nonwandering set

Let $X$ be a metric space, and $f:X\to X$ a continuous surjection. An element $x$ of $X$ is a wandering point if there is a neighborhood $U$ of $x$ and an integer $N$ such that, for all $n\ge N$, $f^n(U)\cap U=\mathrm{\varnothing }$. 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 $n\ge 1$ such that $f^n(U)\cap U$ is nonempty. The set of all nonwandering points is called the nonwandering set of $f$, and is denoted by $\mathrm{\Omega }(f)$.

If $X$ is compact, then $\mathrm{\Omega }(f)$ is compact, nonempty, and forward invariant; if, additionally, $f$ is an homeomorphism, then $\mathrm{\Omega }(f)$ 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