Ratner's Theorems On Unipotent Flows

by Morris, Dave Witte

• ISBN13: 9780226539843
• ISBN10: 0226539849
• Format: Paperback
• Copyright: 2005-08-15
• Publisher: Univ of Chicago Pr

Summary

The theorems of Berkeley mathematician Marina Ratner have guided key advances in the understanding of dynamical systems. Unipotent flows are well-behaved dynamical systems, and Ratner has shown that the closure of every orbit for such a flow is of a simple algebraic or geometric form. In Ratner's Theorems on Unipotent Flows, Dave Witte Morris provides both an elementary introduction to these theorems and an account of the proof of Ratner's measure classification theorem. A collection of lecture notes aimed at graduate students, the first four chapters of Ratner's Theorems on Unipotent Flows can be read independently. The first chapter, intended for a fairly general audience, provides an introduction with examples that illustrate the theorems, some of their applications, and the main ideas involved in the proof. In the following chapters, Morris introduces entropy, ergodic theory, and the theory of algebraic groups. The book concludes with a proof of the measure-theoretic version of Ratner's Theorem. With new material that has never before been published in book form, Ratner's Theorems on Unipotent Flows helps bring these important theorems to a broader mathematical readership.

Author Biography

Dave Witte Morris is professor of mathematics at the University of Lethbridge.

Table of Contents

Abstract	ix
Possible lecture schedules	x
Acknowledgments	xi
Chapter 1. Introduction to Ratner's Theorems	1	(72)
§1.1. What is Ratner's Orbit Closure Theorem?	1	(12)
§1.2. Margulis, Oppenheim, and quadratic forms	13	(6)
§1.3. Measure-theoretic versions of Ratner's Theorem	19	(6)
§1.4. Some applications of Ratner's Theorems	25	(6)
§1.5. Polynomial divergence and shearing	31	(17)
§1.6. The Shearing Property for larger groups	48	(5)
§1.7. Entropy and a proof for G = SL(2, R)	53	(3)
§1.8. Direction of divergence and a joinings proof	56	(3)
§1.9. From measures to orbit closures	59	(3)
Brief history of Ratner's Theorems	62	(2)
Notes	64	(3)
References	67	(6)
Chapter 2. Introduction to Entropy	73	(32)
§2.1. Two dynamical systems	73	(3)
§2.2. Unpredictability	76	(3)
§2.3. Definition of entropy	79	(6)
§2.4. How to calculate entropy	85	(4)
§2.5. Stretching and the entropy of a translation	89	(5)
§2.6. Proof of the entropy estimate	94	(6)
Notes	100	(2)
References	102	(3)
Chapter 3. Facts from Ergodic Theory	105	(18)
§3.1. Pointwise Ergodic Theorem	105	(3)
§3.2. Mautner Phenomenon	108	(5)
§3.3. Ergodic decomposition	113	(3)
§3.4. Averaging sets	116	(2)
Notes	118	(2)
References	120	(3)
Chapter 4. Facts about Algebraic Groups	123	(42)
§4.1. Algebraic groups	123	(4)
§4.2. Zariski closure	127	(3)
§4.3. Real Jordan decomposition	130	(5)
§4.4. Structure of almost-Zariski closed groups	135	(5)
§4.5. Chevalley's Theorem and applications	140	(2)
§4.6. Subgroups that are almost Zariski closed	142	(4)
§4.7. Borel Density Theorem	146	(5)
§4.8. Subgroups defined over Q	151	(3)
§4.9. Appendix on Lie groups	154	(6)
Notes	160	(3)
References	163	(2)
Chapter 5. Proof of the Measure-Classification Theorem	165	(28)
§5.1. An outline of the proof	166	(2)
§5.2. Shearing and polynomial divergence	168	(3)
§5.3. Assumptions and a restatement of 5.2.4'	171	(2)
§5.4. Definition of the subgroup S	173	(5)
§5.5. Two important consequences of shearing	178	(2)
§5.6. Comparing S_ with S_	180	(2)
§5.7. Completion of the proof	182	(2)
§5.8. Some precise statements	184	(5)
§5.9. How to eliminate Assumption 5.3.1	189	(2)
Notes	191	(1)
References	192	(1)
List of Notation	193	(4)
Index	197