LocallyMaximal 2004-01-20 11:38:43 2004-01-20 11:41:31 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}} Let $f:M\to M$ be a diffeomorphism of a smooth manifold $M$, and let $\Lambda$ be a hyperbolic set for $f$. We say that $\Lambda$ is \emph{locally maximal} or \emph{basic} if there exists a neighborhood $U$ of $\Lambda$ such that $$\Lambda=\bigcap_{n=-\infty}^\infty f^n(\overline{U}).$$