Stability of Unfoldings
- PDF / 7,547,397 Bytes
- 173 Pages / 482.4 x 720 pts Page_size
- 94 Downloads / 159 Views
393
Gordon Wassermann
Stability of Unfoldings
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Gordon Wassermann Abteilung fur Mathematik, Universitat Bochum Universitatsstr. 150,4630 Bochum, Federal Republic of Germany
1st Edition 1974 2nd Printing 1986 Mathematics Subject Classification (1970): 57045, 58C25 ISBN 3-540-06794-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06794-9 Springer-Verlag New York Heidelberg Berlin Tokyo
Thiswork is subject to copyright. 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 1974 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS
1
§ 1.
Preliminaries
§ 2.
Finitely determined germs
35
Universal unfoldings
54
§ 4.
Stable unfoldings
84
§ 5.
The seven elementary catastrophes:
§
3.
a classification theorem
116
Appendix: Thom's catastrophe theory
153
References
162
INTRODUCTION
The concept of stability plays a major role in the theory of singularities. There are several reasons for the importance of this notion. For one, usually the problem of classifying the objects being studied is extremely difficult; it becomes much simpler if one tries to classify only the stable objects. For another, in many cases (though not in all) the stable objects are generic, that is, they form an open and dense set; so in these cases almost every object is stable and every object is near to a stable one; the nonstable objects are peculiar exceptions. But a third reason for the importance of stability is that the theory of singularities has in recent years, especially through the ideas of R. Thom, acquired important applications to the natural sciences; stability is a natural condition to place upon mathematical models for processes in nature because the conditions under which such processes take place can never be exactly duplicated; therefore what is observed must be invariant under small perturbations and hence stable. stability notions have been defined for a variety of objects occurring in the theory of singularities: for mappings, for mapgerms, for varieties, for vector fields, for attractors of vector fields and so on. In some cases very little is known about the stable objects; in some cases the stable objects have been completely classified. Other cases lie between these extremes; for example, characterizations of stable proper smooth mappings between manifolds and of stable smooth mapgerms have been given by Mather; he has also computed the dimensions in which the stable proper mappings are dense in the set of all proper mappings (see
[ 7J, [8J, [9]). For smooth real valued mapgerms the theory is
Data Loading...
