This book is designed as an introduction into what I call 'abstract' Topological Dynamics (TD): the study of topological transformation groups with respect to problems that can be traced back to the qualitative theory of differential equations. So this book is in the tradition of the books [GH] and [El].
The title ('Elements . . .' rather than 'Introduction . . .') does not mean that this book should be compared, either in scope or in (intended) impact, with the 'Elements' of Euclid or Bourbaki. Instead, it reflects the choice and organisation of the material in this book: elementary and basic (but sufficient to understand recent research papers in this field). There are still many challengeing problems waiting for a solution, and especially among general topologists there is a growing interest in this direction. However, the technical inaccessability of many research papers makes it almost impossible for an outsider to understand what is going on. To a large extend, this inaccessability is caused by the lack of a good and systematic exposition of the fundamental methods and techniques of abstract TD. This book is an attempt to fill this gap.
The guiding principle for the organization of the material in this book has been the exposition of methods and techniques rather than a discussion of the leading problems and their solutions, though the latter are certainly not neglected: they are used as a motivation wherever possible. As a rule, clarity of the exposition has had a higher priority than the completeness of the included material (though it was hard to resist my natural inclination to be encyclopedic). In addition, I have included an abundance of examples, as illustration of results, as 'testcases' for techniques, and often also for their own interest.
The book can be divided in two parts: the Chapters I-III (actions of R and Z only, and not exclusively concerned with minimality) form part one, and the Chapters IV-VI (actions of arbitrary topological groups, with the accent on minimal flows and their extensions) form part two. The theory in the second part is independent of the first part, but for many examples in part two we refer to part one. A description of the contents of the various chapters can be found in the introductory remarks in each chapter. However, at this place I want to make a remark about Chapter I. This chapter is not needed for the understanding of the rest of the book in a technical sense. But mathematics is more than just formal techniques and results. Though I am not enough of a philosopher to be able to explain briefly what exactly there is more to it, it is my opinion that a good mathematician should know at least how his specialism came into being: what are its roots, what are its connections with other fields of mathematics? Chapter I gives a concise answer to these questions: it describes in a bird's flight a large part of the field of Dynamical Systems and contains historically orientated motivations for various problems and notions studied in this field.
In every chapter (except Chapter I) the material is organized at three levels: first there is a systematic and essentially self-contained exposition of the theory, then there is a collection of 'Illustrations', containing miscellaneous results, applications and examples (presented as exercises with hints), and finally there is a set of 'Notes', containing references to the sources, additional results (usually without proofs) and references to related material. In this way I hope the book is also of value for specialists. To increase its value as a reference work there are many cross-references in the book; in addition, I have included an extensive subject index.
Prerequisites for reading the book are a working knowledge of general topology, and familiarity with the elements of (Lebesgue) integration theory; also some functional analysis will be useful. For easy reference I have included, in four appendices, some material from these fields; in a fifth appendix I present the elements of the theory of topological transformation groups as needed in this book.
This book was conceived and written at the CWI (the Centre for Mathematics and Computer Science) in Amsterdam, at the end of the paradisiac period in which justification of mathematical research was not yet restricted to industrial and computational applicability. The idea to write this book was born in the early eighties, when Jaap van der Woude and I organized a seminar on Topological Dynamics at the Mathematical Centre (as the CWI was then called).
In this book one will find no revolutionary new results, but for many details the presentation is new. In particular, my presentation of the structure theorems for distal and point-distal extensions of compact minimal flows is entirely based on relatively invariant measures and avoids the use of tau-topologies. However, I could not always trace back whose ideas I used: those of my own, those of Jaap van der Woude, or those of other participants of our seminar at the MC (in particular, the late Jan Aarts and the late Ietje Paalman-de Miranda). But of course, the full responsability for all mistakes is for me.