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. ↩︎


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