Computational ergodic theory
type: Item
instance of: scholarly article
title: Computational ergodic theory
zbMATH Open document ID: 1064.37004
author: Geon Ho Choe
published in: Algorithms and Computation in Mathematics
publication date: 2005-02-25T00:00:00Z
review text: This substantial book contains a basic course in ergodic theory and related ideas in smooth ergodic theory, low-dimensional dynamics, recurrence and data compression. The title is unduly modest: it might better be described as `Ergodic theory with a huge number of examples illustrated using a computer'. In particular, the results presented here are developed rigorously and often proved; where the proof is not given or only sketched references are provided to the literature. The novel feature is that essentially all the results are illustrated with well-chosen examples and attractive pictures and graphs. Thus, for example, after proving that an ergodic circle rotation is not mixing, this is illustrated by a pair of graphs showing the failure of convergence of the inner-product for a pair of explicit functions as opposed to the Cesàro convergence due to ergodicity. In addition to many familiar pictures, the reader is repeatedly presented with imaginative illustrations (often using the `pointillism' of simply plotting orbits) of results usually taught without numerical examples. Some measure of the extent of this effort to illustrate results is shown by the fact that there are some 250 figures. The other -- related -- theme of the book is careful explanation of theory and attempts to orient the reader in new subjects. Thus there are tables presenting `dictionaries' for translation between measure theory, probability theory and information theory. Little background is assumed, and in particular (from one angle) there is a section of basic MAPLE commands and (from another) there are sections of mathematical background without proofs on topology, measure theory, character theory and Perron-Frobenius theory much in the spirit of the introductory chapter of [\textit{P. Walters}, An introduction to ergodic theory. Graduate Texts in Mathematics, Vol. 79. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0475.28009)]. Ergodic theory and dynamical systems are part of mathe…
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 37-01
zbMATH DE Number: 2138652
zbMATH Keywords: ergodic theory
zbMATH Keywords: data compression
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: b138894
OpenAlex ID: W606609856
DOI: 10.1007/B138894
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description: scientific article; zbMATH DE number 2138652