Ratner's Theorems on Unipotent Flows
by Dave Witte Morris

Published in the Chicago Lectures in Mathematics Series of the University of Chicago Press (August, 2005).

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Click HERE for PDF file of the entire book (1.6 MB).
217 pages (Final version posted 2 November 2004; minor corrections 11 February 2005)

The Latex source files are available at http://arxiv.org/abs/math.DS/0310402

Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. After presenting some consequences of this important theorem, these lectures explain the main ideas of the proof. Some algebraic technicalities will be pushed to the background.

Chapter 1 is the main part of the book. It is intended for a fairly general audience, and provides an elementary introduction to the subject, by presenting examples that illustrate the theorem, some of its applications, and the main ideas involved in the proof.

Chapter 2 gives an elementary introduction to the theory of entropy, and proves an estimate used in the proof of Ratner's Theorem. It is of independent interest.

Chapters 3 and 4 are utilitarian. They present some basic facts of ergodic theory and the theory of algebraic groups that are needed in the proof. The reader (or lecturer) may wish to skip over them, and refer back as necessary.

Chapter 5 presents a fairly complete (but not entirely rigorous) proof of the measure-theoretic version of Ratner's Theorem. (We follow the approach of G.A.Margulis and G.Tomanov.) Unlike the other chapters, it is rather technical.

The first four chapters can be read independently, and should be largely accessible to second-year graduate students. All four are needed for Chapter 5. A reader who is familiar with ergodic theory and algebraic groups, but not unipotent flows, may skip Chapters 2, 3, and 4 entirely, and read only Sections 1.4-1.7 of Chapter 1 before beginning Chapter 5.