## Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous SpacesThe 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 |

198 | |

### Other editions - View all

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces M Bachir Bekka,Matthias Mayer No preview available - 2013 |

### Common terms and phrases

action acts algebraic application assume bounded called Chap Chapter claim classification Clearly closure conjecture connected Consider constant contains continuous contradiction Corollary corresponding Dani decomposition defined Definition denote dense diagonal discuss domain element ergodic theorem Example exists fact finite fixed formula function fundamental G-invariant geodesic flow geometry give given group G Hence holds homogeneous spaces horocycle flow hyperbolic identified implies infinity integer invariant invariant measures lattice Lemma Let G linear manifold mapping Margulis Math measure minimal mixing negative normalized Notes Observe one-parameter orbit plane positive probability measure Proof Proof Let properties Proposition proved quadratic forms Recall Remark respect result Riemannian sequence shows simple SL(n strongly mixing subgroup subgroup of G subset Suppose surface symmetric space tangent theory transformation translations unipotent unique unit vanish vector vertices