Education activities of karma: Masters level.
Ergodic Theory 2012/2013.
This course forms part of the Masters
in Stochastics and Financial Mathematics.
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.
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
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.
exam (40%), an oral exam (40%) and Hand-In exercises (20%).
Introduction to Ergodic Theory by Peter Walters, Springer
Theory by Karl Petersen, Cambridge University Press.
Topics in Ergodic Theory, Cambridge University Press.
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
Theory: with a view towards Number Theory by Manfred
Einsiedler and Thomas Ward.
- Invitation to ergodic theory by Cesar
Mathematical Library, 42. American Mathematical
Society, Providence, RI, 2008. x+262 pp. ISBN:
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
- 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.
November 7: there is
- Wednesday November 14:
section 4.3 +4.4. Hand
in exercises: 4.3.1, 4.3.2, 4.4.2 due date November 28.
November 21: section 6.1
November 28: section 6.2.
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