**Translations of Mathematical Monographs**

2000;
243 pp;
Hardcover

MSC: Primary 37;
Secondary 11; 22

Print ISBN: 978-0-8218-1389-8

Product Code: MMONO/190

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-4604-8
Product Code: MMONO/190.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

# Dynamical Systems on Homogeneous Spaces

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*Alexander N. Starkov*

A homogeneous flow is a dynamical system generated by the action of a closed
subgroup \(H\) of a Lie group \(G\) on a homogeneous space of
\(G\). The study of such systems is of great significance because they
constitute an algebraic model for more general and more complicated systems.
Also, there are abundant applications to other fields of mathematics, most
notably to number theory.

The present book gives an extensive survey of the subject. In the first chapter
the author discusses ergodicity and mixing of homogeneous flows. The second
chapter is focused on unipotent flows, for which substantial progress has been
made during the last 10–15 years. The culmination of this progress was M.
Ratner's celebrated proof of far-reaching conjectures of Raghunathan and Dani.
The third chapter is devoted to the dynamics of nonunipotent flows. The final
chapter discusses applications of homogeneous flows to number theory, mainly to
the theory of Diophantine approximations. In particular, the author describes
in detail the famous proof of the Oppenheim-Davenport conjecture using
ergodic properties of homogeneous flows.

#### Readership

Graduate students and research mathematicians working in dynamical systems and ergodic theory.

#### Reviews & Endorsements

The book would be very useful to experts as well as those who wish to learn the topic. While experts would benefit from the breadth of the coverage and find it a convenient reference, the learners would relish many proofs that are more palatable compared to the original sources.

-- Mathematical Reviews

This book provides a thorough discussion of many of the main topics in the field. Theorems are stated precisely, references are provided when proofs are omitted and the historical development of the subject id described, so the book is a very useful reference.

-- Bulletin of the LMS

#### Table of Contents

# Table of Contents

## Dynamical Systems on Homogeneous Spaces

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction ix10
- List of notations xv16
- Preliminaries 118
- Ergodicity and mixing of homogeneous flows 2340
- Dynamics of unipotent flows 93110
- Dynamics of nonunipotent flows 147164
- Applications to number theory 189206
- References 227244
- Index 239256
- Back Cover Back Cover1263