Shadowing in Dynamical Systems
This book is an introduction to the theory of shadowing of approximate trajectories in dynamical systems by exact ones. This is the first book completely devoted to the theory of shadowing. It shows the importance of shadowing theory for both the qualitat
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1706
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1706
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Sergei Yu. Pilyugin
Shadowing in Dynamical Systems
Springer
Author Sergei Yu. Pilyugin Faculty of Mathematics and Mechanics St. Petersburg State University Bibliotechnaya pI., 2, Petrodvorets 198904 St. Petersburg, Russia E-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Piljugin, Sergej Ju.:
Shadowing in dynamical systems / Sergei Yu. Pilyugin. - Berlin; Heidelberg; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture Dotes in mathematics; 1706) ISBN 3-540-66299-5
Mathematics Subject Classification (1991): 58Fxx, 34Cxx, 65Lxx, 65Mxx ISSN 0075-8434 ISBN 3-540-66299-5 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, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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 Berlin Heidelberg 1999 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 specific 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: 10650213 41/3143-543210 - Printed on acid-free paper
To my sons Sergei and Kirill
Preface Let (X, r) be a metric space and let ( Xk-l + Vk-l) - Xk·
Since 14>(Xk) - Xk+ll :S d, it follows from (1.23) and from the estimates above that for v E B(dl ) we have
hence F is an operator from B(dl ) to B. By (1.22), 'fJ is a trajectory of only if v is a fixed point of F. Now we represent
B
=
B S EB BU ,
4> if and (1.24)
where
(recall that we denote by S(x), U(x) the extended families on W). It follows from (1.23) that F is differentiable at 0 with (DF(O)vh+I D4>(Xk)Vk. Take
Vk
=
vk + vi:, Vk E S(Xk), vi: E U(Xk),
and represent Wk+I = D4>(Xk)Vk as
Denote
Z
= Xk+l. Then we have
where Similarly, where Take d < inequality
= II: D4>(Xk)vL {j
(where
Iz -
(j
= II:D4>(Xk)V'k.
is given by Lemma 1.2.1 for the fixed A, to), so that the {j holds. Then it follows from Lemma 1.2.1 that
4>(xk)1
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