Ergodic theory / Karl Petersen

For an introduction and general survey, Petersen's book is an excellent choice. Chapter One provides a general introduction to ergodic theory including the basic facts needed from measure theory and functional analysis. In Chapter Two the fundamental mean and pointwise ergodic theorems are proved. Recurrence and mixing are also introduced. In Chapter Three, additional results for pointwise convergence are obtained including the dominated ergodic theorem, local ergodic theorem, and the Chacon-Ornstein theorem. In Chapter Four additional results for recurrence are discussed and topological dynamics is introduced. Furstenberg's approach to Szemerédi's theorem is then considered as well as topological multiple recurrence, van der Waerden's theorem, and Hindman's theorem. Entropy and generators are introduced in Chapter Five and the Kolmogorov-Sinaĭ theorem is proved. Entropy is discussed further in Chapter Six, and the Ornstein theory is introduced. The book concludes with a detailed discussion of the Keane-Smorodinsky construction of a finitary isomorphism between Bernoulli shifts on finite alphabets with the same entropy.
The above is a brief description of the main topics covered. The book also has an abundance of intuitive detailed discussions of additional results and examples. In particular, there is a discussion of the Hilbert transform in Chapter Three, the Jewett-Krieger representation theory in Chapter Four, a proof that Chacon's transformation is prime in Chapter Four, and the computation of entropies in Chapter Six. Exercises are also provided at the end of each chapter. It is clear that a lot of effort went into making the presentation suitable for a graduate level course following an introduction to real analysis. (MathSciNet)

Bibliogr. p. [302]-321. Index