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Full Description
The purpose of this book is to provide an invitation to the beautiful and important subject of ergodic theorems, both classical and modern, which lies at the intersection of many fundamental mathematical disciplines: dynamical systems, probability theory, topology, algebra, number theory, analysis and functional analysis. The book is suitable for undergraduate and graduate students as well as non-specialists with basic knowledge of functional analysis, topology and measure theory.
Starting from classical ergodic theorems due to von Neumann and Birkhoff, the state-of-the-art of modern ergodic theorems such as subsequential, multiple and weighted ergodic theorems are presented. In particular, two deep connections between ergodic theorems and number theory are discussed: Furstenberg's famous proof of Szemerédi's theorem on existence of arithmetic progressions in large sets of integers, and the Sarnak conjecture on the random behavior of the Möbius function.
An extensive list of references to other literature for readers wishing to deepen their knowledge is provided.
Contents
Chapter 1. Preliminaries.- Part I. The Fundamentals.- Chapter 2. Measure-preserving systems.- Chapter 3. Minimality and ergodicity.- Chapter 4. Some Fourier analysis.- Chapter 5. The spectral theorem.- Chapter 6. Decompositions in Hilbert spaces.- Part II. Classical Ergodic Theorems.- Chapter 7. Classical ergodic theorems and more.- Chapter 8. Factors of measure-preserving systems.- Chapter 9. First applications of ergodic theorems.- Chapter 10. Equidistribution.- Chapter 11. Groups, semigroups and homogeneous spaces.- Part III. More Ergodic Theorems.- Chapter 12. Subsequential ergodic theorems.- Chapter 13. Multiple recurrence.- Chapter 14. Nilsystems.- Chapter 15. Gowers-Host-Kra seminorms and multiple convergence.- Chapter 16. Weighted ergodic theorems.- Chapter 17. Sarnak's conjecture.
