A dynamical system $f \colon M \to M$ of a compact manifold $M$ is said to satisfy Axiom A if:

• the non-wandering set $\mathsf{nw}(f)$ is hyperbolic,
• the non-wandering set is the closure of the periodic points.

This is not such a great name, since (i) it is not really an Axiom (ii) there is no Axiom B (for us) and (iii) it is completely non-descriptive. Oh well. ¯\_(ツ)_/¯

In this lecture we show that if the chain recurrent set is hyperbolic, then $f$ satisfies Axiom A. The importance of this statement (together with a partial converse) will become clear by the end of the course.

Along the way we use the Shadowing Theorem from the last lecture to prove two more foundational results in hyperbolic dynamics: the Anosov Closing Lemma and the In Phase Theorem.