nonwandering set NonwanderingSet 2003-05-29 15:19:12 2003-05-29 15:19:12 Definition wandering point nonwandering point % 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}} Let $X$ be a metric space, and $f:X\rightarrow X$ a continuous surjection. An element $x$ of $X$ is a \emph{wandering point} if there is a neighborhood $U$ of $x$ and an integer $N$ such that, for all $n\geq N$, $f^n(U)\cap U=\emptyset$. If $x$ is not wandering, we call it a \emph{nonwandering point}. Equivalently, $x$ is a nonwandering point if for every neighborhood $U$ of $x$ there is $n\geq 1$ such that $f^n(U)\cap U$ is nonempty. The set of all nonwandering points is called the \emph{nonwandering set} of $f$, and is denoted by $\Omega(f)$. If $X$ is compact, then $\Omega(f)$ is compact, nonempty, and forward invariant; if, additionally, $f$ is an homeomorphism, then $\Omega(f)$ is invariant.