CONTEMPORARY MATHEMATICS 532 Dynamical Numbers Interplay between Dynamical Systems and Number Theory A Special Program May 1–July 31, 2009 International Conference July 20–24, 2009 Max Planck Institute for Mathematics, Bonn, Germany Sergiy Kolyada Yuri Manin Martin Möller Pieter Moree Thomas Ward Editors American Mathematical Society http://dx.doi.org/10.1090/conm/532 Dynamical Numbers Interplay between Dynamical Systems and Number Theory CONTEMPORARY MATHEMATICS 532 Dynamical Numbers Interplay between Dynamical Systems and Number Theory A Special Program May 1–July 31, 2009 International Conference July 20–24, 2009 Max Planck Institute for Mathematics, Bonn, Germany Sergiy Kolyada Yuri Manin Martin Möller Pieter Moree Thomas Ward Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 11J70, 20F65, 22D40, 30E05, 37A15, 37A20, 37A30, 37A35, 54H20, 60B15. Library of Congress Cataloging-in-Publication Data Dynamical numbers : interplay between dynamical systems and number theory : a special pro- gram, May 1–July 31, 2009 : international conference, July 20–24, 2009, Max Planck Institute for Mathematics, Bonn, Germany / Sergiy Kolyada ...[et al.], editors. p. cm. — (Contemporary mathematics ; v. 532) Includes bibliographical references. ISBN 978-0-8218-4958-3 (alk. paper) 1. Number theory—Congresses. 2. Ergodic theory—Congresses. 3. Topological algebras— Congresses. I. Koliada, S. F. II. Max-Planck-Institut f¨ur Mathematik. QA241.D96 2010 512.7—dc22 2010027232 Copying and reprinting. 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Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface vii List of Participants ix Stationary dynamical systems Hillel Furstenberg & Eli Glasner 1 Geodesic flow on the Teichm¨uller disk of the regular octagon, cutting sequences and octagon continued fractions maps John Smillie & Corinna Ulcigrai 29 Expansion of rational numbers in M¨obius number systems Petr K˚urka 67 Localisation of algebraic integers and polynomial iteration Pierre Moussa 83 Horospheres and Farey fractions Jens Marklof 97 Exponential mixing for automorphisms on compact K¨ahlermanifolds Tien-Cuong Dinh & Nessim Sibony 107 Orbit theory, locally finite permutations and Morse arithmetic A. M. Vershik 115 Ergodic abelian actions with homogeneous spectrum Alexandre I. Danilenko & Anton V. Solomko 137 Geometric entropy of geodesic currents on free groups Ilya Kapovich & Tatiana Nagnibeda 149 Laurent polynomial moment problem: A case study Fedor Pakovich, Christian Pech & Alexander K. Zvonkin 177 Entropy and growth rate of periodic points of algebraic Zd-actions Douglas Lind, Klaus Schmidt & Evgeny Verbitskiy 195 Statistics of matrix products in hyperbolic geometry Mark Pollicott & Richard Sharp 213 Infinite sequence of fixed point free pseudo-Anosov homeomorphisms on a family of genus two surface Erwan Lanneau 231 v Preface This volume contains the proceedings of the activity “Dynamical Numbers: Interplay between Dynamical Systems and Number Theory” held at the Max Planck Institute for Mathematics (MPIM) in Bonn, from 1 May to 31 July, 2009, and the conference of the same title, also held at the Max Planck Institute, from 20 to 24 July, 2009 and financed by the Clay Mathematics Institute, the Hausdorff Center for Mathematics, and the Max Planck Institute for Mathematics. The activities brought together researchers from many different branches of dynamical systems and number theory, and some of the papers in this volume reflect the collaborative research that took place during the activity. The papers and the talks presented at MPIM also reflect the extraordinary vitality of dynamical systems in its interaction with algebra, number theory, combinatorics, geometry, probability, analysis and other fields. The topics of the activity covered both dynamical systems and number theory in a broad sense, together with some aspects of their many interactions, and some of the many applications of dynamical systems to the natural sciences. The re- search areas discussed during the activity were diverse, and included the following: asymptotic geometric analysis and (topological) transformation groups; arithmetic dynamics; polynomials and pointwise ergodic theorems; Bernoulli convolutions; ac- tions of Polish groups; low-dimensional dynamics: graph theory, rotation theory, smooth interval dynamics, area-preserving diffeomorphisms and time-one maps on surfaces; complex and real dynamics; interval-exchange transformations and trans- lation flows; billiards; leaf-wise cohomology of algebraic Anosov diffeomorphisms; symbolic dynamics; multi-fractal analysis and Diophantine approximations; dynam- ics and moduli spaces; rigidity of invariant measures and Littlewood’s conjecture; M¨obius number systems; flows on manifolds; translation surfaces and Abelian dif- ferentials; symbolic representations of quasihyperbolic toral automorphisms; non- commutative Mahler measures, Fuglede–Kadison determinants, and entropy; repre- sentations of integers; statistical properties of dynamical systems; transfer operators for geodesic flows and Hecke operators; transfer operators for Anosov diffeomor- phisms; shift operators on buildings and non-commutative spaces; topological orbit equivalence; the theory of entropy and chaos. In addition to the hundred or so mathematicians from many different countries who visited MPIM for the activity, many of the long-term visitors at the MPIM also participated in the research activities and seminars. Seminars and discussions took place in an informal and constructive atmosphere, and it was a pleasure to see research taking place between groups of participants all over the Institute and at all times of day. vii viii PREFACE The editors wish to record their thanks to the staff at the Max-Planck Institute for Mathematics, and to the many researchers who participated, for all their efforts in making this such a stimulating, productive, and enjoyable experience. Sergiy Kolyada, Kiev Yuri Manin, Bonn Martin M¨oller, Frankfurt Pieter Moree, Bonn Thomas Ward, Norwich List of Participants Oleg Ageev Tien-Cuong Dinh Moscow State Technical University, Institut de Mathematiques de Jussieu, Russia France Matthew Bainbridge Matus Dirbak University of Chicago, U.S.A. Matej Bel University, Slovak Republic Yves Benoist Manfred Einsiedler Universit´e Paris-Sud, France Ohio State University, U.S.A. & ETH, Switzerland Jozeph Bernstein Tel Aviv University, Israel Tanja Eisner University of Tubingen, Germany Michael Bjorglund KTH, Sweden Alexander Felshtyn University of Szczecin, Poland Alexander Blokh University of Alabama, U.S.A. Nikos Frantzikinakis University of Memphis, U.S.A. Alexey Bondal Anna Gierzkiewicz Steklov Mathematical Institute, Jagiellonian University in Krakow, Moscow, Russia Poland Roelf Bruggeman Thierry Giordano University of Utrecht, Netherlands University of Ottawa, Canada Alexander Bufetov Eli Glasner Rice University, U.S.A. Tel Aviv University, Israel Carlo Carminati Arek Goetz Universita di Pisa, Italy San Francisco State University, U.S.A. Francesco Cellarosi Alexander Gorodnik Princeton University, U.S.A. University of Bristol, U.K. John Chaika Rostislav Grigorchuk Rice University, U.S.A. Texas A&M University, U.S.A. Alexandre Danilenko Dmitry Grigoriev ILT, Kharkov, Ukraine Universit´e de Lille 1, France Christopher Deninger Roland Gunesch Universit¨at M¨unster, Germany University of Hamburg, Germany ix x PARTICIPANTS Maryam Hosseini Gregory Margulis University of Guilan, Iran Yale University, U.S.A. Roman Hric Jens Marklof Matej Bel University, Slovak Republic Bristol University, U.K. Pascal Hubert Stefano Marmi Universit´e Aix-Marseille III, France Scuola Normale Superiore di Pisa, Italy Tobias Jaeger Carlos Matheus College de France, France College de France, France Rafe Jones Dieter Mayer College of the Holy Cross, U.S.A. TU Clausthal, Germany Chris Judge Curtis McMullen Indiana University, U.S.A. Harvard University, U.S.A. Shirali Kadyrov Michal