1. Nonwandering set is hyperbolic
2. Periodic points are dense in the nonwandering set
1. All periodic points are hyperbolic
2. For each pair of periodic points , of , the intersection between the stable manifold of $p$ and the unstable manifold of is transversal
The set of Kupka-Smale diffeomorphisms is residual in under topology.
1.Axiom A with only finitely many periodic points (hence is just the set of periodic points)
2.For each pair of periodic points , of , the intersection between the stable manifold of and the unstable manifold of is transversal.
All points are hyperbolic, i.e. there is a splitting of the whole tangent bundle such that under the diffeo, stable directions are exponentially contracted and unstable directions are exponentially expanded.
Morse-Smale Axiom A
Anosov Axiom A