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 [math]\displaystyle{ X }[/math] be a Hausdorff space and [math]\displaystyle{ f\colon X\to X }[/math] a function. A point [math]\displaystyle{ x\in X }[/math] is said to be recurrent (for [math]\displaystyle{ f }[/math]) if [math]\displaystyle{ x\in \omega(x) }[/math], i.e. if [math]\displaystyle{ x }[/math] belongs to its [math]\displaystyle{ \omega }[/math]-limit set. This means that for each neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] there exists [math]\displaystyle{ n\gt 0 }[/math] such that [math]\displaystyle{ f^n(x)\in U }[/math].

The set of recurrent points of [math]\displaystyle{ f }[/math] is often denoted [math]\displaystyle{ R(f) }[/math] and is called the recurrent set of [math]\displaystyle{ f }[/math]. Its closure is called the Birkhoff center of [math]\displaystyle{ f }[/math], and appears in the work of George David Birkhoff on dynamical systems.<ref>{{citation

| last1 = Coven | first1 = Ethan M.
| doi = 10.1090/S0002-9939-1980-0565362-0 | doi-access=free
| jstor=2043258
| issue = 2
| journal = Proceedings of the American Mathematical Society
| mr = 565362
| pages = 316–318
| title = [math]\displaystyle{ \bar P=\bar R }[/math] for maps of the interval
| volume = 79

Every recurrent point is a nonwandering point, hence if [math]\displaystyle{ f }[/math] is a homeomorphism and [math]\displaystyle{ X }[/math] is compact, then [math]\displaystyle{ R(f) }[/math] is an invariant subset of the non-wandering set of [math]\displaystyle{ f }[/math] (and may be a proper subset).