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.