Basic Ergodic Theory

This book treats mainly some basic topics of ergodic theory in a revised form, bringing into focus its interactions with classical descriptive set theory more than is normally the practice. The presentation has a slow pace and can be read by anyone with a background in measure theory and point set topology. In particular, the first two chapters, the core of ergodic theory, can form a course of four to six lectures at third year B. Sc. , M. Sc. , or M. Phil. level in Indian Universities. I have borrowed freely from existing texts ( with acknowledgements) but the overall theme of the book falls in the complement of these. G. W. Mackey has emphasised the need to look at group actions also from a purely descriptive standpoint. This helps clarify ideas and leads to sharper theo- rems even for the case of a single transformation. With this in view, basic topics of ergodic theory such as the Poincare recurrence lemma, induced automorphisms and Kakutani towers, compressibility and HopPs theorem, the Ambrose represen- tation of flows etc. are treated at the descriptive level before appearing in their measure theoretic or topological versions. In addition, topics centering around the Glimm-Effros theorem are discussed. These topics have so far not found a place in texts on ergodic theory. Dye's theorem, proved at the measure theoretic level in Chapter 11, when combined with some descriptive results of earlier chapters, becomes a very neat theorem of descriptive set theory.