Iterates of Piecewise Monotone Mappings on an Interval
Piecewise monotone mappings on an interval provide simple examples of discrete dynamical systems whose behaviour can be very complicated. These notes are concerned with the properties of the iterates of such mappings. The material presented can be underst
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1347 Chris Preston
Iterates of Piecewise Monotone Mappings on an Interval
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author Chris Preston FSP Mathematisierung, Universitat Bielefeld 4800 Bielefeld 1, Federal Republic of Germany
Mathematics Subject Classification (1980): 58F08, 54H20, 26A 18
ISBN 3-540-50329-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50329-3 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 other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Piecewise monotone mappings on an interval provide simple examples of discrete dynamical systems whose behaviour can be very complicated. These notes are concerned with some of the properties of such mappings. It is hoped that the material presented can be understood by anyone who has had a basic course in (one-dimensional) real analysis. This account is self-contained, but it can be regarded as a sequel to Iterates of maps on an interval (Springer Lecture Notes in Mathematics, Vol. 999). I would like to thank Lai-Sang Young, Richard Hohmann-Damaschke and JOrgen Willms for their suggestions and comments during the writing of these notes. My thanks go also to the staff of the FSP Mathematisierung at the University of "Bielefeld for technical assistance in the preparation of the text.
Bielefeld May 1987
Chris Preston
ITERATES OF PIECEWISE MONOTONE MAPPINGS ON AN INTERVAL - CONTENTS
Section 1
Introduction
.
Section 2
Piecewise monotone mappings
20
Section 3
Proof of Theorems 2.4 and 2.5
34
Section 4
Sinks and homtervals
41
Section 5
Examples of register-shifts
50
Section 6
A proof of Parry's theorem (Theorem 2.6)
57
Section 7
Reduct ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Section 8
The structure of the set D(f)
82
Section 9
Countable closed invariant sets
103
Section 10 Extensions
112
Section 11 Refinements
126
Section 12 Mappings with one turning point
135
Section 13 Some miscellaneous results from real analysis
153
References Index
•••••••••• f
.
160 163
1. INTRODUCTION Let I = [a,b]
be a closed, bounded interval and let C(I) denote the set of
continuous functions f
E
f: I
I which map the interval
C(I) we define
n > 1)
back into itself. For x, f 1(x) = f(x) and (for
fn E C(I) inductively by fO(x) n-1(x» fn(x) = f(f . fn is called the nth. iterate of f. The set of
it
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