Then f is an axiom A diffeomorphism if the following two conditions hold: For surfaces, hyperbolicity of the nonwandering set implies the density of periodic points, but this is no longer true in higher dimensions.
Axiom A diffeomorphisms generalize Morse–Smale systems, which satisfy further restrictions (finitely many periodic points and transversality of stable and unstable submanifolds).
Rufus Bowen showed that the non-wandering set Ω(f) of any axiom A diffeomorphism supports a Markov partition.
The density of the periodic points in the non-wandering set implies its local maximality: there exists an open neighborhood U of Ω(f) such that An important property of Axiom A systems is their structural stability against small perturbations.
A diffeomorphism f is omega stable if and only if it satisfies axiom A and the no-cycle condition (that an orbit, once having left an invariant subset, does not return).