Axiom A

Let $M$ be a smooth manifold. We say that a diffeomorphism $f:M\to M$ satisfies(Smale’s) Axiom A (or that $f$ is an Axiom A diffeomorphism) if

1. the nonwandering set $\mathrm{\Omega }(f)$ has a hyperbolic structure;

2. the set of periodic points of $f$ is dense in $\mathrm{\Omega }(f)$: $\overline{\mathrm{Per}(f)}=\mathrm{\Omega }(f)$.

Sometimes, Axiom A diffeomorphisms are called hyperbolic diffeomorphisms, because the portion of $M$ where the “interesting” dynamics occur (namely, $\mathrm{\Omega }(f)$) has a hyperbolic behaviour.