Information and randomnes

This book presents a sample of exciting topics under development at the forefront of probability theory and ergodic theory. It benefit from the authors's research and teaching experience, and should help to make the detailed mathematical techniques of the subject accessible to a much wider audience (master or Ph.D. students). It is relevant for researchers and students in probability, analysis, dynamical systems and information sciences.

The first chapter presents recent progress on the fundamental isomorphism problem between noninvertible measure preserving maps. This new theory is applied to four natural classes of measure preserving maps, including rational maps of the Riemann sphere.

The second chapter is devoted to presenting the main ingredients of the theory that is allowing researchers to tackle long standing questions about the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. The results presented here represent a major advance in ergodic theory over the last five years.

The final chapter deals with scenery reconstruction, a subject which has been an extremely hot topic in recent years in probabilty theory. This chapter provides an introduction to scenery reconstruction and gives an overview of the large variety of techniques that have been employed to solve this problem in different settings.