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# Axiom A

Let $M$ be a smooth manifold. We say that a diffeomorphism $f\colon M\to M$ satisfies
(Smale’s) *Axiom A* (or that $f$ is an Axiom A diffeomorphism) if

1. the nonwandering set $\Omega(f)$ has a hyperbolic structure;

2. the set of periodic points of $f$ is dense in $\Omega(f)$: $\overline{\operatorname{Per}(f)}=\Omega(f)$.

Sometimes, Axiom A diffeomorphisms are called hyperbolic diffeomorphisms, because the portion of $M$ where the “interesting” dynamics occur (namely, $\Omega(f)$) has a hyperbolic behaviour.

Synonym:

hyperbolic diffeomorphism

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

37D20*no label found*

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## Comments

## Suggestions

Why not add some observations to the definition as is done, for instance, in the page about Anosov diffeomorphisms? For instance, adding that all Anosov diffeomorphisms are Axiom A, (as well as Morse-Smale for instance, or Smale horseshoe’s); or that the conditions in the definition are not redundant, in the sense that the first condition doesn’t imply the second (it does in dimension 2);