Introduction to Ergodic Theory and Applications in Number Theory
This course is given as DIAMANT mastermath course and as SFM course.
Time and Place: uesday, 10:15-13:00 hrs, Vrije Universiteit Amsterdam:
wk 38-42: WN-S655
wk 43: WN-KC137
wk 44-50: WN-S655
wk 51: WN-KC137
Starting date: The course begins in week 38, so first lecture hour is on Tuesday September 20.
Lecturers: Karma Dajani (firstname.lastname@example.org) and Charlene Kalle (email@example.com )
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, in particular
number theory will be given.
Recommended (Additional) 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.
After the completion of the course, the student is able to
1) model the evolution of random phenomena in the set up of ergodic theory
2) apply the tools of ergodic theory to predict the future behavior of stationary ergodic systems
3) identify the relationship between two ergodic systems
4) quantify the amount of information transmitted by an ergodic systems via the notion of entropy
5) construct stationary ergodic measures and to identify systems for
which one can make exact predictions instead of probabilistic ones.
6) apply ergodic theory to understand the typical behavior of various number theoretic expansions of numbers
Testing: The grade of the course is determined by Hand-in exercises (20%) one take-home exam ( 40%) and a written exam (40%).