Recurrent point

From Wikipedia, the free encyclopedia

You have new messages (last change).

In mathematics, a recurrent point for function f is a point that is in the limit set of the iterated function f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.

[edit] 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 ω-limit set. This means that for each neighborhood U of x there exists n > 0 such that f^n(x)\in U.

The closure of the set of recurrent points of f is often denoted R(f) and is called the recurrent set of f.

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).

This article incorporates material from Recurrent point on PlanetMath, which is licensed under the GFDL.

Retrieved from "http://en.wikipedia.org../../../r/e/c/Recurrent_point.html"