**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 , of , the intersection between the stable manifold of $p$ and the unstable manifold of is transversal

**Kupka-Smale theorem**

The set of Kupka-Smale diffeomorphisms is residual in under topology.

**Morse-Smale**

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.

**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 Axiom A

Morse-Smale Kupka-Smale

Anosov Axiom A

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This entry was posted on January 12, 2010 at 10:38 pm and is filed under Uncategorized.

Tags: Axiom A, generic, hyperbolicity, Smale

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