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.Literature
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%).
Exam Dates: Final Exam is on Tuesday January 6 from 10:00-13:00 at the VU room 02A00 (main building). The exam is open book, but you can ONLY use the Lecture Notes of this course.
Retake Final is on Tuesday
January 20 from 14:00-17:00 in WN-M655 (VU).