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
accessible to second-year graduate students. All four are needed
5. A reader who is familiar with ergodic theory and algebraic
not unipotent flows, may skip Chapters 2, 3, and 4 entirely, and
1.4-1.7 of Chapter 1 before beginning Chapter 5.