| Version current |
Version 3 |
| \PMlinkescapeword{satisfies} |
\PMlinkescapeword{satisfies} |
| Let $M$ be a smooth manifold. We say that a diffeomorphism $f\colon M\to M$ satisfies |
Let $M$ be a smooth manifold. We say that a diffeomorphism $f\colon M\to M$ satisfies |
| (Smale's) \emph{Axiom A} (or that $f$ is an Axiom A diffeomorphism) if |
(Smale's) \emph{Axiom A} (or that $f$ is an Axiom A diffeomorphism) if |
| \begin{enumerate} |
\begin{enumerate} |
| \item the nonwandering set $\Omega(f)$ has a hyperbolic structure; |
\item the nonwandering set $\Omega(f)$ has a hyperbolic structure; |
| \item the set of periodic points of $f$ is dense in $\Omega(f)$: $\overline{\Per(f)} = \Omega(f)$. |
\item the set of periodic points of $f$ is dense in $\Omega(f)$: $\overline{\Per(f)} = \Omega(f)$. |
| \end{enumerate} |
\end{enumerate} |
| 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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