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:

1. The set $\mathsf{fix}(f)$ of fixed points.
2. The set $\mathsf{per}(f)$ of periodic points.
3. The set $\mathsf{rec}(f)$ of recurrent points.
4. The set $\mathsf{nw}(f)$ of non-wandering points.
5. 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:

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

1. 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. ↩︎