<s>
In	O
dynamical	O
systems	O
theory	O
,	O
the	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
a	O
chaotic	O
map	O
from	O
the	O
unit	O
square	O
into	O
itself	O
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
can	O
be	O
understood	O
as	O
the	O
bilateral	B-Algorithm
shift	I-Algorithm
operator	O
of	O
a	O
bi-infinite	O
two-state	O
lattice	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
topologically	O
conjugate	O
to	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
In	O
physics	O
,	O
a	O
chain	O
of	O
coupled	O
baker	B-Algorithm
's	I-Algorithm
maps	I-Algorithm
can	O
be	O
used	O
to	O
model	O
deterministic	O
diffusion	O
.	O
</s>
<s>
As	O
with	O
many	O
deterministic	O
dynamical	O
systems	O
,	O
the	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
studied	O
by	O
its	O
action	O
on	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
defined	O
on	O
the	O
unit	O
square	O
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
defines	O
an	O
operator	O
on	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
,	O
known	O
as	O
the	O
transfer	O
operator	O
of	O
the	O
map	O
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
an	O
exactly	O
solvable	O
model	O
of	O
deterministic	O
chaos	O
,	O
in	O
that	O
the	O
eigenfunctions	B-Algorithm
and	O
eigenvalues	O
of	O
the	O
transfer	O
operator	O
can	O
be	O
explicitly	O
determined	O
.	O
</s>
<s>
There	O
are	O
two	O
alternative	O
definitions	O
of	O
the	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
which	O
are	O
in	O
common	O
use	O
.	O
</s>
<s>
One	O
definition	O
folds	O
over	O
or	O
rotates	O
one	O
of	O
the	O
sliced	O
halves	O
before	O
joining	O
it	O
(	O
similar	O
to	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
)	O
and	O
the	O
other	O
does	O
not	O
.	O
</s>
<s>
while	O
the	O
unfolded	O
map	O
is	O
analogous	O
to	O
the	O
Bernoulli	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
Bernoulli	B-Algorithm
map	I-Algorithm
can	O
be	O
understood	O
as	O
the	O
map	O
that	O
progressively	O
lops	O
digits	O
off	O
the	O
dyadic	O
expansion	O
of	O
x	O
.	O
</s>
<s>
Unlike	O
the	O
tent	B-Algorithm
map	I-Algorithm
,	O
the	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
invertible	O
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
preserves	O
the	O
two-dimensional	O
Lebesgue	O
measure	O
.	O
</s>
<s>
The	O
transfer	O
operator	O
is	O
unitary	B-Algorithm
on	O
the	O
Hilbert	O
space	O
of	O
square-integrable	B-Algorithm
functions	I-Algorithm
on	O
the	O
unit	O
square	O
.	O
</s>
<s>
The	O
spectrum	O
is	O
continuous	O
,	O
and	O
because	O
the	O
operator	O
is	O
unitary	B-Algorithm
the	O
eigenvalues	O
lie	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
The	O
transfer	O
operator	O
is	O
not	O
unitary	B-Algorithm
on	O
the	O
space	O
of	O
functions	O
polynomial	O
in	O
the	O
first	O
coordinate	O
and	O
square-integrable	B-Algorithm
in	O
the	O
second	O
.	O
</s>
<s>
On	O
this	O
space	O
,	O
it	O
has	O
a	O
discrete	O
,	O
non-unitary	O
,	O
decaying	O
spectrum	O
.	O
</s>
<s>
The	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
can	O
be	O
understood	O
as	O
the	O
two-sided	O
shift	B-Algorithm
operator	I-Algorithm
on	O
the	O
symbolic	O
dynamics	O
of	O
a	O
one-dimensional	O
lattice	O
.	O
</s>
<s>
which	O
is	O
seen	O
to	O
be	O
the	O
unfolded	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
given	O
above	O
.	O
</s>
