# Types of hyperbolicity

Axiom A

1. Nonwandering set is hyperbolic

2. Periodic points are dense in the nonwandering set

Kupka-Smale

1. All periodic points are hyperbolic

2. For each pair of periodic points $p$, $q$ of $f$, the intersection between the stable manifold of \$p\$ and the unstable manifold of $q$ is transversal

Kupka-Smale theorem

The set of Kupka-Smale diffeomorphisms is residual in $\mbox{Diff}^r(M)$ under $C^r$ topology.

Morse-Smale

1.Axiom A with only finitely many periodic points (hence $\Omega(f)$ is just the set of periodic points)

2.For each pair of periodic points $p$, $q$ of $f$, the intersection between the stable manifold of $p$ and the unstable manifold of $q$ is transversal.

Anosov

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.

Relations:

Morse-Smale $\subseteq$ Axiom A

Morse-Smale $\subseteq$ Kupka-Smale

Anosov $\subseteq$ Axiom A