Ergodic Theory and Additive Combinatorics

Master Lecture: Department of Mathematics, SNSB (Scoala Normala Superioara Bucuresti), Summer Semester 2013

Lecturer: Laurentiu Leustean


Lecture Notes version 05.07.2013

Lectures :

Lecture 1: a general presentation of the course.

Lecture 2: Topological Dynamical Systems: definitions, examples.

Lecture 3: Basic constructions continued: homomorphisms, (strongly) invariant sets, subsystems, direct products, disjoint unions. Transitivity.

Lecture 4: Minimality. Recurrence.

Lecture 5: Application to a result of Hilbert, presumably the first result of Ramsey Theory.

Lecture 6: Multiple Recurrence Theorem.

Lecture 7: Ramsey Theory: van der Waerden Theorem.

Lecture 8: Ultrafilter approach to Ramsey Theory.

Lecture 9: Ergodic Theory: measure-preserving systems, induced operator.

Lecture 10: Bernoulli shift. Recurrence.

Lecture 11: Ergodicity. Maximal ergodic theorems. Birkhoff ergodic theorem


Seminars:

Seminar Sheets: [1], [2], [3], [4], [5], [6], [7], [8], [9]

Solutions: [1], [2], [3], [4], [5], [6], [7], [8], [9]



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