Rotation Sets and Complex Dynamics
This monograph examines rotation sets under the multiplication by d (mod 1) map and their relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs of the corresponding sets under the angle-doubling map of t
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Saeed Zakeri
Rotation Sets and Complex Dynamics
Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg
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More information about this series at http://www.springer.com/series/304
Saeed Zakeri
Rotation Sets and Complex Dynamics
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Saeed Zakeri Department of Mathematics Queens College of CUNY Queens, NY USA Department of Mathematics Graduate Center of CUNY New York, NY USA
ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-78809-8 ISBN 978-3-319-78810-4 (eBook) https://doi.org/10.1007/978-3-319-78810-4 Library of Congress Control Number: 2018939069 Mathematics Subject Classification (2010): 37E10, 37E15, 37E45, 37F10 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
For an integer d ≥ 2, let md : R/Z → R/Z denote the multiplication by d map of the circle defined by md (t) = dt (mod Z). A rotation set for md is a compact subset of R/Z on which md acts in an order-preserving fashion and therefore has a well-defined rotation number. Rotation sets for the doubling map m2 seem to have first appeared under the disguise of Sturmian sequences in a 1940 paper of Morse and Hedlund on symbolic dynamics [17] (the equivalence with the rotation set condition was later shown by Gambaudo et al. [10] and Veerman [28]). Fertile ground for their
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