In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
Let
X
be a Hausdorff space and
f
:
X
→
X
a function. A point
x
∈
X
is said to be recurrent (for
f
) if
x
∈
ω
(
x
)
, i.e. if
x
belongs to its
ω
-limit set. This means that for each neighborhood
U
of
x
there exists
n
>
0
such that
f
n
(
x
)
∈
U
.
The set of recurrent points of
f
is often denoted
R
(
f
)
and is called the recurrent set of
f
. Its closure is called the Birkhoff center of
f
, and appears in the work of George David Birkhoff on dynamical systems.
Every recurrent point is a nonwandering point, hence if
f
is a homeomorphism and
X
is compact, then
R
(
f
)
is an invariant subset of the non-wandering set of
f
(and may be a proper subset).
