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单词 NonwanderingSet
释义

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 Uof 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.

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更新时间:2025/5/4 3:14:44