<s>
In	O
mathematics	O
,	O
specifically	O
in	O
the	O
study	O
of	O
dynamical	O
systems	O
,	O
an	O
orbit	B-Algorithm
is	O
a	O
collection	O
of	O
points	O
related	O
by	O
the	O
evolution	O
function	O
of	O
the	O
dynamical	O
system	O
.	O
</s>
<s>
It	O
can	O
be	O
understood	O
as	O
the	O
subset	O
of	O
phase	O
space	O
covered	O
by	O
the	O
trajectory	O
of	O
the	O
dynamical	O
system	O
under	O
a	O
particular	O
set	O
of	O
initial	B-Algorithm
conditions	I-Algorithm
,	O
as	O
the	O
system	O
evolves	O
.	O
</s>
<s>
As	O
a	O
phase	O
space	O
trajectory	O
is	O
uniquely	O
determined	O
for	O
any	O
given	O
set	O
of	O
phase	O
space	O
coordinates	O
,	O
it	O
is	O
not	O
possible	O
for	O
different	O
orbits	B-Algorithm
to	O
intersect	O
in	O
phase	O
space	O
,	O
therefore	O
the	O
set	O
of	O
all	O
orbits	B-Algorithm
of	O
a	O
dynamical	O
system	O
is	O
a	O
partition	O
of	O
the	O
phase	O
space	O
.	O
</s>
<s>
Understanding	O
the	O
properties	O
of	O
orbits	B-Algorithm
by	O
using	O
topological	O
methods	O
is	O
one	O
of	O
the	O
objectives	O
of	O
the	O
modern	O
theory	O
of	O
dynamical	O
systems	O
.	O
</s>
<s>
For	O
discrete-time	O
dynamical	O
systems	O
,	O
the	O
orbits	B-Algorithm
are	O
sequences	O
;	O
for	O
real	O
dynamical	O
systems	O
,	O
the	O
orbits	B-Algorithm
are	O
curves	O
;	O
and	O
for	O
holomorphic	O
dynamical	O
systems	O
,	O
the	O
orbits	B-Algorithm
are	O
Riemann	O
surfaces	O
.	O
</s>
<s>
is	O
called	O
orbit	B-Algorithm
through	O
x	O
.	O
</s>
<s>
An	O
orbit	B-Algorithm
which	O
consists	O
of	O
a	O
single	O
point	O
is	O
called	O
constant	O
orbit	B-Algorithm
.	O
</s>
<s>
is	O
called	O
negative	O
semi-orbit	O
through	O
x	O
.	O
</s>
<s>
forward	O
orbit	B-Algorithm
of	O
x	O
is	O
a	O
set	O
:	O
</s>
<s>
backward	O
orbit	B-Algorithm
of	O
x	O
is	O
a	O
set	O
:	O
</s>
<s>
and	O
orbit	B-Algorithm
of	O
x	O
is	O
a	O
set	O
:	O
</s>
<s>
is	O
an	O
evolution	O
function	O
which	O
is	O
here	O
an	O
iterated	B-Algorithm
function	I-Algorithm
,	O
</s>
<s>
For	O
a	O
general	O
dynamical	O
system	O
,	O
especially	O
in	O
homogeneous	O
dynamics	O
,	O
when	O
one	O
has	O
a	O
"	O
nice	O
"	O
group	O
acting	O
on	O
a	O
probability	O
space	O
in	O
a	O
measure-preserving	O
way	O
,	O
an	O
orbit	B-Algorithm
will	O
be	O
called	O
periodic	O
(	O
or	O
equivalently	O
,	O
closed	O
)	O
if	O
the	O
stabilizer	O
is	O
a	O
lattice	O
inside	O
.	O
</s>
<s>
In	O
addition	O
,	O
a	O
related	O
term	O
is	O
a	O
bounded	O
orbit	B-Algorithm
,	O
when	O
the	O
set	O
is	O
pre-compact	O
inside	O
.	O
</s>
<s>
The	O
classification	O
of	O
orbits	B-Algorithm
can	O
lead	O
to	O
interesting	O
questions	O
with	O
relations	O
to	O
other	O
mathematical	O
areas	O
,	O
for	O
example	O
the	O
Oppenheim	O
conjecture	O
(	O
proved	O
by	O
Margulis	O
)	O
and	O
the	O
Littlewood	O
conjecture	O
(	O
partially	O
proved	O
by	O
Lindenstrauss	O
)	O
are	O
dealing	O
with	O
the	O
question	O
whether	O
every	O
bounded	O
orbit	B-Algorithm
of	O
some	O
natural	O
action	O
on	O
the	O
homogeneous	O
space	O
is	O
indeed	O
periodic	O
one	O
,	O
this	O
observation	O
is	O
due	O
to	O
Raghunathan	O
and	O
in	O
different	O
language	O
due	O
to	O
Cassels	O
and	O
Swinnerton-Dyer	O
.	O
</s>
