In this course we study measurable dynamical systems and especially their long term behavior. This concept has initially emerged in statistical physics but by now is connected with other areas such as combinatorics, number theory and geometry. 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

· Continued fractions and Diophantine approximation

· Conditional measures, factors and joinings

· Furstenberg’s Multiple recurrence and Szemeredi’s Theorem

· Dynamics on quotients of the Hyperbolic Plane

We will develop most concepts however basic familiarity with measure theory and functional analysis is helpful. There is no official textbook but will follow relevant chapters from the book: “Ergodic Theory: With a view towards Number Theory” by Einsiedler and Ward.