Laws of Chaos Invariant Measures and Dynamical Systems in One Dimens
A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda tions of Newtonian physics and opened a window
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Abraham Boyarsky Pawel Göra
Laws of Chaos Invariant Measures and Dynamical Systems in One Dimension
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Abraham Boyarsky Pawel Göra Department of Mathematics and Statistics Concordia University Montreal, Canada H4B 1R6
Library of Congress Cataloging-in-Publication Data
Boyarsky, Abraham. Laws of chaos : invariant measures and dynamical systems in one dimension / Abraham Boyarsky, Pawel Göra. p. cm. -- (Probability and its applications) Includes bibliographical references and index. ISBN 978-1-4612-7386-8 ISBN 978-1-4612-2024-4 (eBook) DOI 10.1007/978-1-4612-2024-4
1. Chaotic behavior in systems. 2. Dynamics . 3. Nonlinear theories. 4. Invarient measures. 5. Probabilities. I. Göra, Pawel. II. Title. III. Series. Q172. 5. C45B69 1997 97-22134 003' .857~dc21 CIP A M S Classification: 28D05, 58FXX, 58F, 58F11 Printed on acid-free paper © Springer Science+Business Media New York 1997 Originally published by Birkhäuser Boston 1997 Softcover reprint of the hardcover 1st edition 1997
Birkhäuser
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ISBN 0-8176-4003-7 ISBN 3-7643-4003-7 Typesetting by the authors in AMSTgL
987654321
A.B. dedicates his work on this book to hisfather, David Boyarsky.
P. G. dedicates his work on this book
to his parents, Alfreda and wtadysfaw Gora.
Preface A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the foundations of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated behavior of deterministic systems remained no more than a mathematical curiosity. Not until the late 1970s, with the advent of fast and cheap computers, was it recognized that chaotic behavior was prevalent in almost all domains of science and technology. Smale horseshoes began appearing in many scientific fields. In 1971, the phrase 'strange attractor' was coined to describe complicated long-term behavior of deterministic systems, and the term quickly became a paradigm of nonline
