Topics in ergodic theory

   Math 542, Tu-Th 11-12:15, MATX 1118
Description:  The aim of the course is to develop the methods of topological and mearure preserving dynamics used in applications to Ramsey theory and additive combinatorics. The prerequisite for the course is a solid knowledge of measure theory, Hilbert space theory and some functional analysis (at the level of Rudin's: Real and Complex Analysis)
Textbook:  There is no strict textbook for the course. We'll use B. Green's notes  and to a lesser extent T. Tao's Blog/Notes . We'll also use Furstenberg's book: Recurrence in Ergodic Theory and Combinatorial Number theory to a large extent. (feel free to borrow it from me).  Notes will also be uploaded later for parts of the material not covered in the references given.
        
          Instructor: Akos Magyar, Office Hrs: Tue- Thrs 3:30-4:30, Office: Math 229E, Ph.: 822-3045, Email: magyar@math.ubc.ca

I. Toplogical dynamics

We will study topological dynamical systems, discuss the Bernoulli systems, and the existence of minimal systems. Prove the multiple recurrence theorem of Birkhoff and use it to show various versions of  van der Waerden's theorem on the existence of monochromatic arithmetic progressions. Finally discuss group extensions and further applications to number theory.

Basic Notes                                                                                                    Supplementary  Notes                                                       

Green's notes 1: Topological dynamics                                                             Combinatorial proof of van der Waerden's Theorem                                                           
Green's notes 2: Topological dynamics                                                             Monochromatic 2-dimensional corners
                                                                                                                       
             

II. Basic ergodic and recurrence theorems


We will study measure preserving dynamical system, discuss ergodicity, the basic mean and pointwise ergodic theorems and give applications to number theory. We introduce the notion of a factor discuss the "invariant" and the "rational" factors and use them to prove the basic recurrence theorems of Poincare, Khinchin and Sarkozy-Furstenberg. Finally discuss the Fursetenberg correspondence principle, compact and weak mixing systems, and prove Roth' theorem on 3-term arithmetic progressions.

Green's notes 3: Basic ergodic theorems                                                      
Green's notes 4: Applications of ergodic theorems
Green's notes 6: The Furstenberg-Sarkozy theorem
Roth' Theorem: The ergodic approach



III. Structure of measure preserving systems and multiple recurrence
We'll discuss the notion of factors, regular factors, disintegration  of measures, relative products, compact and weak mixing extensions and prove the Furstenberg structure theorem and Szemeredi's theorem on the existence of arithmetic progressions in sets of positive density. We'll follow Furstenberg's book, lecture notes will be uploaded later.

 
IV. Further results on multiple recurrence

If time permits we'll also discuss the chracteristic factors for mutiple recurrence, such as nilsystems. A reference for this more recent point of view is:

B. Kra: Ergodic methods in additive combinatorics