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

Ideal for researchers in all aspects of dynamical systems and a useful introduction for graduate students entering the field.

Anatole Katok (Author), Viorel Ni?ic? (Author)

9780521879095, Cambridge University Press

Hardback, published 16 June 2011

320 pages, 3 b/w illus.
23.5 x 15.7 x 1.9 cm, 0.58 kg

"This very welcome addition to the literature is the first book-length introduction to the rigidity of higher rand abelian group actions."
David Michael Fisher for Mathematical Reviews

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.

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
References
Index.

Subject Areas: Nonlinear science [PBWR], Geometry [PBM], Algebra [PBF]