Recurrent point explained

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

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

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 ,^[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

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).

Notes and References

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