Tame Geometry with Application in Smooth Analysis
The Morse-Sard theorem is a rather subtle result and the interplay between the high-order analytic structure of the mappings involved and their geometry rarely becomes apparent. The main reason is that the classical Morse-Sard theorem is basically qu
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Yosef Yomdin Georges Comte
Tame Geometry with Application in Smooth Analysis
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Authors Yosef Yomdin Department of Mathematics Weizmann Institute of Science Rehovot 76100, Israel e-mail: [email protected] Georges Comte Laboratoire J. A. Dieudonn´e UMR CNRS 6621 Universit´e de Nice Sophia-Antipolis 28, avenue de Valrose 06108 Nice Cedex 2, France e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 28A75, 14Q20, 14P10, 26B5, 26B15, 32S15 ISSN 0075-8434 ISBN 3-540-20612-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 is a part of Springer Science + Business Media http://www.springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10973455
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Preface
This book presents results and methods developed during quite a long period of time and many people helped in this work. We would like to thank Y. Kannai for pointing out, in the very beginning of the work on quantitative transversality, the relevance of metric entropy. Since 1983 M. Gromov encouraged this research and helped us in many fruitful discussions of qualitative transversality and Semialgebraic Geometry in Dynamics and Analysis. We would like to thank him especially for suggesting a problem of quantitative Kupka-Smale, for his contribution to C k -reparametrization of semialgebraic sets and applications to dynamics, for providing a central (for this book) reference to Multidimensional Variations and to books of Vitushkin and Ivanov and for encouraging writing preliminary texts, which were used in this book. This book would not have be written without the help and encouragement of J. -J. Risler and B. Teissier and numerous fruitful discussions with them during all the long period of the book’s preparation. It i
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