Invitation to Ergodic Theory
Another introductory book on ergodic theory enters the rich collection that was started by P. R. Halmos and his excellent Lectures on Ergodic Theory in 1956 with more recent additions from M.G. Nadkarni, Basic Ergodic Theory, Birkhäuser, 1995, and M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge, 2002. This book is meant as a ‘special topics’ course aimed at students fairly familiar with elements of real analysis, while Lebesgue measure and integration are developed as needed in chapters 1 and 4, to be later on applied in measurable dynamics. Some metric space topology (including the Baire category theorem) is presented in appendix B to support a treatment of topological dynamics. The core of the text is to be found in chapters 2, 5 and 6, where concepts such as recurrence, ergodicity and mixing are treated. The author develops in detail several examples to illustrate concepts of classical ergodic theory (such as the baker’s transformation, irrational rotations, the dyadic odometer and the Gauss, Kakutani and Chacon transformations). The text is accompanied by some exercises and open problems formulated with the aim of providing the reader with an orientation to current research in ergodic theory.