Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

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Cambridge University Press, May 11, 2000 - Mathematics - 200 pages
The study of geodesic flows on homogeneous spaces is an area of research that has recently yielded some fascinating developments. This book focuses on many of these, with one of its highlights an elementary and complete proof by Margulis and Dani of Oppenheim's conjecture. Other features are self-contained treatments of 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; and Ledrappier's example of a mixing action which is not a mixing of all orders.
 

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Contents

Ergodic Systems
1
2 Ergodic Theory and Unitary Representations
13
3 Invariant Measures and Unique Ergodicity
30
The Geodesic Flow of Riemannian Locally Symmetric Spaces
36
1 Some Hyperbolic Geometry
37
2 Lattices and Fundamental Domains
42
3 The Geodesic Flow of Compact Riemann Surfaces
57
4 The Geodesic Flow of Riemannian Locally Symmmetric Spaces
62
4 Equidistribution of Horocycle Orbits
128
Siegel Sets Mahlers Criterion and Margulis Lemma
139
2 SLnZ is a Lattice in SLnR
144
3 Mahlers Criterion
146
4 Reduction of Positive Definite Quadratic Forms
148
5 Margulis Lemma
150
An Application to Number Theory Oppenheims Conjecture
161
1 Oppenheims Conjecture
162

The Vanishing Theorem of Howe and Moore
80
1 Howe Moores Theorem
81
2 Moores Ergodicity Theorems
89
3 Counting Lattice Points in the Hyperbolic Plane
93
4 Mixing of All Orders
98
The Horocycle Flow
110
1 The Horocycle Flow of a Riemann Surface
111
2 Proof of Hedlunds Theorem Cocompact Case
116
3 Classification of Invariant Measures
120
2 Proof of the Theorem Preliminaries Reduction to the case n 3
163
3 Existence of Minimal Closed Subsets
172
4 Orbits of OneParameter Groups of Unipotent Linear Transformations
177
5 Proof of the Theorem Conclusion
179
6 Ratners Results on the Conjectures of Raghunathan Dani and Margulis
184
Bibliography
189
Index
198
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