MetricEntropy 2004-08-05 06:47:53 2007-07-03 06:40:45 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 $(X,\mathscr{B},\mu)$ be a probability space, and $T\colon X\to X$ a measure-preserving transformation. The entropy of $T$ with respect to a finite measurable partition $\mathcal{P}$ is \[h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right),\] where $H_\mu$ is the entropy of a partition and $\vee$ denotes the join of partitions. The above limit always exists, although it can be $+\infty$. The entropy of $T$ is then defined as \[h_\mu(T) = \sup_{\mathcal{P}} h_\mu(T,\mathcal{P}),\] with the supremum taken over all finite measurable partitions. Sometimes $h_\mu(T)$ is called the metric or measure theoretic entropy of $T$, to differentiate it from topological entropy. \textbf{Remarks.} \begin{enumerate} \item There is a natural correspondence between finite measurable partitions and finite sub-$\sigma$-algebras of $\mathscr{B}$. Each finite sub-$\sigma$-algebra is generated by a unique partition, and clearly each finite partition generates a finite $\sigma$-algebra. Because of this, sometimes $h_\mu(T,\mathcal{P})$ is called the entropy of $T$ with respect to the $\sigma$-algebra $\mathscr{P}$ generated by $\mathcal{P}$, and denoted by $h_\mu(T,\mathscr{P})$. This simplifies the notation in some instances. \end{enumerate}