Ergodic Theory of Expanding Thurston Maps
Thurston maps are topological generalizations of postcritically-finite rational maps. This book provides a comprehensive study of ergodic theory of expanding Thurston maps, focusing on the measure of maximal entropy, as well as a more general class of inv
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Zhiqiang Li
Ergodic Theory of Expanding Thurston Maps
Atlantis Studies in Dynamical Systems Volume 4
Series editors Henk Broer, Groningen, The Netherlands Boris Hasselblatt, Medford, USA
The “Atlantis Studies in Dynamical Systems” publishes monographs in the area of dynamical systems, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications.
More information about this series at http://www.atlantis-press.com
Zhiqiang Li
Ergodic Theory of Expanding Thurston Maps
Zhiqiang Li Institute for Mathematical Sciences Stony Brook University Stony Brook, NY USA
Atlantis Studies in Dynamical Systems ISBN 978-94-6239-173-4 ISBN 978-94-6239-174-1 DOI 10.2991/978-94-6239-174-1
(eBook)
Library of Congress Control Number: 2017930418 © Atlantis Press and the author(s) 2017 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
To the loving memory of my grandmother Fengxian Shen
Preface
This monograph came out of my thesis work under the supervision of my Ph.D. advisor Mario Bonk during my graduate studies at the University of Michigan, Ann Arbor, and later at the University of California, Los Angeles. It focuses on the dynamics, more specifically ergodic theory, of some continuous branched covering maps on the 2-sphere, called expanding Thurston maps. More than 15 years ago, Mario Bonk and Daniel Meyer became independently interested in some basic problems on quasisymmetric parametrization of 2-spheres, related to the dynamics of rational maps. They joined forces during their time together at the University of Michigan and started their investigation of a class of continuous (but not necessarily holomorphic) maps modeling a subclass of rational maps. These maps belong to a bigger class of continuous maps on the 2-sphere studied by William P. Thurston in his famous characterization theorem of rational maps (see [DH93]). As a result, Mario Bonk and Daniel Meyer called their maps expanding Thurston maps. Related studies were carried out by other researchers around the same time, notably Peter Haïssinsky and Kevin Pilgrim [HP09], and James W. Cannon, William J. Floyd, and Walter R. Parry [CFP07]. By late 2010, Mario Bonk and Daniel Meyer had summarized their findings in a reader-friendly arXiv draft [BM10] entitled Expanding Thurston maps, which they initially intended to publish in the AMS Mathematical Surveys and Monographs series. In order to make the material even more accessible, they decided later to expand their draft. This led to a long delay for the final published version [BM17] with almost twice the size of [BM10]. I was introduced to expanding Thurston maps by Mario Bonk soon after I joined in the graduate program at the University of Mic
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