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Michael Shub,Global Stability of,Dynamical Systems. With the Collaboration of,Albert Fathi and Remi Langevin. Translated by Joseph Christy,With 43 Illustrations. Springer Verlag Berlin Heidelberg GmbH,Michael Shub. Thomas 1 Watson Research Center,Yorktown Heights NY 10598. With the collaboration of Albert Fathi and Remi Langevin. Translated by Joseph Christy,AMS Subject Classifications 34C35 58FI0 54H20. Original edition Stabilite globale des systemes dynamiques Asterisque Vol 56 Societe. Mathematique de France 1978, Library of Congress Cataloging in Publication Data. Shub Michael,Global stability of dynamical systems. Translation of Stabilite globale des systemes,dynamiques. Includes bibliographies,l Topological dynamics 2 Stability I Title. QA611 5 S49713 1986 514 86 10164,1987 by Springer Verlag Berlin Heidelberg. Originally published by Springer Verlag Berlin Heidelberg New York in 1987. Softcover reprint of the hardcover 1st edition 1987. All rights reserved No part of this book may be translated or reproduced in any form without. written permission from Springer Verlag,Typeset by Asco Trade Typesetting Ltd Hong Kong. 9 8 7 6 5 4 321, ISBN 978 1 4419 3079 8 ISBN 978 1 4757 1947 5 eBook. DOI 10 1007 978 1 4757 1947 5,To Alex and Steve,Introduction xi. Generalities 1, Wandering and nonwandering points nonwandering set The w limit. set wAx the IX limit set IXAx The limit set L f Pseudo orbits Chain. recurrence and R f The C topology,Commentary,References. Filtrations 8, A filtration adapted to a homeomorphism The invariant set of a filtration. The stable and unstable sets W and W U Filtration orderings. Commentary,References,Sequences of Filtrations 13, Refining a filtration Sequence of filtrations Fine filtrations Fine. sequences of filtrations Function decreasing along orbits Cl explosions. Lyapunov functions,Commentary,References,Hyperbolic Sets 20. Hyperbolic invariant sets Adapted metrics Hyperbolic fixed points. Anosov diffeomorphisms Smale s horseshoe Shifts The solenoid. Commentary,References,viii Contents,Stable Manifolds 33. Hyperbolic automorphisms Lipschitz maps The local unstable manifold. theorem for a point The graph transform Differentiability Adapted. metrics on vector bundles The C section theorem Holder bundles The. global unstable manifold theorem for a point, Appendix I The Local Lipschitz Inverse Function Theorem 49. Appendix II Irwin s Proof of the Stable Manifold Theorem 51. Appendix III Center and Strong Stable Manifolds 61. Commentary,References,Stable Manifolds for Hyperbolic Sets 71. Stable and unstable discs The spaces of bounded and continuous. sections of a bundle The exponential map The local and global. unstable manifolds, Appendix IV Center and Strong Stable Manifolds for Invariant Sets 79. Commentary,References,More Consequences of Hyperbolicity 83. Transversality Expansivity The invariance of hyperbolicity under. perturbation The fixed point of a perturbation of a hyperbolic. automorphism Theorem 7 8 The Grobman Hartman Linearization. Commentary,References,Stability 100, Semiconjugacy topological conjugacy Structural stability l stability. Axiom A Local product structure Topological transitivity and. topological mixing Spectral decomposition Cycles in the spectral. decomposition The shadowing lemma The l stability theorem The. central problem,Commentary,References,A Potpourri of Stability Results 117. Axiom A Strong transversality Morse Smale diffeomorphisms The. stability conjecture,Commentary,References,Contents ix. CHAPTER 10,Markov Partitions 122, Symbolic dynamics Subshift of finite type Isolated periodic points The. zeta function Rectangles and proper rectangles Markov partitions. Unstable bands The finiteness of the coding by a Markov partition. The rationality of the zeta function,Commentary,References. List of Notation 147,Introduction, These notes are the result of a course in dynamical systems given at Orsay. during the 1976 77 academic year I had given a similar course at the Gradu. ate Center of the City University of New York the previous year and came to. France equipped with the class notes of two of my students there Carol. Hurwitz and Michael Maller My goal was to present Smale s n Stability. Theorem as completely and compactly as possible and in such a way that. the students would have easy access to the literature I was not confident that. I could do all this in lectures in French so I decided to distribute lecture. notes I wrote these notes in English and Remi Langevin translated them. into French His work involved much more than translation He consistently. corrected for style clarity and accuracy Albert Fathi got involved in reading. the manuscript His role quickly expanded to extensive rewriting and writing. Fathi wrote 5 1 and 5 2 and rewrote Theorem 7 8 when I was in despair of. ever getting it right with all the details He kept me honest at all points and. played a large role in the final form of the manuscript He also did the main. work in getting the manuscript ready when I had left France and Langevin. was unfortunately unavailable I ran out of steam by the time it came to. Chapter 10 M Yoccoz wrote this chapter based on a preliminary version of. class notes by M Lebasque, The original English notes have disappeared Anyway much writing and. revising was done directly in French Joe Christy undertook to translate the. French book into English I have used the opportunity to correct some small. errors and sloppiness that made certain points of the French original seem. extraterrestrial I have also added some appendices concerning the C center. center unstable and strong unstable manifolds These theorems have gained. currency in recent years but it is not ea sy to find clear presentations of them. in the literature The stable manifold theorem was written to make generaliza. xii Introduction, tion easy and these new appendices are written with references back to the. chapters on stable manifolds Indeed the entire approach is taken from. Hirsch Pugh and Shub Invariant Manifolds Springer Lecture Notes No. 583 and lends itself easily to the theory of normally hyperbolic invariant. manifolds as carried out there, The chapters are meant to be read in order except that the appendices and. Chapter 9 may be omitted There is a commentary and a bibliography for. each chapter These commentaries are subjective they reflect how I learned. the subject and do not pretend in any way to provide a historical account of. its development Certain notions may be introduced in the commentary of. some chapter which will only be justified in the commentary of a later chapter. Reference i j is the jth reference in the bibliography of Chapter i The. chapters themselves are narrowly focused on the mathematical results The. commentaries sometimes give some perspective in which to view these results. and should be read together with or just after the corresponding chapter. Translator s Note, I would like to thank first of all Mary Ann Finnerty for typing the manuscript. and devoting countless hours to typesetting and producing this book My. debt to her is immeasurable as it is to Mike Shub to them my gratitude is. boundless I should also like to thank Kristen Harber for the lovely pictures. of the solenoid and Paul Blanchard Pat McSwiggen Alec Norton and Jose.
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