viii Preface notes for my course on topological dynamics and ergodic theory, which origi- nated in part from Poincar´ e’s pioneering work in chaotic dynamical systems. Many situations in mathematics, physics, or other sciences can be modeled by a discrete or continuous dynamical system, which at its most abstract level is simply a space X, together with a shift T : X X (or family of shifts) acting on that space, and possibly preserving either the topological or measure-theoretic structure of that space. At this level of generality, there are a countless variety of dynamical systems available for study, and it may seem hopeless to say much of interest without specialising to much more concrete systems. Nevertheless, there is a remarkable phenomenon that dy- namical systems can largely be classified into “structured” (or “periodic”) components, and “random” (or “mixing”) components,1 which then can be used to prove various recurrence theorems that apply to very large classes of dynamical systems, not the least of which is the Furstenberg multiple re- currence theorem (Theorem 2.10.3). By means of various correspondence principles, these recurrence theorems can then be used to prove some deep theorems in combinatorics and other areas of mathematics, in particular yielding one of the shortest known proofs of Szemer´ edi’s theorem (Theorem 2.10.1) that all sets of integers of positive upper density contain arbitrarily long arithmetic progressions. The road to these recurrence theorems, and several related topics (e.g. ergodicity, and Ratner’s theorem on the equidis- tribution of unipotent orbits in homogeneous spaces) will occupy the bulk of this course. I was able to cover all but the last two sections in a 10-week course at UCLA, using the exercises provided within the notes to assess the students (who were generally second or third-year graduate students, having already taken a course or two in graduate real analysis). Finally, I close this volume with a third (and largely unrelated) topic (Chapter 3), namely a series of lectures on recent developments in additive prime number theory, both by myself and my coauthors, and by others. These lectures are derived from a lecture I gave at the annual meeting of the AMS at San Diego in January of 2007, as well as a lecture series I gave at Penn State University in November 2007. A remark on notation For reasons of space, we will not be able to define every single mathematical term that we use in this book. If a term is italicised for reasons other than emphasis or definition, then it denotes a standard mathematical object, result, or concept, which can be easily looked up in any number of references. 1 One also has to consider extensions of systems of one type by another, e.g. mixing extensions of periodic systems see Section 2.15 for a precise statement.
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