This book is just what its subtitle says: an introduction. It is addressed primarily to graduate students who want to learn the basic ideas of topological dynamics. Thus, the fundamental notions of (topological) dynamical systems are defined and their elementary properties are discussed. Students who have mastered this book will have a firm basis to start research related with the topics discussed here. Unfortunately, in order to keep this book reasonably sized many important topics could not be included (or even mentioned). The choice of which topics to include is to a great extend determined by my wish to concentrate on the purely topological aspects of the theory, in the spirit of the work by Birkhoff.
When I wrote my previous book Elements of Topological Dynamics, '(topological) dynamics' was a relatively unknown topic. At present the situation is completely different and one might say that this topic is quite popular, to the extend that introductory courses in differential equations are sometimes called courses in dynamical systems. On the other hand, the purely topological approach to dynamical systems theory gets less attention than it deserves. Therefore I decided to publish the lecture notes of a course in 'applied topology' that I gave at the Free University in Amsterdam from 1995 until my retirement in 2002.
This book treats systems consisting of a topological space (the phase space) and iterations of a single continuous mapping of this space into itself (the phase mapping). The assumption that the phase space is metrizable is avoided as much as possible.
The book is organized as follows. In the Introduction the 'dynamical systems approach' is explained: the philosophy behind the material in the book and the red thread through the subsequent chapters. The examples at the end of the Introduction are heavily used later on (but reading them can be postponed until they are needed).
The Chapters 1 and 2 together with the examples in the Introduction are used throughout the remainder of the book. In Chapter 1 the basic notions of the theory are defined, the elementary properties of dynamical systems are discussed and illustrated by examples. Chapter 2 treats dynamical systems on intervals in R; it culminates in a proof of Šarkovskij's Theorem.
The Chapters 3 and 4 are about stability: stability of invariant sets and variants of Poisson-stability (which we call 'recurrence'), including almost periodic and non-wandering points. Chapter 3 also discusses attraction, though we refrain from giving a formal definition of an 'attractor'. The chapter ends with a discussion of the space of components of a transitive (asymptotically) stable set in a locally connected locally compact phase space --- providing full proofs of statements that have unconvincing proofs elsewhere. The discussion of recurrence and almost periodicity in Chapter 4 is restricted to a bare minimum because there is much other literature about these topics.
Next, in Chapter 5 we discuss shift systems (spaces of sequences with the shift operator) and in Chapter 6 we investigate how such systems can be used to represent other systems by means of a suitable coding (symbolic dynamics). The study of shift systems has a strong algebraic/combinatorial flavour and it has many applications in and points of contact with other parts of mathematics, from artificial languages to coding theory and from automata theory to probability theory. We discuss none of these applications; the interested reader is referred to the books "An introduction to symbolic dynamics and coding" by D. Lind and B. Marcus and "Symbolic Dynamics" by B. Kitchens. We give only some applications of symbolic dynamics to 1-dimensional systems; these are used in Chapter 8 to compute the topological entropy of those systems.
Chapter 7 deals with notions of chaos. There are many definitions of this notion, all about 'erratic behaviour'. We concentrate on sensitive dependence on initial conditions and the existence of large so-called 'scrambled' sets. In Chapter 8 we discuss the notion of topological entropy. We include a proof of the fact that, for maps of an interval into itself, positive entropy is equivalent to the existence of a point with odd primitive period greater than 1 which, by the results of Chapter 7, implies chaos.
The results of the Chapters 3 and 4 are not needed for a good understanding of the later chapters, though in Chapter 5 some examples are given that illustrate notions and results from these previous chapters. Similarly, the Chapters 5 and 6 are not needed for an understanding of the Chapters 7 and 8, though also here examples in the latter chapters are taken from the former. So possible courses can be be based on the Chapters 1 and 2 followed by either the Chapters 3 and/or 4, or 5 and 6, or 7 and 8.
Every chapter concludes with a set of exercises. Most of them are routine applications of the material in the chapter, others deal with extensions of the theory. For the more challenging ones one may find hints (or if you prefer: telegramm-style answers) at the end of the book. The exercises are followed by a section of Notes in which also references to the literature are given. These Notes are rather sketchy and the references are far from complete. In particular, they are not meant as complete historical introductions; rather, they tell how the results came to me. In point of fact, many results included in the book are common knowledge and are known already for decennia. My knowledge of dynamical systems grew over a rather long period and often I lost track of where I read or heard the various results. I have tried to give credit to whom it deserves, but for many results in this book references to the original sources are missing.
The prerequisites for understanding this book are are rather modest. A reader who has mastered a course in General Topology and has a working knowledge of Calculus should be able to follow all arguments. For a good understanding of the Introduction some familiarity with the theory of differential equations is useful. For easy reference there are two Appendices at the end of the book: one with the preliminaries from general topology and a second one about the Cantor set.
