<s>
In	O
the	O
mathematics	O
of	O
chaos	O
theory	O
,	O
a	O
horseshoe	B-Algorithm
map	I-Algorithm
is	O
any	O
member	O
of	O
a	O
class	O
of	O
chaotic	O
maps	O
of	O
the	O
square	O
into	O
itself	O
.	O
</s>
<s>
The	O
map	O
was	O
introduced	O
by	O
Stephen	O
Smale	O
while	O
studying	O
the	O
behavior	O
of	O
the	O
orbits	B-Algorithm
of	O
the	O
van	O
der	O
Pol	O
oscillator	O
.	O
</s>
<s>
The	O
points	O
that	O
remain	O
in	O
the	O
square	O
under	O
repeated	O
iteration	O
form	O
a	O
fractal	B-Application
set	O
and	O
are	O
part	O
of	O
the	O
invariant	O
set	O
of	O
the	O
map	O
.	O
</s>
<s>
The	O
squishing	O
,	O
stretching	O
and	O
folding	O
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
are	O
typical	O
of	O
chaotic	O
systems	O
,	O
but	O
not	O
necessary	O
or	O
even	O
sufficient	O
.	O
</s>
<s>
In	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
,	O
the	O
squeezing	O
and	O
stretching	O
are	O
uniform	O
.	O
</s>
<s>
The	O
folding	O
is	O
done	O
neatly	O
,	O
so	O
that	O
the	O
orbits	B-Algorithm
that	O
remain	O
forever	O
in	O
the	O
square	O
can	O
be	O
simply	O
described	O
.	O
</s>
<s>
For	O
a	O
horseshoe	B-Algorithm
map	I-Algorithm
:	O
</s>
<s>
there	O
are	O
an	O
infinite	O
number	O
of	O
periodic	B-Algorithm
orbits	I-Algorithm
;	O
</s>
<s>
periodic	B-Algorithm
orbits	I-Algorithm
of	O
arbitrarily	O
long	O
period	O
exist	O
;	O
</s>
<s>
close	O
to	O
any	O
point	O
of	O
the	O
fractal	B-Application
invariant	O
set	O
there	O
is	O
a	O
point	O
of	O
a	O
periodic	O
orbit	B-Algorithm
.	O
</s>
<s>
The	O
horseshoe	B-Algorithm
map	I-Algorithm
is	O
a	O
diffeomorphism	O
defined	O
from	O
a	O
region	O
of	O
the	O
plane	O
into	O
itself	O
.	O
</s>
<s>
To	O
keep	O
the	O
class	O
of	O
horseshoe	B-Algorithm
maps	I-Algorithm
simple	O
,	O
the	O
curved	O
region	O
of	O
the	O
horseshoe	O
should	O
not	O
map	O
back	O
into	O
the	O
square	O
.	O
</s>
<s>
The	O
horseshoe	B-Algorithm
map	I-Algorithm
is	O
one-to-one	O
,	O
which	O
means	O
that	O
an	O
inverse	O
f−1	O
exists	O
when	O
restricted	O
to	O
the	O
image	O
of	O
under	O
.	O
</s>
<s>
By	O
folding	O
the	O
contracted	O
and	O
stretched	O
square	O
in	O
different	O
ways	O
,	O
other	O
types	O
of	O
horseshoe	B-Algorithm
maps	I-Algorithm
are	O
possible	O
.	O
</s>
<s>
The	O
horseshoe	B-Algorithm
map	I-Algorithm
is	O
an	O
Axiom	O
A	O
diffeomorphism	O
that	O
serves	O
as	O
a	O
model	O
for	O
the	O
general	O
behavior	O
at	O
a	O
transverse	O
homoclinic	O
point	O
,	O
where	O
the	O
stable	O
and	O
unstable	O
manifolds	O
of	O
a	O
periodic	O
point	O
intersect	O
.	O
</s>
<s>
The	O
horseshoe	B-Algorithm
map	I-Algorithm
was	O
designed	O
to	O
reproduce	O
the	O
chaotic	O
dynamics	O
of	O
a	O
flow	O
in	O
the	O
neighborhood	O
of	O
a	O
given	O
periodic	O
orbit	B-Algorithm
.	O
</s>
<s>
The	O
neighborhood	O
is	O
chosen	O
to	O
be	O
a	O
small	O
disk	O
perpendicular	O
to	O
the	O
orbit	B-Algorithm
.	O
</s>
<s>
As	O
the	O
system	O
evolves	O
,	O
points	O
in	O
this	O
disk	O
remain	O
close	O
to	O
the	O
given	O
periodic	O
orbit	B-Algorithm
,	O
tracing	O
out	O
orbits	B-Algorithm
that	O
eventually	O
intersect	O
the	O
disk	O
once	O
again	O
.	O
</s>
<s>
Other	O
orbits	B-Algorithm
diverge	O
.	O
</s>
<s>
The	O
behavior	O
of	O
all	O
the	O
orbits	B-Algorithm
in	O
the	O
disk	O
can	O
be	O
determined	O
by	O
considering	O
what	O
happens	O
to	O
the	O
disk	O
.	O
</s>
<s>
The	O
intersection	O
of	O
the	O
disk	O
with	O
the	O
given	O
periodic	O
orbit	B-Algorithm
comes	O
back	O
to	O
itself	O
every	O
period	O
of	O
the	O
orbit	B-Algorithm
and	O
so	O
do	O
points	O
in	O
its	O
neighborhood	O
.	O
</s>
<s>
The	O
set	O
of	O
points	O
that	O
never	O
leaves	O
the	O
neighborhood	O
of	O
the	O
given	O
periodic	O
orbit	B-Algorithm
form	O
a	O
fractal	B-Application
.	O
</s>
<s>
A	O
symbolic	O
name	O
can	O
be	O
given	O
to	O
all	O
the	O
orbits	B-Algorithm
that	O
remain	O
in	O
the	O
neighborhood	O
.	O
</s>
<s>
Knowing	O
the	O
sequence	O
in	O
which	O
the	O
orbit	B-Algorithm
visits	O
these	O
regions	O
allows	O
the	O
orbit	B-Algorithm
to	O
be	O
pinpointed	O
exactly	O
.	O
</s>
<s>
The	O
visitation	O
sequence	O
of	O
the	O
orbits	B-Algorithm
provide	O
a	O
symbolic	O
representation	O
of	O
the	O
dynamics	O
,	O
known	O
as	O
symbolic	O
dynamics	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
describe	O
the	O
behavior	O
of	O
all	O
initial	O
conditions	O
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Its	O
iterate	O
is	O
the	O
point	O
u2	O
=	O
f(u1 )	O
=	O
f	O
2(u0 )	O
,	O
and	O
repeated	O
iteration	O
generates	O
the	O
orbit	B-Algorithm
u0	O
,	O
u1	O
,	O
u2	O
,	O
...	O
</s>
<s>
Under	O
repeated	O
iteration	O
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
,	O
most	O
orbits	B-Algorithm
end	O
up	O
at	O
the	O
fixed	O
point	O
in	O
the	O
left	O
cap	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
horseshoe	B-Algorithm
maps	I-Algorithm
the	O
left	O
cap	O
into	O
itself	O
by	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
that	O
has	O
exactly	O
one	O
fixed	O
point	O
.	O
</s>
<s>
Any	O
orbit	B-Algorithm
that	O
lands	O
on	O
the	O
left	O
cap	O
never	O
leaves	O
it	O
and	O
converges	O
to	O
the	O
fixed	O
point	O
in	O
the	O
left	O
cap	O
under	O
iteration	O
.	O
</s>
<s>
Under	O
iteration	O
,	O
most	O
points	O
will	O
be	O
part	O
of	O
orbits	B-Algorithm
that	O
converge	O
to	O
the	O
fixed	O
point	O
in	O
the	O
left	O
cap	O
,	O
but	O
some	O
points	O
of	O
the	O
square	O
never	O
leave	O
.	O
</s>
<s>
Under	O
forward	O
iterations	O
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
,	O
the	O
original	O
square	O
gets	O
mapped	O
into	O
a	O
series	O
of	O
horizontal	O
strips	O
.	O
</s>
<s>
A	O
dot	O
is	O
used	O
to	O
separate	O
the	O
region	O
the	O
point	O
of	O
an	O
orbit	B-Algorithm
is	O
in	O
from	O
the	O
region	O
the	O
point	O
came	O
from	O
.	O
</s>
<s>
The	O
notation	O
can	O
be	O
extended	O
to	O
higher	O
iterates	O
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Any	O
one	O
of	O
the	O
intersections	O
of	O
a	O
horizontal	O
strip	O
with	O
a	O
vertical	O
strip	O
,	O
where	O
P	O
and	O
F	O
are	O
sequences	O
of	O
As	O
and	O
Bs	O
,	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
of	O
a	O
small	O
region	O
in	O
V1	O
.	O
</s>
<s>
It	O
converges	O
to	O
a	O
point	O
that	O
is	O
part	O
of	O
a	O
periodic	O
orbit	B-Algorithm
of	O
the	O
horseshoe	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
periodic	O
orbit	B-Algorithm
can	O
be	O
labeled	O
by	O
the	O
simplest	O
sequence	O
of	O
s	O
and	O
s	O
that	O
labels	O
one	O
of	O
the	O
regions	O
the	O
periodic	O
orbit	B-Algorithm
visits	O
.	O
</s>
<s>
For	O
every	O
sequence	O
of	O
s	O
and	O
s	O
there	O
is	O
a	O
periodic	O
orbit	B-Algorithm
.	O
</s>
