<s>
In	O
mathematics	O
,	O
an	O
iterated	B-Algorithm
function	I-Algorithm
is	O
a	O
function	O
(	O
that	O
is	O
,	O
a	O
function	O
from	O
some	O
set	O
to	O
itself	O
)	O
which	O
is	O
obtained	O
by	O
composing	B-Application
another	O
function	O
with	O
itself	O
a	O
certain	O
number	O
of	O
times	O
.	O
</s>
<s>
The	O
process	O
of	O
repeatedly	O
applying	O
the	O
same	O
function	O
is	O
called	O
iteration	B-Algorithm
.	O
</s>
<s>
Iterated	B-Algorithm
functions	I-Algorithm
are	O
objects	O
of	O
study	O
in	O
computer	B-General_Concept
science	I-General_Concept
,	O
fractals	B-Application
,	O
dynamical	O
systems	O
,	O
mathematics	O
and	O
renormalization	O
group	O
physics	O
.	O
</s>
<s>
The	O
formal	O
definition	O
of	O
an	O
iterated	B-Algorithm
function	I-Algorithm
on	O
a	O
set	O
X	O
follows	O
.	O
</s>
<s>
Defining	O
as	O
the	O
n-th	O
iterate	B-Algorithm
of	O
(	O
a	O
notation	O
introduced	O
by	O
Hans	O
Heinrich	O
Bürmann	O
and	O
John	O
Frederick	O
William	O
Herschel	O
)	O
,	O
where	O
n	O
is	O
a	O
non-negative	O
integer	O
,	O
by	O
:	O
</s>
<s>
where	O
is	O
the	O
identity	O
function	O
on	O
and	O
denotes	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
Because	O
the	O
notation	O
may	O
refer	O
to	O
both	O
iteration	B-Algorithm
(	O
composition	O
)	O
of	O
the	O
function	O
or	O
exponentiation	O
of	O
the	O
function	O
(	O
the	O
latter	O
is	O
commonly	O
used	O
in	O
trigonometry	O
)	O
,	O
some	O
mathematicians	O
choose	O
to	O
use	O
to	O
denote	O
the	O
compositional	O
meaning	O
,	O
writing	O
for	O
the	O
-th	O
iterate	B-Algorithm
of	O
the	O
function	O
,	O
as	O
in	O
,	O
for	O
example	O
,	O
meaning	O
.	O
</s>
<s>
The	O
sequence	O
of	O
functions	O
is	O
called	O
a	O
Picard	B-Algorithm
sequence	I-Algorithm
,	O
named	O
after	O
Charles	O
Émile	O
Picard	O
.	O
</s>
<s>
For	O
a	O
given	O
in	O
,	O
the	O
sequence	O
of	O
values	O
is	O
called	O
the	O
orbit	B-Algorithm
of	O
.	O
</s>
<s>
If	O
for	O
some	O
integer	O
,	O
the	O
orbit	B-Algorithm
is	O
called	O
a	O
periodic	O
orbit	B-Algorithm
.	O
</s>
<s>
The	O
smallest	O
such	O
value	O
of	O
for	O
a	O
given	O
is	O
called	O
the	O
period	O
of	O
the	O
orbit	B-Algorithm
.	O
</s>
<s>
The	O
cycle	B-Algorithm
detection	I-Algorithm
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
is	O
the	O
algorithmic	O
problem	O
of	O
finding	O
the	O
first	O
periodic	O
point	O
in	O
an	O
orbit	B-Algorithm
,	O
and	O
the	O
period	O
of	O
the	O
orbit	B-Algorithm
.	O
</s>
<s>
If	O
for	O
some	O
in	O
(	O
that	O
is	O
,	O
the	O
period	O
of	O
the	O
orbit	B-Algorithm
of	O
is	O
)	O
,	O
then	O
is	O
called	O
a	O
fixed	O
point	O
of	O
the	O
iterated	B-Algorithm
sequence	O
.	O
</s>
<s>
There	O
are	O
several	O
techniques	O
for	O
convergence	B-Algorithm
acceleration	I-Algorithm
of	O
the	O
sequences	O
produced	O
by	O
fixed	O
point	O
iteration	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Aitken	B-Algorithm
method	I-Algorithm
applied	O
to	O
an	O
iterated	B-Algorithm
fixed	O
point	O
is	O
known	O
as	O
Steffensen	O
's	O
method	O
,	O
and	O
produces	O
quadratic	O
convergence	O
.	O
</s>
<s>
Upon	O
iteration	B-Algorithm
,	O
one	O
may	O
find	O
that	O
there	O
are	O
sets	O
that	O
shrink	O
and	O
converge	O
towards	O
a	O
single	O
point	O
.	O
</s>
<s>
Conversely	O
,	O
iteration	B-Algorithm
may	O
give	O
the	O
appearance	O
of	O
points	O
diverging	O
away	O
from	O
a	O
single	O
point	O
;	O
this	O
would	O
be	O
the	O
case	O
for	O
an	O
unstable	O
fixed	O
point	O
.	O
</s>
<s>
When	O
the	O
points	O
of	O
the	O
orbit	B-Algorithm
converge	O
to	O
one	O
or	O
more	O
limits	O
,	O
the	O
set	O
of	O
accumulation	O
points	O
of	O
the	O
orbit	B-Algorithm
is	O
known	O
as	O
the	O
limit	O
set	O
or	O
the	O
ω-limit	O
set	O
.	O
</s>
<s>
The	O
ideas	O
of	O
attraction	O
and	O
repulsion	O
generalize	O
similarly	O
;	O
one	O
may	O
categorize	O
iterates	B-Algorithm
into	O
stable	O
sets	O
and	O
unstable	O
sets	O
,	O
according	O
to	O
the	O
behaviour	O
of	O
small	O
neighborhoods	O
under	O
iteration	B-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
visualized	O
as	O
the	O
behavior	O
of	O
a	O
point-cloud	O
or	O
dust-cloud	O
under	O
repeated	O
iteration	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
because	O
repeated	O
iteration	B-Algorithm
corresponds	O
to	O
a	O
shift	O
,	O
the	O
transfer	O
operator	O
,	O
and	O
its	O
adjoint	O
,	O
the	O
Koopman	B-Algorithm
operator	I-Algorithm
can	O
both	O
be	O
interpreted	O
as	O
shift	B-Algorithm
operators	I-Algorithm
action	O
on	O
a	O
shift	O
space	O
.	O
</s>
<s>
The	O
theory	O
of	O
subshifts	O
of	O
finite	O
type	O
provides	O
general	O
insight	O
into	O
many	O
iterated	B-Algorithm
functions	I-Algorithm
,	O
especially	O
those	O
leading	O
to	O
chaos	O
.	O
</s>
<s>
The	O
roots	O
chosen	O
are	O
normally	O
the	O
ones	O
belonging	O
to	O
the	O
orbit	B-Algorithm
under	O
study	O
.	O
</s>
<s>
Fractional	B-Algorithm
iteration	I-Algorithm
of	O
a	O
function	O
can	O
be	O
defined	O
:	O
for	O
instance	O
,	O
a	O
half	O
iterate	B-Algorithm
of	O
a	O
function	O
is	O
a	O
function	O
such	O
that	O
.	O
</s>
<s>
This	O
idea	O
can	O
be	O
generalized	O
so	O
that	O
the	O
iteration	B-Algorithm
count	O
becomes	O
a	O
continuous	O
parameter	O
,	O
a	O
sort	O
of	O
continuous	O
"	O
time	O
"	O
of	O
a	O
continuous	O
orbit	B-Algorithm
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
one	O
refers	O
to	O
the	O
system	O
as	O
a	O
flow	B-Algorithm
.	O
</s>
<s>
Negative	O
iterates	B-Algorithm
correspond	O
to	O
function	O
inverses	O
and	O
their	O
compositions	O
.	O
</s>
<s>
For	O
example	O
,	O
is	O
the	O
normal	O
inverse	O
of	O
,	O
while	O
is	O
the	O
inverse	O
composed	B-Application
with	O
itself	O
,	O
i.e.	O
</s>
<s>
Fractional	O
negative	O
iterates	B-Algorithm
are	O
defined	O
analogously	O
to	O
fractional	O
positive	O
ones	O
;	O
for	O
example	O
,	O
is	O
defined	O
such	O
that	O
,	O
or	O
,	O
equivalently	O
,	O
such	O
that	O
.	O
</s>
<s>
One	O
of	O
several	O
methods	O
of	O
finding	O
a	O
series	O
formula	O
for	O
fractional	B-Algorithm
iteration	I-Algorithm
,	O
making	O
use	O
of	O
a	O
fixed	O
point	O
,	O
is	O
as	O
follows	O
.	O
</s>
<s>
This	O
,	O
in	O
some	O
ways	O
,	O
is	O
the	O
most	O
natural	O
extra	O
condition	O
to	O
place	O
upon	O
the	O
fractional	O
iterates	B-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
two	O
iterated	B-Algorithm
functions	I-Algorithm
,	O
and	O
there	O
exists	O
a	O
homeomorphism	O
such	O
that	O
,	O
then	O
and	O
are	O
said	O
to	O
be	O
topologically	O
conjugate	O
.	O
</s>
<s>
Clearly	O
,	O
topological	O
conjugacy	O
is	O
preserved	O
under	O
iteration	B-Algorithm
,	O
as	O
.	O
</s>
<s>
Thus	O
,	O
if	O
one	O
can	O
solve	O
for	O
one	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
,	O
one	O
also	O
has	O
solutions	O
for	O
all	O
topologically	O
conjugate	O
systems	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
tent	B-Algorithm
map	I-Algorithm
is	O
topologically	O
conjugate	O
to	O
the	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
its	O
iteration	B-Algorithm
orbit	I-Algorithm
,	O
or	O
flow	B-Algorithm
,	O
under	O
suitable	O
provisions	O
(	O
e.g.	O
,	O
)	O
,	O
amounts	O
to	O
the	O
conjugate	O
of	O
the	O
orbit	B-Algorithm
of	O
the	O
monomial	O
,	O
</s>
<s>
where	O
in	O
this	O
expression	O
serves	O
as	O
a	O
plain	O
exponent	O
:	O
functional	O
iteration	B-Algorithm
has	O
been	O
reduced	O
to	O
multiplication	O
!	O
</s>
<s>
Here	O
,	O
however	O
,	O
the	O
exponent	O
no	O
longer	O
needs	O
be	O
integer	O
or	O
positive	O
,	O
and	O
is	O
a	O
continuous	O
"	O
time	O
"	O
of	O
evolution	O
for	O
the	O
full	O
orbit	B-Algorithm
:	O
the	O
monoid	O
of	O
the	O
Picard	B-Algorithm
sequence	I-Algorithm
(	O
cf	O
.	O
</s>
<s>
transformation	B-Algorithm
semigroup	O
)	O
has	O
generalized	O
to	O
a	O
full	O
continuous	O
group	O
.	O
</s>
<s>
This	O
method	O
(	O
perturbative	O
determination	O
of	O
the	O
principal	O
eigenfunction	B-Algorithm
Ψ	O
,	O
cf	O
.	O
</s>
<s>
Carleman	B-Algorithm
matrix	I-Algorithm
)	O
is	O
equivalent	O
to	O
the	O
algorithm	O
of	O
the	O
preceding	O
section	O
,	O
albeit	O
,	O
in	O
practice	O
,	O
more	O
powerful	O
and	O
systematic	O
.	O
</s>
<s>
If	O
the	O
function	O
is	O
linear	O
and	O
can	O
be	O
described	O
by	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
,	O
that	O
is	O
,	O
a	O
matrix	O
whose	O
rows	O
or	O
columns	O
sum	O
to	O
one	O
,	O
then	O
the	O
iterated	B-Algorithm
system	O
is	O
known	O
as	O
a	O
Markov	O
chain	O
.	O
</s>
<s>
There	O
are	O
many	B-Algorithm
chaotic	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Well-known	O
iterated	B-Algorithm
functions	I-Algorithm
include	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
and	O
iterated	B-Algorithm
function	I-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
Ernst	O
Schröder	O
,	O
in	O
1870	O
,	O
worked	O
out	O
special	O
cases	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
,	O
such	O
as	O
the	O
chaotic	O
case	O
,	O
so	O
that	O
,	O
hence	O
.	O
</s>
<s>
If	O
is	O
the	O
action	O
of	O
a	O
group	O
element	O
on	O
a	O
set	O
,	O
then	O
the	O
iterated	B-Algorithm
function	I-Algorithm
corresponds	O
to	O
a	O
free	O
group	O
.	O
</s>
<s>
Most	O
functions	O
do	O
not	O
have	O
explicit	O
general	O
closed-form	O
expressions	O
for	O
the	O
n-th	O
iterate	B-Algorithm
.	O
</s>
<s>
Iterated	B-Algorithm
functions	I-Algorithm
can	O
be	O
studied	O
with	O
the	O
Artin	O
–	O
Mazur	O
zeta	O
function	O
and	O
with	O
transfer	O
operators	O
.	O
</s>
<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
iterated	B-Algorithm
functions	I-Algorithm
occur	O
as	O
a	O
special	O
case	O
of	O
recursive	O
functions	O
,	O
which	O
in	O
turn	O
anchor	O
the	O
study	O
of	O
such	O
broad	O
topics	O
as	O
lambda	B-Language
calculus	I-Language
,	O
or	O
narrower	O
ones	O
,	O
such	O
as	O
the	O
denotational	B-Application
semantics	I-Application
of	O
computer	O
programs	O
.	O
</s>
<s>
Two	O
important	O
functionals	O
can	O
be	O
defined	O
in	O
terms	O
of	O
iterated	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
functional	O
derivative	O
of	O
an	O
iterated	B-Algorithm
function	I-Algorithm
is	O
given	O
by	O
the	O
recursive	O
formula	O
:	O
</s>
<s>
Iterated	B-Algorithm
functions	I-Algorithm
crop	O
up	O
in	O
the	O
series	O
expansion	O
of	O
combined	O
functions	O
,	O
such	O
as	O
.	O
</s>
<s>
Given	O
the	O
iteration	B-Algorithm
velocity	O
,	O
or	O
beta	O
function	O
(	O
physics	O
)	O
,	O
</s>
<s>
Consequently	O
,	O
,	O
action	O
by	O
a	O
plain	O
shift	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
For	O
continuous	O
iteration	B-Algorithm
index	O
,	O
then	O
,	O
now	O
written	O
as	O
a	O
subscript	O
,	O
this	O
amounts	O
to	O
Lie	O
's	O
celebrated	O
exponential	O
realization	O
of	O
a	O
continuous	O
group	O
,	O
</s>
<s>
The	O
initial	O
flow	B-Algorithm
velocity	O
suffices	O
to	O
determine	O
the	O
entire	O
flow	B-Algorithm
,	O
given	O
this	O
exponential	O
realization	O
which	O
automatically	O
provides	O
the	O
general	O
solution	O
to	O
the	O
translation	O
functional	O
equation	O
,	O
</s>
