vi Contents §2.2. Three categories of dynamical systems 81 §2.3. Minimal dynamical systems, recurrence, and the Stone-Cechˇ compactification 88 §2.4. Multiple recurrence 98 §2.5. Other topological recurrence results 105 §2.6. Isometric systems and isometric extensions 119 §2.7. Structural theory of topological dynamical systems 134 §2.8. The mean ergodic theorem 141 §2.9. Ergodicity 152 §2.10. The Furstenberg correspondence principle 163 §2.11. Compact systems 172 §2.12. Weakly mixing systems 181 §2.13. Compact extensions 195 §2.14. Weakly mixing extensions 205 §2.15. The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem 212 §2.16. A Ratner-type theorem for nilmanifolds 217 §2.17. A Ratner-type theorem for SL2(R) orbits 227 Chapter 3. Lectures in Additive Prime Number Theory 239 §3.1. Structure and randomness in the prime numbers 239 §3.2. Linear equations in primes 248 §3.3. Small gaps between primes 259 §3.4. Sieving for almost primes and expanders 267 Bibliography 277 Index 291

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