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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
- the nonwandering set $\Omega(f)$ has a hyperbolic structure;
- the set of periodic points of $f$ is dense in $\Omega(f)$ $\overline{\Per(f)} = \Omega(f)$
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.
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"Axiom A" is owned by Koro.
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| Other names: |
hyperbolic diffeomorphism |
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Cross-references: dense in, periodic points, hyperbolic structure, nonwandering set, diffeomorphism, smooth manifold
There are 7 references to this entry.
This is version 4 of Axiom A, born on 2003-06-11, modified 2004-01-12.
Object id is 4340, canonical name is AxiomA.
Accessed 3566 times total.
Classification:
| AMS MSC: | 37D20 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Uniformly hyperbolic systems ) |
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Pending Errata and Addenda
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