Computational Ergodic Theory
Ergodic theory is hard to study because it is based on measure theory, which is a technically difficult subject to master for ordinary students, especially for physics majors. Many of the examples are introduced from a different perspective than in other
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Geon Ho Choe
Computational Ergodic Theory With 250 Figures and 10 Tables
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Geon Ho Choe Korea Advanced Institute of Science and Technology Department of Mathematics Guseong-dong 373-1, Yuseong-gu Daejeon 305-701, Korea e-mail: [email protected]
Mathematics Subject Classification (2000): 11Kxx, 28Dxx, 37-01, 37Axx
Library of Congress Control Number: 2004117450
ISSN 1431-1550 ISBN 3-540-23121-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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In Memory of My Father
Preface
This book is about probabilistic aspects of discrete dynamical systems and their computer simulations. Basic measure theory is used as a theoretical tool to describe probabilistic phenomena. All of the simulations are done using the mathematical software Maple, which enables us to translate theoretical ideas into computer programs word by word avoiding any other technical details. The level of familiarity with computer programming is kept minimal. Thus even theoretical mathematicians can appreciate the experimental nature of one of the fascinating areas in mathematics. A new philosophy is introduced even for those who are already familiar with numerical simulations of dynamical systems. Most computer experiments in this book employ many significant digits depending on the entropy of the given transformation so that they can still produce meaningful results after many iterations. This is why we can do experiments that have not been tried elsewhere before. It is the main point of the book. Readers should regard the Maple programs as important as the theoretical explanations of mathematical facts. The book is designed in such a way that theoretical and experimental parts complement each other. A dynamical system is a set of points together with a transformation rule that describes the movement of points. Sometimes the points lose the geometrical implication and
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