Cohomological Theory of Dynamical Zeta Functions
Dynamical zeta functions are associated to dynamical systems with a countable set of periodic orbits. The dynamical zeta functions of the geodesic flow of lo cally symmetric spaces of rank one are known also as the generalized Selberg zeta functions. The
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Series Editors H. Bass J. Oesterle A. Weinstein
Andreas Juhl
Cohomological Theory of Dynamical Zeta Functions
Springer Basel AG
Author: Andreas Juhl Matematiska Institutionen Universitet Uppsala P.O. Box 480 S-75 \06 Uppsala e-mail: [email protected] 2000 Mathematics Subject Classification llF70, IlF72, IlM36, 22E46, 43A85; 58-02, 58F05, 58F06, 58F15,58F17, 58GIO, 58G25 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data Juhl, Andreas: Cohomological theory of dynamical Zeta functions / Andreas Juhl. - Basel; Boston; Berlin : Birkhäuser, 2001 (Progress in mathematics ; Vol. 194) ISBN 978-3-0348-9524-8 ISBN 978-3-0348-8340-5 (eBook) DOI 10.1007/978-3-0348-8340-5
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ISBN 978-3-0348-9524-8 987654321
Contents Preface....................................................................
ix
Chapter 1. Introduction................................... . . . . . . . . . . . . . . . 1.1. The dynamical zeta functions ..................................... 1.2. The motivations of the cohomological theory ...................... 1.2.1. Quantization of chaos ........................................... 1.2.2. Uniform descriptions of the divisors of zeta functions ............ 1.3. The contents of the book .......................................... 1.3.1. Spectral theory on X, Lefschetz formulas on SX and r-invariant distributions on the ideal boundary sn-l ............ 1.3.2. Harmonic currents and divisors of the zeta functions. The main ideas .................................................. 1.3.3. Harmonic currents and divisors of the zeta functions. The results and the conjectures ....... . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 4 4 7 13
Chapter 2. Preliminaries.................................. . . . . . . . . . . . . . . . 2.1. General notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. Lie theory related to the conformal group ......................... 2.3. Hyperbolic spaces as Riemannian manifolds and symmetric spaces ............................................. 2.4. n- -homology, n- -cohomology and Osborne's character formula 2.5. Induced representations and differential intertwining operators 2.6. The classification of the unitary irreducible representations of the Lorentz group SO(l, n)O ....................................
13 25 30 63 63 63 67 75 76 79
Chapter 3. Zeta Functions of the Geodesic Flow of Compact Locally Symmetric Manifolds ...
