<s>
The	O
name	O
bit	B-Algorithm
shift	I-Algorithm
map	I-Algorithm
arises	O
because	O
,	O
if	O
the	O
value	O
of	O
an	O
iterate	O
is	O
written	O
in	O
binary	O
notation	O
,	O
the	O
next	O
iterate	O
is	O
obtained	O
by	O
shifting	O
the	O
binary	O
point	O
one	O
bit	O
to	O
the	O
right	O
,	O
and	O
if	O
the	O
bit	O
to	O
the	O
left	O
of	O
the	O
new	O
binary	O
point	O
is	O
a	O
"	O
one	O
"	O
,	O
replacing	O
it	O
with	O
a	O
zero	O
.	O
</s>
<s>
The	O
dyadic	B-Algorithm
transformation	I-Algorithm
provides	O
an	O
example	O
of	O
how	O
a	O
simple	O
1-dimensional	O
map	O
can	O
give	O
rise	O
to	O
chaos	O
.	O
</s>
<s>
The	O
word	O
"	O
infinite	O
"	O
is	O
qualified	O
with	O
"	O
semi-	O
"	O
,	O
as	O
one	O
can	O
also	O
define	O
a	O
different	O
space	O
consisting	O
of	O
all	O
doubly-infinite	O
(	O
double-ended	O
)	O
strings	O
;	O
this	O
will	O
lead	O
to	O
the	O
Baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Since	O
one	O
can	O
easily	O
see	O
that	O
For	O
the	O
doubly-infinite	O
sequence	O
of	O
bits	O
the	O
induced	O
homomorphism	O
is	O
the	O
Baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
gives	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
,	O
as	O
conventionally	O
defined	O
.	O
</s>
<s>
This	O
illustrates	O
sensitive	O
dependence	O
on	O
initial	O
conditions	O
—	O
the	O
mapping	B-Algorithm
from	O
the	O
truncated	O
initial	O
condition	O
has	O
deviated	O
exponentially	O
from	O
the	O
mapping	B-Algorithm
from	O
the	O
true	O
initial	O
condition	O
.	O
</s>
<s>
The	O
dyadic	B-Algorithm
transformation	I-Algorithm
is	O
topologically	O
semi-conjugate	O
to	O
the	O
unit-height	O
tent	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
conjugate	O
to	O
the	O
chaotic	O
r	O
=	O
4	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
There	O
is	O
also	O
a	O
semi-conjugacy	O
between	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
(	O
here	O
named	O
angle	O
doubling	B-Algorithm
map	I-Algorithm
)	O
and	O
the	O
quadratic	B-Algorithm
polynomial	I-Algorithm
.	O
</s>
<s>
If	O
x0	O
is	O
rational	O
the	O
image	O
of	O
x0	O
contains	O
a	O
finite	O
number	O
of	O
distinct	O
values	O
within	O
[	O
0	O
,	O
1	O
)	O
and	O
the	O
forward	B-Algorithm
orbit	I-Algorithm
of	O
x0	O
is	O
eventually	O
periodic	O
,	O
with	O
period	O
equal	O
to	O
the	O
period	O
of	O
the	O
binary	O
expansion	O
of	O
x0	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
forward	B-Algorithm
orbit	I-Algorithm
of	O
11/24	O
is	O
:	O
</s>
<s>
Within	O
any	O
subinterval	O
of	O
[	O
0	O
,	O
1	O
)	O
,	O
no	O
matter	O
how	O
small	O
,	O
there	O
are	O
therefore	O
an	O
infinite	O
number	O
of	O
points	O
whose	O
orbits	B-Algorithm
are	O
eventually	O
periodic	O
,	O
and	O
an	O
infinite	O
number	O
of	O
points	O
whose	O
orbits	B-Algorithm
are	O
never	O
periodic	O
.	O
</s>
<s>
This	O
sensitive	O
dependence	O
on	O
initial	O
conditions	O
is	O
a	O
characteristic	O
of	O
chaotic	B-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
The	O
periodic	O
and	O
non-periodic	O
orbits	O
can	O
be	O
more	O
easily	O
understood	O
not	O
by	O
working	O
with	O
the	O
map	O
directly	O
,	O
but	O
rather	O
with	O
the	O
bit	B-Algorithm
shift	I-Algorithm
map	I-Algorithm
defined	O
on	O
the	O
Cantor	O
space	O
.	O
</s>
<s>
It	O
is	O
a	O
surjection	B-Algorithm
:	O
every	O
dyadic	O
rational	O
has	O
not	O
one	O
,	O
but	O
two	O
distinct	O
representations	O
in	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
The	O
initial	O
sequence	O
corresponds	O
to	O
the	O
non-periodic	O
part	O
of	O
the	O
orbit	B-Algorithm
,	O
after	O
which	O
iteration	O
settles	O
down	O
to	O
all	O
zeros	O
(	O
equivalently	O
,	O
all-ones	O
)	O
.	O
</s>
<s>
Expressed	O
as	O
bit	O
strings	O
,	O
the	O
periodic	B-Algorithm
orbits	I-Algorithm
of	O
the	O
map	O
can	O
be	O
seen	O
to	O
the	O
rationals	O
.	O
</s>
<s>
That	O
is	O
,	O
after	O
an	O
initial	O
"	O
chaotic	O
"	O
sequence	O
of	O
,	O
a	O
periodic	O
orbit	B-Algorithm
settles	O
down	O
into	O
a	O
repeating	O
string	O
of	O
length	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
periodic	B-Algorithm
orbits	I-Algorithm
of	O
the	O
map	O
are	O
in	O
one-to-one	O
correspondence	O
with	O
the	O
rationals	O
.	O
</s>
<s>
For	O
example	O
,	O
geodesics	O
on	O
compact	O
manifolds	B-Architecture
can	O
have	O
periodic	B-Algorithm
orbits	I-Algorithm
that	O
behave	O
in	O
this	O
way	O
.	O
</s>
<s>
Almost	O
all	O
orbits	B-Algorithm
are	O
not	O
periodic	O
!	O
</s>
<s>
The	O
aperiodic	O
orbits	B-Algorithm
correspond	O
to	O
the	O
irrational	O
numbers	O
.	O
</s>
<s>
An	O
open	O
question	O
is	O
to	O
what	O
degree	O
the	O
behavior	O
of	O
the	O
periodic	B-Algorithm
orbits	I-Algorithm
constrain	O
the	O
behavior	O
of	O
the	O
system	O
as	O
a	O
whole	O
.	O
</s>
<s>
Instead	O
of	O
looking	O
at	O
the	O
orbits	B-Algorithm
of	O
individual	O
points	O
under	O
the	O
action	O
of	O
the	O
map	O
,	O
it	O
is	O
equally	O
worthwhile	O
to	O
explore	O
how	O
the	O
map	O
affects	O
densities	O
on	O
the	O
unit	O
interval	O
.	O
</s>
<s>
The	O
denominator	O
in	O
the	O
above	O
is	O
the	O
Jacobian	O
determinant	O
of	O
the	O
transformation	O
,	O
here	O
it	O
is	O
just	O
the	O
derivative	B-Algorithm
of	O
and	O
so	O
.	O
</s>
<s>
The	O
map	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
as	O
one	O
easily	O
sees	O
that	O
and	O
for	O
all	O
functions	O
on	O
the	O
unit	O
interval	O
,	O
and	O
all	O
constants	O
.	O
</s>
<s>
Viewed	O
as	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
the	O
most	O
obvious	O
and	O
pressing	O
question	O
is	O
:	O
what	O
is	O
its	O
spectrum	O
?	O
</s>
<s>
The	O
uniform	O
density	O
is	O
,	O
in	O
fact	O
,	O
nothing	O
other	O
than	O
the	O
invariant	O
measure	O
of	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
To	O
explore	O
the	O
spectrum	O
of	O
in	O
greater	O
detail	O
,	O
one	O
must	O
first	O
limit	O
oneself	O
to	O
a	O
suitable	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
(	O
on	O
the	O
unit	O
interval	O
)	O
to	O
work	O
with	O
.	O
</s>
<s>
This	O
might	O
be	O
the	O
space	O
of	O
Lebesgue	O
measurable	O
functions	O
,	O
or	O
perhaps	O
the	O
space	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
,	O
or	O
perhaps	O
even	O
just	O
polynomials	O
.	O
</s>
<s>
This	O
linear	B-Architecture
operator	I-Architecture
is	O
called	O
the	O
transfer	O
operator	O
or	O
the	O
Ruelle	O
–	O
Frobenius	O
–	O
Perron	O
operator	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
operator	O
has	O
a	O
discrete	O
spectrum	O
,	O
and	O
the	O
eigenfunctions	B-Algorithm
are	O
(	O
curiously	O
)	O
the	O
Bernoulli	O
polynomials	O
!	O
</s>
<s>
Another	O
complete	O
basis	O
is	O
provided	O
by	O
the	O
Takagi	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
fractal	O
eigenfunctions	B-Algorithm
show	O
an	O
explicit	O
symmetry	O
under	O
the	O
fractal	O
groupoid	O
of	O
the	O
modular	O
group	O
;	O
this	O
is	O
developed	O
in	O
greater	O
detail	O
in	O
the	O
article	O
on	O
the	O
Takagi	B-Algorithm
function	I-Algorithm
(	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
)	O
.	O
</s>
<s>
and	O
the	O
resulting	O
renormalization	O
group	O
transformation	O
for	O
will	O
be	O
precisely	O
the	O
dyadic	B-Algorithm
map	I-Algorithm
:	O
</s>
