recurrent point

Let X be a Hausdorff space and f:XX a function. A point xX is said to be recurrent (for f) if xω(x), i.e. if x belongs to its ω-limit ( set. This means that for each neighborhoodMathworldPlanetmathPlanetmath U of x there exists n>0 such that fn(x)U.

The closureMathworldPlanetmathPlanetmath 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 homeomorphismPlanetmathPlanetmath and X is compactPlanetmathPlanetmath, R(f) is an invariant subset of Ω(f), which may be a proper subsetMathworldPlanetmathPlanetmath.

Title recurrent point
Canonical name RecurrentPoint
Date of creation 2013-03-22 14:29:53
Last modified on 2013-03-22 14:29:53
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Definition
Classification msc 37B20
Related topic NonwanderingSet
Defines recurrent set