## Math 8330 - Ergodic Theory

Textbook:      There is no textbook for the course, but here are some recommended books. I will post notes and links to the various topics.
•    M. Ward: Ergodic Theory with a view toward number theory
•   H. Furstenberg: Recurrence in Ergodic Theory and Combinatorial Number Theory

Description:

In this course we study measurable dynamical systems and especially their long term behavior.  The first part of the course is devoted to introduce the basic concepts and to prove the classical recurrence and ergodic theorems due to Poincare, Birkhoff and von Neumann. Then we discuss some of the more modern developments in ergodic theory and their applications. The topics discussed may include but may not be limited to

·         Ergodicity, examples and motivation

·         The Poincare Recurrence Theorem

·         The Mean and Pointwise Ergodic Theorems

·         Weak mixing, almost periodic systems and their recurrence properties

·         Conditional measures, factors and joinings

·         Furstenberg’s Multiple recurrence and Szemeredi’s Theorem

·         Dynamics on quotients of the Nilpotent Lie groups

### We introduce the basic notions and classical results. We discuss, ergodicity, weak mixing, and almost periodicity and prove the basic ergodic and recurrence theorems, as well as some staring results for non-conventional ergodic averages.

Lecture  Notes                                                                                                                                                                       Problems and projects

Lecture 1   Motivation and examples. the Poincare recurrence theorem

Lecture 2  The mean and pointwise ergodic theorems, Khinchine's recurrence theorem

Lecture 3  Unique ergodicity, polynomial ergodic and recurrence theorems

Lecture 4  Weak mixing and multiple recurrence

II. Disintegration of Measures, Factors and Joinings

We discuss the Kronecker factor and prove the double recurrence theorem of Furstenberg. We introduce the basic technical tools, such as the disintegration of measures, ergodic decomposition, factors and the notion of relative independent joining needed for higher order multiple recurrence results.

Lecture Notes                                                                                                                                                   Problems and Projects

Lecture 5  The Kronecker factor and the ergodic Rothe theorem

Lecture 6   Disintegration of measures and the ergodic decomposition theorem

III. Multiple recurrence and nilsystems

We sketch the classical proof of the Furstenberg recurrence theorem and give an introduction to the modern theory of multiple recurrence,
based on the Host-Kra characteristic factors and the dynamics of quotients of nilpotent Lie groups.

Lecture Notes                                                                                                                                                   Problems and Projects