Introduction to Ergodic Theory

This is the course page for the first year master level course "Introduction to Ergodic Theory."  This course is listed as "MM13: Grundmodul Geometry un Topologie" and is a course in "Pure mathematics."  LSF

We will meet on Tuesday and Thursday from 11:00 - 13:00 in SR4 **Note that this room is currently over capactiy** starting April 18 and ending [July 27].  Exams will take place on the weeks of [weeks of exams].  There will be exercise sessions on Wednesdays 16-18 in the Hörsaal (except for May 3 and May 24) lead by Fernando Camacho Cadena and Colin Davalo.

Exercise sheet 1.  Due May 3 to Colin and Fernando.  You may work in small groups and submit solutions together.  No more than 4 per group.

Exercise sheet 2 Pick 5 of 6 prolblems to turn in (note: the first part of problem 5 is incorrect as stated, so perhaps avoid this one). Due May 10

Exercise sheet 3 Due May 17

Exercise sheet 4 Due May 24

No exercises due May 31

Exercise sheet 5 Due June 7

Solutions to selected exercises sheets 1-4

Exercise sheet 6 Due June 14

Exercise sheet 7 Due June 28 (note that no exercises are due June 21)

Exercise sheet 8 Due July 5

Exercise sheet 9 Due July 12

Exercise sheet 10 Due in class July 20

Some more solutions (sheets 6, 7, 8 and 10)

Here's a brief abstract for the course:

We are interested in the long term behavior of measure preserving dynamical systems, i.e., measurable maps T: X -> X preserving a (probability) measure m on X. That is, we want to answer the question, "what happens to the typical point x in X after n applications of T as n goes to infinity?"

The Poincaré Recurrence Theorem says that almost every point recurrs infinity often to a set of positive measure under iteration.  Thus there are non-trivial dynamics (as long as T preserves a probability measure). 

An ergodic probability system T: X ->X does not break up into smaller measurable dynamical systems.  They are the basic building blocks for measure preserving dynamics, and every probability measure preserving dynamical system can be decomposed into it ergodic components, which then can be studied individually. 

 The Birkhoff Ergodic Theorem (analogous to the Strong Law of Large Numbers from probability theory) says that the average value of a function sampled along the orbit of a point converges to the average value of the function for ergodic prorbabily systems.  This theorem is extremely useful, and we will see many interesting and wide ranging applications (to seeminly unrelated areas of mathematics).

After establishing the basic thoery of (ergodic) probability measure preserving dynamical systems and proving the ergodic theorems, we will study continuous dynamical systems with many exmaples and applications in number theory and geometry.

A selection of topics of a more geometrical and topological nature will be presented, as well.  Highlights may include dynamics of interval exchange transformations, (rigidity of) circle rotations, geodesic flow on a hyperbolic surface, and the dynamics of the Gauss map and continued fraction expansions (relationship also with hyperbolic geometry and Teichmüller space of the torus).  

Prerequisites: 

I will assume a working knowledge of basic measure theory and functional analysis (L^p spaces, dual spaces, and linear operators).  We will review many of these concepts in the first few weeks of the course, however.  We may also dive into some more differential geometric topics related to hyperbolic geometry, the modular surface, and complex structures on the torus.  These topics would require some familiarity with algebra, complex analysis, differential geometry, etc. (but none of these at a very high level).  

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