nonwandering set

Let X be a metric space, and f:XX a continuousPlanetmathPlanetmath surjection. An element x of X is a wandering point if there is a neighborhoodMathworldPlanetmathPlanetmath U of x and an integer N such that, for all nN, fn(U)U=. If x is not wandering, we call it a nonwandering point. Equivalently, x is a nonwandering point if for every neighborhood U of x there is n1 such that fn(U)U is nonempty. The set of all nonwandering points is called the nonwandering set of f, and is denoted by Ω(f).

If X is compactPlanetmathPlanetmath, then Ω(f) is compact, nonempty, and forward invariant; if, additionally, f is an homeomorphismPlanetmathPlanetmath, then Ω(f) is invariant.

Title nonwandering set
Canonical name NonwanderingSet
Date of creation 2013-03-22 13:39:31
Last modified on 2013-03-22 13:39:31
Owner Koro (127)
Last modified by Koro (127)
Numerical id 4
Author Koro (127)
Entry type Definition
Classification msc 37B20
Related topic OmegaLimitSet3
Related topic RecurrentPoint
Defines wandering point
Defines nonwandering point