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.
Title  Axiom A 

Canonical name  AxiomA 
Date of creation  20130322 13:40:27 
Last modified on  20130322 13:40:27 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  7 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 37D20 
Synonym  hyperbolic diffeomorphism 