Dynamics in One Dimension
The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interv
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L. S. Block
W. A. Coppel
Dynamics in One Dimension
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Louis Stuart Block Department of Mathematics University of Florida Gainesville, Florida 32611, USA William Andrew Coppel Department of Theoretical Physics Institute of Advanced Studies Australian National University GPO Box4 Canberra 260 I, Australia
Mathematics Subject Classification (1991): 26A18, 54H20, 58F08
ISBN 3-540-55309-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55309-6 Springer-Verlag New York Berlin Heidelberg 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
Preface There has recently been an explosion of interest in one-dimensional dynamics. The extremely complicated -
and yet orderly -
behaviour exhibited by the logistic map, and by unimodal
maps in general, has attracted particular attention. The ease with which such maps can be explored with a personal computer, or even with a pocket calculator, has certainly been a contributing factor. The unimodal case is extensively studied in the book of Collet and Eckmann [49], for example. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. It is quite remarkable how many strong, general properties can be established, considering that such maps may be either real-analytic or nowhere differentiable. The purpose of the present book is to give a clear, connected account of this subject. Thus it updates and extends the survey article of Nitecki [96]. The two books [112], [113] by Sarkovskii and his collaborators contain material on the same subject. However, they are at present available only in Russian and in general omit proofs. Here complete proofs are given. In many cases these have previously been difficult of access, and in some cases no complete proof has hitherto appeared in print. Our standpoint is topological. We do not discuss questions of a measure-theoretical nature or connections with ergodic theory. This is not to imply that such matters are without interest, merely that they are outside our scope. [A forthcoming book by de Melo and van Strien discusses these matters, and also the theory of smooth maps.] The material here could indeed form the basis for a course in topologi
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