Ergodic Theory and Topological Dynamics of

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.

Preface

vii

I. Ergodic Systems

1

(35)

Examples and Basic Results

1

(12)

Ergodic Theory and Unitary Representations

13

(17)

Invariant Measures and Unique Ergodicity

30

(6)

II. The Geodesic Flow of Riemannian Locally Symmetric Spaces

36

(44)

Some Hyperbolic Geometry

38

(4)

Lattices and Fundamental Domains

42

(15)

The Geodesic Flow of Compact Riemann Surfaces

57

(5)

The Geodesic Flow on Riemannian Locally Symmetric Spaces

62

(18)

III. The Vanishing Theorem of Howe and Moore

80

(30)

Howe-Moore's Theorem

81

(8)

Moore's Ergodicity Theorems

89

(4)

Counting Lattice Points in the Hyperbolic Plane

93

(5)

Mixing of All Orders

98

(12)

IV. The Horocycle Flow

110

(29)

The Horocycle Flow of a Riemann Surface

111

(5)

Proof of Hedlund's Theorem-Cocompact Case

116

(4)

Classification of Invariant Measures

120

(8)

Equidistribution of Horocycle Orbits

128

(11)

V. Siegel Sets, Mahler's Criterion and Margulis' Lemma

139

(22)

Siegel Sets in SL(n, R)

139

(5)

SL(n, Z) is a lattice in SL(n, R)

144

(2)

Mahler's Criterion

146

(2)

Reduction of Positive Definite Quadratic Forms

148

(2)

Margulis' Lemma

150

(11)

VI. An Application to Number Theory: Oppenheim's Conjecture

161

(28)

Oppenheim's Conjecture

162

(1)

Proof of the Theorem-Preliminaries

163

(9)

Existence of Minimal Closed Subsets

172

(5)

Orbits of One-Parameter Groups of Unipotent Linear Transformations

177

(2)

Proof of the Theorem-Conclusion

179

(5)

Ratner's Results on the Conjectures of Raghunathan, Dani and Margulis

184

(5)

Bibliography

189

(9)

Index

198

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Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

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