hyperbolic fixed point HyperbolicFixedPoint 2003-07-27 13:05:51 2003-07-29 12:17:59 Definition hyperbolic periodic point source sink saddle % 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 $M$ be a smooth manifold. A fixed point $x$ of a diffeomorphism $f\colon M\to M$ is said to be a \textbf{hyperbolic fixed point} if $Df(x)$ is a linear hyperbolic isomorphism. If $x$ is a periodic point of least period $n$, it is called a \textbf{hyperbolic periodic point} if it is a hyperbolic fixed point of $f^n$ (the $n$-th iterate of $f$). If the dimension of the stable manifold of a fixed point is zero, the point is called a \textbf{source}; if the dimension of its unstable manifold is zero, it is called a \textbf{sink}; and if both the stable and unstable manifold have nonzero dimension, it is called a \textbf{saddle}.