Iterates of Maps on an Interval
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999 Chris Preston
Iterates of Maps on an Interval
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Chris Preston Universitiit Bielefeld, USP-Mathematisierung 4800 Bielefeld, Federal Republic of Germany
AMS Subject Classifications (1980): 26A 18,54 H 20 ISBN 3-540-12322-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12322-9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to cqpyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140·543210
for Os and Tam
The elegant body of mathematiaal theory pertaining to linear
systems (Fourier analysis, orthogonal funations, and so on), and its suaaessful appliaation to many fundamentally linear problems in the physiaal saienaes, tends to dominate even moderately advanaed University aourses in mathematias and theoretiaal physias. The mathematiaal intuition so developed ill equips the student to aonfront the bizarre behaviour exhibited by the simplest of disarete nonlinear systems, suah x + = axn(l-X . Yet suah nonlinear systems n 1 n) are surely the rule, not the exaeption, outside the physiaal
as the equation saienaes.
Bob May in Simple mathematiaal models with very aompliaated dynamias in Nature, Vol. 261, June 1976.
These are some notes on the iterates of maps on an interval, which we hope can be understood by anyone who has had a basic course in (one-dimensional) real analysis. The main reason for writing this account is as an attempt to make the very beautiful mathematics behind the bizarre behaviour exhibited by the simplest of discrete nonlinear systems
accessible to as wide an audience as possible. Parts of these notes have appeared as Volumes 34 and 37 in the series: Materialien des Universitatssahwerpunktes Mathematisierung from the
Universitat Bielefeld, and I would like to thank the USP Mathematisierung for their support. Thanks also to David Griffeath for some pertinent comments on the text and to Bob May and Alister Mees for getting me interested in this subject. Bielefeld October 1982
Chris Preston
ITERATES OF MAPS ON AN INTERVAL - CONTENTS Section 1.
Introduction .•........••.••.••.•.••.••..........• 1
Section 2.
Piecewise monotone functions •.......•.....••..•... 19
Section 3.
Well-behaved piecewise monotone functions·········41
Section 4.
Property R and negative Schwarzian derivatives····60
Section 5.
The iterates of functions in
Section 6.
Reductions .....•••...•••..••••..•••••••••.•••••. 109
Section 7.
Getting rid of homtervals ...........•........... 143
Section 8.
Kneading sequences
Section 9.
An "alm
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