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.
Let be a Hausdorff space and a function. A point is said to be recurrent (for ) if , i.e. if belongs to its - limit set. This means that for each neighborhood of there exists such that . [1]
The set of recurrent points of is often denoted and is called the recurrent set of . Its closure is called the Birkhoff center of , [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 is a homeomorphism and is compact, then is an invariant subset of the non-wandering set of (and may be a proper subset).
This article incorporates material from Recurrent point on
PlanetMath, which is licensed under the
Creative Commons Attribution/Share-Alike License.
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.
Let be a Hausdorff space and a function. A point is said to be recurrent (for ) if , i.e. if belongs to its - limit set. This means that for each neighborhood of there exists such that . [1]
The set of recurrent points of is often denoted and is called the recurrent set of . Its closure is called the Birkhoff center of , [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 is a homeomorphism and is compact, then is an invariant subset of the non-wandering set of (and may be a proper subset).
This article incorporates material from Recurrent point on
PlanetMath, which is licensed under the
Creative Commons Attribution/Share-Alike License.