Preface

Summary

Ergodic theory today is a large and rapidly developing subject. The aim of this book is to introduce the reader first to the fundamentals of the ergodic theory of point transformations and then to several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, a student can quickly approach the specialized literature and indeed the frontier of the area of interest.

Of course the number of interesting topics that we have neglected is necessarily far greater than that of those we have been able to include. Thus we have to refer the reader elsewhere for discussions of, for example, operator ergodic theory, the existence of invariant measures, nonsingular transformations, orbit equivalence, differentiable dynamics, subadditive ergodic theorems, etc. Unfortunately, there do not exist coherent expositions of all of these topics; I invite those of my colleagues who are more expert than I in the areas I have omitted to do some more expository writing.

It should also be understood that, even for the advanced topics that we do discuss, their treatment here cannot be more than an entryway to the rapidly expanding specialized literature. Thus our presentations of multiple recurrence and the Ornstein theory, to mention two examples, are intended as introductions to the books of Furstenberg (1981) and Ornstein (1974), respectively.