### 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).

A hardcopy (US $26 paperback) can be purchased
online
from:

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

Abstract. 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.