<s>
In	O
mathematics	O
,	O
Arnold	B-Algorithm
's	I-Algorithm
cat	I-Algorithm
map	I-Algorithm
is	O
a	O
chaotic	O
map	O
from	O
the	O
torus	O
into	O
itself	O
,	O
named	O
after	O
Vladimir	O
Arnold	O
,	O
who	O
demonstrated	O
its	O
effects	O
in	O
the	O
1960s	O
using	O
an	O
image	O
of	O
a	O
cat	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
That	O
is	O
,	O
with	O
a	O
unit	O
equal	O
to	O
the	O
width	O
of	O
the	O
square	O
image	O
,	O
the	O
image	O
is	O
sheared	B-Algorithm
one	O
unit	O
up	O
,	O
then	O
two	O
units	O
to	O
the	O
right	O
,	O
and	O
all	O
that	O
lies	O
outside	O
that	O
unit	O
square	O
is	O
shifted	O
back	O
by	O
the	O
unit	O
until	O
it	O
is	O
within	O
the	O
square	O
.	O
</s>
<s>
Γ	O
is	O
invertible	O
because	O
the	O
matrix	B-Architecture
has	O
determinant	O
1	O
and	O
therefore	O
its	O
inverse	O
has	O
integer	O
entries	O
,	O
</s>
<s>
The	O
eigenspaces	O
are	O
orthogonal	O
because	O
the	O
matrix	B-Architecture
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
Arnold	B-Algorithm
's	I-Algorithm
cat	I-Algorithm
map	I-Algorithm
is	O
a	O
particularly	O
well-known	O
example	O
of	O
a	O
hyperbolic	O
toral	O
automorphism	O
,	O
which	O
is	O
an	O
automorphism	O
of	O
a	O
torus	O
given	O
by	O
a	O
square	O
unimodular	B-Algorithm
matrix	I-Algorithm
having	O
no	O
eigenvalues	O
of	O
absolute	O
value	O
1	O
.	O
</s>
<s>
The	O
number	O
of	O
points	O
with	O
period	O
is	O
exactly	O
(	O
where	O
and	O
are	O
the	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
)	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
define	O
a	O
discrete	O
analogue	O
of	O
the	O
cat	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
in	O
the	O
adjacent	O
picture	O
,	O
the	O
original	O
image	O
of	O
the	O
cat	O
is	O
sheared	B-Algorithm
and	O
then	O
wrapped	O
around	O
in	O
the	O
first	O
iteration	O
of	O
the	O
transformation	O
.	O
</s>
<s>
The	O
discrete	O
cat	B-Algorithm
map	I-Algorithm
describes	O
the	O
phase	O
space	O
flow	O
corresponding	O
to	O
the	O
discrete	O
dynamics	O
of	O
a	O
bead	O
hopping	O
from	O
site	O
qt	O
(	O
0	O
≤	O
qt	O
<	O
N	O
)	O
to	O
site	O
qt+1	O
on	O
a	O
circular	O
ring	O
with	O
circumference	O
N	O
,	O
according	O
to	O
the	O
second	O
order	O
equation	O
:	O
</s>
<s>
This	O
Arnold	O
cat	B-Algorithm
mapping	I-Algorithm
shows	O
mixing	O
behavior	O
typical	O
for	O
chaotic	O
systems	O
.	O
</s>
<s>
Such	O
an	O
integer	O
cat	B-Algorithm
map	I-Algorithm
is	O
commonly	O
used	O
to	O
demonstrate	O
mixing	O
behavior	O
with	O
Poincaré	O
recurrence	O
utilising	O
digital	O
images	O
.	O
</s>
