Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem

By Anatole Katok and Viorel Nitica

Cambridge University Press, June 2011. ISBN: 9780521879095


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This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.

Table of Contents

Introduction: an overview

Part I. Preliminaries from Dynamics and Analysis:

1. Definitions and properties of abelian group actions

2. Principal classes of algebraic actions

3. Preparatory results from analysis

Part II. Cocycles, Cohomology and Rigidity:

4. First cohomology and rigidity for vector-valued cocycles

5. First cohomology and rigidity for general cocycles

6. Higher order cohomology