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

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

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.

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.