Types of hyperbolicity

January 12, 2010

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

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