AxiomA 2003-06-11 16:39:04 2004-01-12 11:21:06 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} \PMlinkescapeword{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 \begin{enumerate} \item the nonwandering set $\Omega(f)$ has a hyperbolic structure; \item the set of periodic points of $f$ is dense in $\Omega(f)$: $\overline{\Per(f)} = \Omega(f)$. \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.