Recurrent point

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. 
Definition.
Let formula_1 be a Hausdorff space and formula_2 a function. A point formula_3 is said to be recurrent (for formula_4) if formula_5, "i.e." if formula_6 belongs to its formula_7-limit set. This means that for each neighborhood formula_8 of formula_6 there exists formula_10 such that formula_11.
The set of recurrent points of formula_4 is often denoted formula_13 and is called the recurrent set of formula_4. Its closure is called the Birkhoff center of formula_4, and appears in the work of George David Birkhoff on dynamical systems.
Every recurrent point is a nonwandering point, hence if formula_4 is a homeomorphism and formula_1 is compact, then formula_13 is an invariant subset of the non-wandering set of formula_4 (and may be a proper subset).