Let 
be a 
diffeomorphism on a compact Riemannian
manifold 
.
Then 
satisfies Axiom A if the nonwandering set 
of 
is hyperbolic and the periodic
points of 
are dense in 
. Although it was conjectured that the first of these
conditions implies the second, they were shown to be independent in or around 1977.
Examples include the Anosov diffeomorphisms
and Smale horseshoe map.
In some cases, Axiom A can be replaced by the condition that the diffeomorphism is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990).
