<s>
the	O
name	O
being	O
due	O
to	O
the	O
tent-like	O
shape	O
of	O
the	O
graph	B-Application
of	O
fμ	O
.	O
</s>
<s>
In	O
particular	O
,	O
iterating	B-Algorithm
a	O
point	O
x0	O
in	O
 [ 0 , 1 ] 	O
gives	O
rise	O
to	O
a	O
sequence	O
:	O
</s>
<s>
Iterating	B-Algorithm
the	O
procedure	O
,	O
any	O
point	O
x0	O
of	O
the	O
interval	O
assumes	O
new	O
subsequent	O
positions	O
as	O
described	O
above	O
,	O
generating	O
a	O
sequence	O
xn	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
case	O
of	O
the	O
tent	B-Algorithm
map	I-Algorithm
is	O
a	O
non-linear	O
transformation	O
of	O
both	O
the	O
bit	B-Algorithm
shift	I-Algorithm
map	I-Algorithm
and	O
the	O
r	O
=	O
4	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
tent	B-Algorithm
map	I-Algorithm
with	O
parameter	O
μ	O
=	O
2	O
and	O
the	O
logistic	B-Algorithm
map	I-Algorithm
with	O
parameter	O
r	O
=	O
4	O
are	O
topologically	O
conjugate	O
,	O
and	O
thus	O
the	O
behaviours	O
of	O
the	O
two	O
maps	O
are	O
in	O
this	O
sense	O
identical	O
under	O
iteration	O
.	O
</s>
<s>
Depending	O
on	O
the	O
value	O
of	O
μ	O
,	O
the	O
tent	B-Algorithm
map	I-Algorithm
demonstrates	O
a	O
range	O
of	O
dynamical	O
behaviour	O
ranging	O
from	O
predictable	O
to	O
chaotic	O
.	O
</s>
<s>
This	O
set	O
of	O
intervals	O
is	O
the	O
Julia	B-Language
set	I-Language
of	O
the	O
map	O
–	O
that	O
is	O
,	O
it	O
is	O
the	O
smallest	O
invariant	O
subset	O
of	O
the	O
real	O
line	O
under	O
this	O
map	O
.	O
</s>
<s>
If	O
μ	O
is	O
greater	O
than	O
the	O
square	O
root	O
of	O
2	O
,	O
these	O
intervals	O
merge	O
,	O
and	O
the	O
Julia	B-Language
set	I-Language
is	O
the	O
whole	O
interval	O
from	O
μ−	O
μ2/2	O
to	O
μ/2	O
(	O
see	O
bifurcation	O
diagram	O
)	O
.	O
</s>
<s>
If	O
μ	O
is	O
between	O
1	O
and	O
2	O
the	O
interval	O
 [ μ−μ2/2 , μ/2 ] 	O
contains	O
both	O
periodic	O
and	O
non-periodic	O
points	O
,	O
although	O
all	O
of	O
the	O
orbits	B-Algorithm
are	O
unstable	O
(	O
i.e.	O
</s>
<s>
nearby	O
points	O
move	O
away	O
from	O
the	O
orbits	B-Algorithm
rather	O
than	O
towards	O
them	O
)	O
.	O
</s>
<s>
Orbits	B-Algorithm
with	O
longer	O
lengths	O
appear	O
as	O
μ	O
increases	O
.	O
</s>
<s>
There	O
are	O
now	O
periodic	O
points	O
with	O
every	O
orbit	B-Algorithm
length	O
within	O
this	O
interval	O
,	O
as	O
well	O
as	O
non-periodic	O
points	O
.	O
</s>
<s>
If	O
μ	O
is	O
greater	O
than	O
2	O
the	O
map	O
's	O
Julia	B-Language
set	I-Language
becomes	O
disconnected	O
,	O
and	O
breaks	O
up	O
into	O
a	O
Cantor	O
set	O
within	O
the	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
Julia	B-Language
set	I-Language
still	O
contains	O
an	O
infinite	O
number	O
of	O
both	O
non-periodic	O
and	O
periodic	O
points	O
(	O
including	O
orbits	B-Algorithm
for	O
any	O
orbit	B-Algorithm
length	O
)	O
but	O
almost	O
every	O
point	O
within	O
 [ 0 , 1 ] 	O
will	O
now	O
eventually	O
diverge	O
towards	O
infinity	O
.	O
</s>
<s>
The	O
canonical	O
Cantor	O
set	O
(	O
obtained	O
by	O
successively	O
deleting	O
middle	O
thirds	O
from	O
subsets	O
of	O
the	O
unit	O
line	O
)	O
is	O
the	O
Julia	B-Language
set	I-Language
of	O
the	O
tent	B-Algorithm
map	I-Algorithm
for	O
μ	O
=3	O
.	O
</s>
<s>
A	O
closer	O
look	O
at	O
the	O
orbit	B-Algorithm
diagram	O
shows	O
that	O
there	O
are	O
4	O
separated	O
regions	O
at	O
μ	O
≈	O
1	O
.	O
</s>
<s>
The	O
asymmetric	O
tent	B-Algorithm
map	I-Algorithm
is	O
essentially	O
a	O
distorted	O
,	O
but	O
still	O
piecewise	O
linear	O
,	O
version	O
of	O
the	O
case	O
of	O
the	O
tent	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
case	O
of	O
the	O
tent	B-Algorithm
map	I-Algorithm
is	O
the	O
present	O
case	O
of	O
.	O
</s>
<s>
A	O
sequence	O
 {  } 	O
will	O
have	O
the	O
same	O
autocorrelation	O
function	O
as	O
will	O
data	O
from	O
the	O
first-order	O
autoregressive	B-Algorithm
process	I-Algorithm
with	O
 {  } 	O
independently	O
and	O
identically	O
distributed	O
.	O
</s>
<s>
Thus	O
data	O
from	O
an	O
asymmetric	O
tent	B-Algorithm
map	I-Algorithm
cannot	O
be	O
distinguished	O
,	O
using	O
the	O
autocorrelation	O
function	O
,	O
from	O
data	O
generated	O
by	O
a	O
first-order	O
autoregressive	B-Algorithm
process	I-Algorithm
.	O
</s>
