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 ${\displaystyle X}$ be a Hausdorff space and ${\displaystyle f\colon X\to X}$ a function. A point ${\displaystyle x\in X}$ is said to be recurrent (for ${\displaystyle f}$) if ${\displaystyle x\in \omega (x)}$, i.e. if ${\displaystyle x}$ belongs to its ${\displaystyle \omega }$-limit set. This means that for each neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ there exists ${\displaystyle n>0}$ such that ${\displaystyle f^{n}(x)\in U}$.^[1]

The set of recurrent points of ${\displaystyle f}$ is often denoted ${\displaystyle R(f)}$ and is called the recurrent set of ${\displaystyle f}$. Its closure is called the Birkhoff center of ${\displaystyle f}$,^[2] and appears in the work of George David Birkhoff on dynamical systems.^[3][4]

Every recurrent point is a nonwandering point,^[1] hence if ${\displaystyle f}$ is a homeomorphism and ${\displaystyle X}$ is compact, then ${\displaystyle R(f)}$ is an invariant subset of the non-wandering set of ${\displaystyle f}$ (and may be a proper subset).