Lecture Notes in Mathematics 2164 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 More information about this series at http://www.springer.com/series/304
Boris Hasselblatt Editor Ergodic Theory and Negative Curvature CIRM Jean-Morlet Chair, Fall 2013 123
Editor Boris Hasselblatt Department of Mathematics Tufts University Medford, Massachusetts USA ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-43058-4 ISBN 978-3-319-43059-1 (ebook) DOI 10.1007/978-3-319-43059-1 Library of Congress Control Number: 2017953454 Mathematics Subject Classification (2010): 37C40, 37D40 Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2017 A copublication with the Société de Mathématique de France (SMF) Sold and distributed to its members by the SMF, Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France; http://smf.emath.fr ISBN SMF: 978-2-85629-860-2 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 Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface This volume consists of notes from minicourses given in workshops held under the auspices of the Jean-Morlet Chair at CIRM in 2013 and 2014 and with substantial core support and funding by CIRM. Most of these courses were given during the workshop Young Mathematicians in Dynamical Systems organized by the GDR Platon under the direction of Françoise Dal Bo of Université Rennes 1; her co-organizers were Louis Funar (Université Grenoble 1), Boris Hasselblatt (Tufts University, CIRM, and Aix- Marseille Université), and Barbara Schapira (Université de Picardie Jules Verne). 1 The event was supported by several ANRs such as GEODE, directed by Barbara Schapira, and by the LABEX Archimede. The centerpieces were minicourses by Keith Burns (Northwestern University), Carlos Matheus (Instituto Nacional de Matemática Pura e Aplicada in Brazil and Université Paris 13), and Boris Hasselblatt. The scientific focus was the ergodicity of the Weil Petersson geodesic flow on Teichmüller space. The course by Burns presented geodesic flows and methods for proving their ergodicity, that by Matheus introduced Teichmüller space and the Weil Petersson metric, and that by Hasselblatt provided an introduction to hyperbolic dynamical systems and ergodic theory. Barbara Schapira presented in a unified way the classical dynamical and ergodic properties of the horocycle flow in the Spring 2014 School on Geometry and Dynamics, 2 which was organized by Nicolas Bedaride (Aix-Marseille Université), Alexander Bufetov (Aix- Marseille Université), Moon Duchin (Tufts University), Boris Hasselblatt, Pascal Hubert (Aix-Marseille Université), and Federico Rodriguez Hertz (Pennsylvania State University) with the scientific committee consisting of Giovanni Forni (the University of Maryland), Boris Hasselblatt, Howard Masur (the University of Illinois at Chicago and University of Chicago), and Grigori Olshanski (Dobrushin 1 The participant list is at http://www.cirm-math.fr/archives/?ex=liste_participants&annee= 2013&id_renc=1097&num_semaine=0. 2 The participant list is at http://www.cirm-math.fr/archives/?ex=liste_participants&annee= 2014&id_renc=1129&num_semaine=0. v
vi Preface Mathematics Laboratory at the Institute for Information Transmission Problems of the Russian Academy of Sciences, the Independent University of Moscow, and the National Research University Higher School of Economics). It was supported among others by the National Science Foundation, A*Midex, and the GDR Platon. The contents of this volume are as follows. It begins with foundational material: an introduction to hyperbolic dynamics and ergodic theory (plus a translation of Hadamard s proof of the stable-manifold theorem) by Boris Hasselblatt followed by one to the dynamics of geodesic and horocyclic flows by Barbara Schapira. The next three chapters cover the motivating mathematics for the workshop as follows: A detailed summary of the Burns Masur Wilkinson paper Ergodicity of the Weil Petersson Geodesic Flow lays out the agenda and core ideas. A text based on the lectures by Burns ( Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds ) then provides all the insights into the ergodicity proof. Finally, the lecture notes by Matheus ( The Dynamics of the Weil Petersson Flow ) provide comprehensive background on the Teichmüller theory at the base of the whole problem. We also included a survey of some arithmetic applications of ergodic theory in negative curvature, which rounds out the volume by adding another topic centered on ergodicity of geodesic flows in negative curvature. There is no need to read this book sequentially. The chapters make it as easy as possible to read any one of them independently while keeping it easy to refer to other chapters for any needed context. The editor gratefully acknowledges partial support by the Committee on Faculty Research Awards of Tufts University and will forever be indebted to CIRM for 6 months of truly exceptional working conditions. Medford, MA, USA April 2017 Boris Hasselblatt
Contents 1 Introduction to Hyperbolic Dynamics and Ergodic Theory... 1 Boris Hasselblatt 2 On Iteration and Asymptotic Solutions of Differential Equations by Jacques Hadamard... 125 Boris Hasselblatt (Translator) 3 Dynamics of Geodesic and Horocyclic Flows... 129 Barbara Schapira 4 Ergodicity of the Weil Petersson Geodesic Flow... 157 Keith Burns, Howard Masur, and Amie Wilkinson 5 Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds... 175 Keith Burns, Howard Masur, Carlos Matheus, and Amie Wilkinson 6 The Dynamics of the Weil Petersson Flow... 209 Carlos Matheus 7 A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature... 293 Jouni Parkkonen and Frédéric Paulin vii