# nonwandering set

Let $X$ be a metric space, and $f:X\rightarrow X$ 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 $n\geq N$, $f^{n}(U)\cap U=\emptyset$. 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 $n\geq 1$ 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 $\Omega(f)$.

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

Title nonwandering set NonwanderingSet 2013-03-22 13:39:31 2013-03-22 13:39:31 Koro (127) Koro (127) 4 Koro (127) Definition msc 37B20 OmegaLimitSet3 RecurrentPoint wandering point nonwandering point