Location

Time:Wednesday 13:15-15:00 WIS 610. On October 24 the lecture will take place in room 503.

The course starts Wednesday September 12.

Announcement: On October 24, the lecture will
take place in room 503 (math building) instead of 610.

Course Description

The roots of ergodic theory go back to Boltzmann's ergodic hypothesis concerning the equality of the time mean and the space mean of molecules in a gas, i.e., the long term time average along a single trajectory should equal the average over all trajectories. The hypothesis was quickly shown to be incorrect, and the concept of ergodicity (`weak average independence') was introduced to give necessary and sufficient conditions for the equality of these averages. Nowadays, ergodic theory is known as the probabilistic (or measurable) study of the average behavior of ergodic systems, i.e., systems evolving in time that are in equilibrium and ergodic. The evolution is represented by the repeated application of a single map (in case of discrete time), and by repeated applications of two (or more) commuting maps in case of `higher dimensional discrete time'. The first major contribution in ergodic theory is the generalization of the strong law of large numbers to stationary and ergodic processes (seen as sequences of measurements on your system). This is known as the Birkhoff ergodic theorem. The second contribution is the introduction of entropy to ergodic theory by Kolmogorov. This notion was borrowed from the notion of entropy in information theory defined by Shannon. Roughly speaking, entropy is a measure of randomness of the system, or the average information acquired under a single application of the underlying map. Entropy can be used to decide whether two ergodic systems are not `the same' (not isomorphic).

With a basic knowledge of measure theory, the notions of measure preserving (stationarity), ergodicity, mixing, isomorphism and entropy will be introduced. Also applications to other fields such as probability theory and number theory will be given.

**Literature**

- An Introduction to Ergodic Theory by Peter Walters, Springer Verlag.
- Ergodic Theory by Karl Petersen, Cambridge University Press.
- Parry, Topics in Ergodic Theory, Cambridge University Press.
- Ergodic theory of numbers by K. Dajani and C. Kraaikamp, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC 2002.
- Introduction to dynamical systems by M. Brin
and Stuck G.
*Cambridge University Press, Cambridge,*2002. xii+240 pp. ISBN: 0-521-80841-3 - Ergodic
Theory: with a view towards Number Theory by
*Manfred Einsiedler and Thomas Ward.*

- Invitation to ergodic theory by Cesar
Silva.Student
Mathematical Library, 42.
*American Mathematical Society, Providence, RI,*2008. x+262 pp. ISBN: 978-0-8218-4420-5. - Lecture Notes on Ergodic Theory by Omri Sarig.

Material Covered + Assignments (To be found in the Lecture Notes):

- Wednesday September 12: sections 1.1-1.3. Hand in exercises: 1.3.1, 1.3.2, 1.3.3, due date September 19.
- Wednesday September 19: sections 1.4-1.6. Hand in exercises: 1.5.1, 1.5.2, 1.5.4, due date September 26.
- Wednesday September 26: sections 1.7, 1.8. Hand in exercises: 1.7.1, 1.8.1, 1.8.2, 1.8.3 (assume in 1.8.3 that beta is the golden mean), due date October 3.
- Wednesday October 3: section 2.1. Hand in exercises: 1.8.4, 2.1.1, 2.1.2, due date October 10.
- Wednesday October 10 : section 2.2, 2.3. Hand in exercises: 2.2.1, 2.3.1, 2.3.2, 2.3.3 due date October 17.
- Wednesday October 17: finished chapter 2. No hand in exercises

- Wednesday October 24: section 3.1 + 3.2. Hand in exercises: 3.1.1, 3.1.2, 3.1.3, 3.2.1 due date October 31.
- Wednesday October 31: section 4.1 + 4.2. Hand in exercises: 4.2.2, 4.2.3, 4.2.4 due date November 21. The take home midterm exam will be e-mailed to you on October 31 (it covers Chapters 1,2 and 3). The due date is November 14.
- Wednesday November 7: there is no lecture.
- Wednesday November 14: section 4.3 +4.4. Hand in exercises: 4.3.1, 4.3.2, 4.4.2 due date November 28.
- Wednesday November 21: section 6.1
- Wednesday
November 28: section 6.2.

k.dajani1@uu.nl