Introduction to Ergodic Theory and Applications in Number Theory
This course is given as DIAMANT mastermath course and as SFM course.

Time and Place: Tuesday from 14:00-16:45 at the VU room WN-C121 Mathematics Building.

Starting date: The course begins in week 38, so first lecture hour is on Tuesday September 16.

Lecturers: Karma Dajani (k.dajani1@uu.nl) and Charlene Kalle (kallecccj@math.leidenuniv.nl )

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, in particular number theory will be given.

Literature

Lecture Notes

Learning Goals:

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).

Important Rules and Dates of Homework and Exams are summarized in the following Slides

Material to be covered (including homework):
- Tuesday September 16: Begin Chapter 1 sections 1.1-1.3.

-Tuesday September 23: Chapter 1 sections 1.4-1.7.  Hand in exercises 1.3.4 p.16 and 1.7.3 p.25 to be handed in on Tuesday September 30.

-Tuesday September 30: Finish Chapter 1 (section 1.8).

-Tuesday October 7: Chapter 2, section 2.1

-Tuesday October 14: Chapter 2, rest of section 2.1 + 2.3. Hand in exercise 2.1.1 p.41 and 2.3.4 p.52  to be handed in on Tuesday October 21.

-Tuesday October 21: Chapter 2 section 2.2.

-Tuesday October 28: Chapter 3 (long lecture). Take-home exam is available for download  (question 3 has been updated, in question3 \epsilon_0=1 and not 0)

-Tuesday November 4: no classes, you should work on your take-home exam.

-Tuesday November 11: Chapter 4 sections 4.1 + 4.2.  Hand in exercise 4.1.1 p.69 and 4.2.1 p.74  to be handed in on Tuesday November 25.

-Tuesday November 18: Chapter 4 rest of section 4.2 + 4.3 (long lecture).

-Tuesday November 25: Chapter 5 sections 5.1+5.2 +5.3.

-Tuesday December 2: Chapter 5 sections 5.4+5.5 (long lecture). Hand in exercise 5.4.5 p.97 and 6.2.1  p.115  to be handed in on Tuesday December 16.

-Tuesday December 9: (rest of chapter 5 if need be) Chapter 6: section 6.1.

-Tuesday December 16: Chapter 6: section 6.2. End of course.

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.