Smooth Ergodic Theory is the study of dynamical systems on smooth manifolds from a probabilistic and statistical perspective.

In this course we will focus on some relatively simple systems which are highly “chaotic” (unpredictable) and show that they are nevertheless very “regular” (predictable) from a statistical point of view. This can be observed in many real-life dynamical systems, from simple coin-tossing to the weather, where short term outcomes are unpredictable but long term averages are very stable. The underlying philosophical purpose of the course is to try to understand the mechanisms which allow these two apparently contradictory features to co-exist.

More technically, the course will begin with a survey of some basic dynamical systems (contraction maps, circle rotations, full branch maps, symbolic systems,..), then introduce various concepts (invariant measures, ergodic measures, physical measures...), results (Poincarè Recurrence, Birkhoff’s Ergodic Theorem, and techniques (distortion calculations, push forward of measures,...) through which their statistical properties can be understood.

It is a first introductory course on the subject and there are no particular prerequisites except for standard elementary notions in topology and measure theory.