# Axiom A

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

1. 1.

the nonwandering set $\Omega(f)$ has a hyperbolic structure;

2. 2.

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

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

Title Axiom A AxiomA 2013-03-22 13:40:27 2013-03-22 13:40:27 Koro (127) Koro (127) 7 Koro (127) Definition msc 37D20 hyperbolic diffeomorphism