Axiom A - Definition

Definition

Let M be a smooth manifold with a diffeomorphism f: M→M. Then f is an axiom A diffeomorphism if the following two conditions hold:

  1. The nonwandering set of f, Ω(f), is a hyperbolic set and compact.
  2. The set of periodic points of f is dense in Ω(f).

For surfaces, hyperbolicity of the nonwandering set implies the density of periodic points, but this is no longer true in higher dimensions. Nonetheless, axiom A diffeomorphisms are sometimes called hyperbolic diffeomorphisms, because the portion of M where the interesting dynamics occurs, namely, Ω(f), exhibits hyperbolic behavior.

Axiom A diffeomorphisms generalize Morse–Smale systems, which satisfy further restrictions (finitely many periodic points and transversality of stable and unstable submanifolds). Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.

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