Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous SpacesCambridge University Press, 11 Mei 2000 - 200 halaman 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. |
Isi
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 |
| 198 | |
Edisi yang lain - Lihat semua
Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces M Bachir Bekka,Matthias Mayer Pratinjau tidak tersedia - 2013 |
Istilah dan frasa umum
abelian action of G assume automorphism Borel Chap closed subgroup closure cocompact compact subset constant contradiction Corollary countable decomposition defined denote diagonal Dirichlet region discrete subgroup DV₁ equidistribution equidistribution theorem exists finite centre function fundamental domain G acts G-invariant G.A. Margulis geodesic flow geometry group acting group G Haar measure Hedlund's Hence homogeneous spaces horocycle flow implies integer invariant measures isometry lattice in G Let G linear locally compact group locally symmetric space mapping Math matrix coefficients minimal negative curvature non-compact non-empty one-parameter group Oppenheim's conjecture polynomial previous lemma probability measure Proof Let Proposition proves the claim quadratic forms Ratner Recall Riemannian manifold Riemannian symmetric space S.G. Dani semisimple Lie group sequence shows Siegel sets SL(n strongly mixing subgroup of G subspace topology Tx,tn unipotent unit tangent unitary representation V₁ V₂ vanish at infinity vector