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Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces : London Mathematical Society Lecture Notes - M. Bachir Bekka

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

London Mathematical Society Lecture Notes

Paperback Published: 10th July 2000
ISBN: 9780521660303
Number Of Pages: 200

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The study of geodesic flows on homogenous spaces is an area of research that has in recent years yielded some fascinating developments. This book focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory.

Industry Reviews

'... a most welcome introduction to the study of group actions on homogenous spaces ... I highly recommend the book.' Sanju Velani, Bulletin of the London Mathematical Society 'This book can be used as a guide to modern ergodic theory and dynamics. It can be used by graduate students and by researchers in different areas, since the contents of the book range from elementary results to modern theories.' EMS

Prefacep. vii
Ergodic Systemsp. 1
Examples and Basic Resultsp. 1
Ergodic Theory and Unitary Representationsp. 13
Invariant Measures and Unique Ergodicityp. 30
The Geodesic Flow of Riemannian Locally Symmetric Spacesp. 36
Some Hyperbolic Geometryp. 38
Lattices and Fundamental Domainsp. 42
The Geodesic Flow of Compact Riemann Surfacesp. 57
The Geodesic Flow on Riemannian Locally Symmetric Spacesp. 62
The Vanishing Theorem of Howe and Moorep. 80
Howe--Moore's Theoremp. 81
Moore's Ergodicity Theoremsp. 89
Counting Lattice Points in the Hyperbolic Planep. 93
Mixing of All Ordersp. 98
The Horocycle Flowp. 110
The Horocycle Flow of a Riemann Surfacep. 111
Proof of Hedlund's Theorem--Cocompact Casep. 116
Classification of Invariant Measuresp. 120
Equidistribution of Horocycle Orbitsp. 128
Siegel Sets, Mahler's Criterion and Margulis' Lemmap. 139
Siegel Sets in SL(n, R)p. 139
SL(n, Z) is a lattice in SL(n, R)p. 144
Mahler's Criterionp. 146
Reduction of Positive Definite Quadratic Formsp. 148
Margulis' Lemmap. 150
An Application to Number Theory: Oppenheim's Conjecturep. 161
Oppenheim's Conjecturep. 162
Proof of the Theorem--Preliminariesp. 163
Existence of Minimal Closed Subsetsp. 172
Orbits of One-Parameter Groups of Unipotent Linear Transformationsp. 177
Proof of the Theorem--Conclusionp. 179
Ratner's Results on the Conjectures of Raghunathan, Dani and Margulisp. 184
Bibliographyp. 189
Indexp. 198
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521660303
ISBN-10: 0521660300
Series: London Mathematical Society Lecture Notes
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 200
Published: 10th July 2000
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 1.22
Weight (kg): 0.32

Earn 172 Qantas Points
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