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

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### Summary

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

#### References

#### Index.