<s>
In	O
mathematics	O
,	O
an	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
is	O
a	O
kind	O
of	O
dynamical	O
system	O
that	O
generalises	O
circle	O
rotation	O
.	O
</s>
<s>
Let	O
and	O
let	O
be	O
a	O
permutation	B-Algorithm
on	O
.	O
</s>
<s>
Define	O
a	O
map	O
called	O
the	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
associated	O
with	O
the	O
pair	O
as	O
follows	O
.	O
</s>
<s>
Thus	O
acts	O
on	O
each	O
subinterval	O
of	O
the	O
form	O
by	O
a	O
translation	B-Algorithm
,	O
and	O
it	O
rearranges	O
these	O
subintervals	O
so	O
that	O
the	O
subinterval	O
at	O
position	O
is	O
moved	O
to	O
position	O
.	O
</s>
<s>
Any	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
is	O
a	O
bijection	B-Algorithm
of	O
to	O
itself	O
that	O
preserves	O
the	O
Lebesgue	O
measure	O
.	O
</s>
<s>
The	O
inverse	O
of	O
the	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
is	O
again	O
an	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
If	O
and	O
(	O
in	O
cycle	B-Algorithm
notation	I-Algorithm
)	O
,	O
and	O
if	O
we	O
join	O
up	O
the	O
ends	O
of	O
the	O
interval	O
to	O
make	O
a	O
circle	O
,	O
then	O
is	O
just	O
a	O
circle	O
rotation	O
.	O
</s>
<s>
If	O
,	O
and	O
provided	O
satisfies	O
certain	O
non-degeneracy	O
conditions	O
(	O
namely	O
there	O
is	O
no	O
integer	O
such	O
that	O
)	O
,	O
a	O
deep	O
theorem	O
which	O
was	O
a	O
conjecture	O
of	O
M.Keane	O
and	O
due	O
independently	O
to	O
William	O
A	O
.	O
Veech	O
and	O
to	O
Howard	O
Masur	O
asserts	O
that	O
for	O
almost	O
all	O
choices	O
of	O
in	O
the	O
unit	O
simplex	O
the	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
is	O
again	O
uniquely	O
ergodic	O
.	O
</s>
<s>
The	O
dyadic	O
odometer	O
can	O
be	O
understood	O
as	O
an	O
interval	B-Algorithm
exchange	I-Algorithm
transformation	I-Algorithm
of	O
a	O
countable	O
number	O
of	O
intervals	O
.	O
</s>
