iterates	O
of	O
the	O
critical	O
point	O
do	O
not	O
converge	O
to	O
infinity	O
.	O
</s>
<s>
Similarly	O
a	O
Mandelbrot	B-Algorithm
set	I-Algorithm
can	O
be	O
defined	O
for	O
the	O
set	O
of	O
rational	O
functions	O
ga	O
,	O
b	O
,	O
c	O
by	O
distinguishing	O
between	O
the	O
values	O
of	O
(	O
a	O
,	O
b	O
,	O
c	O
)	O
in	O
complex	O
3-space	O
for	O
which	O
all	O
the	O
three	O
critical	O
points	O
(	O
i.e.	O
</s>
<s>
For	O
each	O
value	O
of	O
a	O
and	O
b	O
,	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
for	O
ga	O
,	O
b	O
,	O
c	O
can	O
be	O
calculated	O
in	O
the	O
plane	O
of	O
complex	O
values	O
c	O
.	O
When	O
a	O
and	O
b	O
are	O
nearly	O
equal	O
,	O
this	O
set	O
approximates	O
the	O
classic	O
Mandelbrot	B-Algorithm
set	I-Algorithm
for	O
quadratic	O
polynomials	O
,	O
because	O
ga	O
,	O
b	O
,	O
c	O
is	O
equal	O
to	O
x2	O
+	O
c	O
when	O
a	O
=	O
b	O
.	O
</s>
<s>
In	O
the	O
classic	O
Mandelbrot	B-Algorithm
set	I-Algorithm
,	O
Siegel	B-Algorithm
discs	I-Algorithm
can	O
be	O
approximated	O
by	O
choosing	O
points	O
along	O
the	O
edge	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
with	O
irrational	O
winding	O
number	O
having	O
continued	O
fraction	O
expansion	O
with	O
bounded	O
denominators	O
.	O
</s>
<s>
These	O
denominators	O
can	O
be	O
identified	O
by	O
the	O
sequence	O
of	O
nodes	O
along	O
the	O
edge	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
approaching	O
the	O
point	O
.	O
</s>
<s>
Similarly	O
,	O
Herman	B-Algorithm
rings	I-Algorithm
can	O
be	O
identified	O
in	O
a	O
Mandelbrot	B-Algorithm
set	I-Algorithm
of	O
rational	O
functions	O
by	O
observing	O
a	O
series	O
of	O
nodes	O
arranged	O
on	O
both	O
sides	O
of	O
a	O
curve	O
,	O
and	O
choosing	O
points	O
along	O
that	O
curve	O
,	O
avoiding	O
the	O
attached	O
nodes	O
,	O
thereby	O
obtaining	O
a	O
desired	O
sequence	O
of	O
denominators	O
in	O
the	O
continued	O
fraction	O
expansion	O
of	O
the	O
rotation	O
number	O
.	O
</s>
<s>
The	O
following	O
illustrates	O
a	O
planar	O
slice	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
of	O
ga	O
,	O
b	O
,	O
c	O
with	O
|	O
a-b	O
|	O
=	O
.0001	O
,	O
and	O
with	O
c	O
centered	O
at	O
a	O
value	O
of	O
c	O
which	O
identifies	O
a	O
5-cycle	O
of	O
Siegel	B-Algorithm
discs	I-Algorithm
in	O
the	O
classic	O
Mandelbrot	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
image	O
above	O
uses	O
a	O
=	O
0.12601278	O
+	O
.0458649i	O
,	O
b	O
=	O
.12582484	O
+	O
.045796497i	O
,	O
and	O
is	O
centered	O
at	O
a	O
value	O
of	O
c	O
=	O
0.3688	O
-.3578	O
,	O
which	O
is	O
near	O
5-cycles	O
of	O
Siegel	B-Algorithm
discs	I-Algorithm
in	O
the	O
classic	O
Mandelbrot	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
In	O
the	O
above	O
image	O
,	O
a	O
5-cycle	O
of	O
Herman	B-Algorithm
rings	I-Algorithm
can	O
be	O
approximated	O
by	O
choosing	O
a	O
point	O
c	O
along	O
the	O
above	O
illustrated	O
curve	O
having	O
nodes	O
on	O
both	O
sides	O
,	O
for	O
which	O
ga	O
,	O
b	O
,	O
c	O
has	O
approximately	O
the	O
desired	O
winding	O
number	O
,	O
using	O
values	O
as	O
follows	O
:	O
</s>
<s>
The	O
resulting	O
5-cycle	O
of	O
Herman	B-Algorithm
rings	I-Algorithm
is	O
illustrated	O
below	O
:	O
</s>
