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Dynamical Systems on Homogeneous Spaces
Alexander N. Starkov, Moscow State University, Russia
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Translations of Mathematical Monographs
2000; 243 pp; hardcover
Volume: 190
ISBN-10: 0-8218-1389-7
ISBN-13: 978-0-8218-1389-8
List Price: US$109 Member Price: US$87.20
Order Code: MMONO/190

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.

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

Reviews

"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