# 3. The Non-Wandering Set and Its Friends

Let $(X,d)$ be a metric space. In this lecture we define an entire menagerie of invariant sets associated to a dynamical system $f \colon X \to X$. We have already met the first two:

- The set $\mathsf{fix}(f)$ of
*fixed*points. - The set $\mathsf{per}(f)$ of
*periodic*points. - The set $\mathsf{rec}(f)$ of
*recurrent*points. - The set $ \mathsf{nw}(f)$ of
*non-wandering*points. - The set $ \mathsf{cha}_d(f)$ of
*chain-recurrent*points^{[1]}.

The set $ \mathsf{rec}(f)$ is not necessarily closed, but all the others are. Moreover they are nested inside one another:

$$

\mathsf{fix}(f) \subseteq \mathsf{per}(f) \subseteq \mathsf{rec}(f) \subseteq \overline{ \mathsf{rec}(f)} \subseteq \mathsf{nw}(f) \subseteq \mathsf{cha}_d(f).

$$

The chain recurrent set depends on the metric, hence we add it to our notation. If the metric space is compact however then it does not depend on the metric. ↩︎

Comments and questions?