<s>
A	O
complex	B-Algorithm
quadratic	I-Algorithm
polynomial	I-Algorithm
is	O
a	O
quadratic	O
polynomial	O
whose	O
coefficients	O
and	O
variable	O
are	O
complex	O
numbers	O
.	O
</s>
<s>
the	O
orbit	B-Algorithm
of	O
the	O
critical	O
point	O
can	O
be	O
finite	O
,	O
because	O
the	O
critical	O
point	O
is	O
periodic	O
or	O
preperiodic	O
.	O
</s>
<s>
When	O
the	O
quadratic	O
polynomial	O
has	O
only	O
one	O
variable	O
(	O
univariate	B-General_Concept
)	O
,	O
one	O
can	O
distinguish	O
its	O
four	O
main	O
forms	O
:	O
</s>
<s>
The	O
factored	O
form	O
used	O
for	O
the	O
logistic	B-Algorithm
map	I-Algorithm
:	O
</s>
<s>
Since	O
is	O
affine	B-Algorithm
conjugate	O
to	O
the	O
general	O
form	O
of	O
the	O
quadratic	O
polynomial	O
it	O
is	O
often	O
used	O
to	O
study	O
complex	O
dynamics	O
and	O
to	O
create	O
images	O
of	O
Mandelbrot	B-Algorithm
,	O
Julia	B-Language
and	O
Fatou	B-Language
sets	I-Language
.	O
</s>
<s>
There	O
is	O
semi-conjugacy	O
between	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
(	O
the	O
doubling	B-Algorithm
map	I-Algorithm
)	O
and	O
the	O
quadratic	O
polynomial	O
case	O
of	O
c	O
=	O
–	O
2	O
.	O
</s>
<s>
Here	O
denotes	O
the	O
n-th	O
iterate	B-Algorithm
of	O
the	O
function	O
:	O
</s>
<s>
Because	O
of	O
the	O
possible	O
confusion	O
with	O
exponentiation	O
,	O
some	O
authors	O
write	O
for	O
the	O
nth	O
iterate	B-Algorithm
of	O
.	O
</s>
<s>
it	O
is	O
named	O
the	O
quadratic	O
map	B-Algorithm
:	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
the	O
set	O
of	O
values	O
of	O
the	O
parameter	O
c	O
for	O
which	O
the	O
initial	O
condition	O
z0	O
=	O
0	O
does	O
not	O
cause	O
the	O
iterates	O
to	O
diverge	O
to	O
infinity	O
.	O
</s>
<s>
A	O
critical	O
point	O
of	O
is	O
a	O
point	O
on	O
the	O
dynamical	O
plane	O
such	O
that	O
the	O
derivative	B-Algorithm
vanishes	O
:	O
</s>
<s>
is	O
an	O
initial	O
point	O
for	O
Mandelbrot	B-Algorithm
set	I-Algorithm
iteration	B-Algorithm
.	O
</s>
<s>
For	O
the	O
quadratic	O
family	O
the	O
critical	O
point	O
z	O
=	O
0	O
is	O
the	O
center	O
of	O
symmetry	O
of	O
the	O
Julia	B-Language
set	I-Language
Jc	O
,	O
so	O
it	O
is	O
a	O
convex	O
combination	O
of	O
two	O
points	O
in	O
Jc	O
.	O
</s>
<s>
The	O
forward	B-Algorithm
orbit	I-Algorithm
of	O
a	O
critical	O
point	O
is	O
called	O
a	O
critical	O
orbit	B-Algorithm
.	O
</s>
<s>
Critical	O
orbits	B-Algorithm
are	O
very	O
important	O
because	O
every	O
attracting	O
periodic	B-Algorithm
orbit	I-Algorithm
attracts	O
a	O
critical	O
point	O
,	O
so	O
studying	O
the	O
critical	O
orbits	B-Algorithm
helps	O
us	O
understand	O
the	O
dynamics	O
in	O
the	O
Fatou	B-Language
set	I-Language
.	O
</s>
<s>
This	O
orbit	B-Algorithm
falls	O
into	O
an	O
attracting	B-Algorithm
periodic	I-Algorithm
cycle	I-Algorithm
if	O
one	O
exists	O
.	O
</s>
<s>
One	O
can	O
use	O
the	O
Julia-Mandelbrot	O
4-dimensional	O
(	O
4D	O
)	O
space	O
for	O
a	O
global	O
analysis	O
of	O
this	O
dynamical	O
system	O
.	O
</s>
<s>
The	O
phase	O
space	O
of	O
a	O
quadratic	O
map	B-Algorithm
is	O
called	O
its	O
parameter	O
plane	O
.	O
</s>
<s>
There	O
are	O
no	O
orbits	B-Algorithm
on	O
the	O
parameter	O
plane	O
.	O
</s>
<s>
This	O
allows	O
us	O
to	O
study	O
the	O
Mandelbrot	B-Algorithm
and	O
Julia	B-Language
sets	I-Language
combinatorially	O
,	O
replacing	O
the	O
dynamical	O
plane	O
by	O
the	O
unit	O
circle	O
,	O
rays	O
by	O
angles	O
,	O
and	O
the	O
quadratic	O
polynomial	O
by	O
the	O
doubling	O
modulo	O
one	O
map.	O
"	O
</s>
<s>
Virpi	O
KaukoTrees	O
of	O
visible	O
components	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
by	O
Virpi	O
K	O
a	O
u	O
k	O
o	O
,	O
FUNDAM	O
E	O
N	O
TA	O
MATHEMATICAE	O
164	O
(	O
2000	O
)	O
On	O
the	O
dynamical	O
plane	O
one	O
can	O
find	O
:	O
</s>
<s>
This	O
can	O
easily	O
be	O
verified	O
by	O
using	O
the	O
chain	O
rule	O
for	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
This	O
derivative	B-Algorithm
is	O
used	O
in	O
the	O
distance	O
estimation	O
method	O
for	O
drawing	O
a	O
Mandelbrot	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
This	O
derivative	B-Algorithm
is	O
used	O
for	O
computing	O
the	O
external	O
distance	O
to	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
The	O
Schwarzian	O
derivative	B-Algorithm
(	O
SD	O
for	O
short	O
)	O
of	O
f	O
is	O
:	O
</s>
