##
MATH 6180 Ergodic Theory

Fall 2008

### Course Information

Introduction to general ergodic theory, followed by entropy theory and applications to homogeneous spaces and number theory. Topics include ergodicity, mixing, recurrence, Lyapunov exponent, entropy and ergodic theory on homogeneous spaces, and possibly a brief introduction to ergodic Ramsey theory.

We will cover only the very basic examples for ergodic theory on homogeneous spaces (mostly SL_{2}(R)-action on the hyperbolic plane). Professor John Smillie will give a second semester ergodic theory course on dynamics of flows on homogeneous spaces and moduli spaces (based on the example of SL_{2}(R) in Spring 2009. Some of the topics covered in this course will be explored more in Bernstein Seminar, Spring 2009.

** Time: T TH 8:40 a.m. - 9:55 a.m.**

** Classroom: MLT 206**

** Instructor:** Seonhee Lim

E-mail:slim@math.cornell.edu email

**Office:** Malott 593

** Office Hours:** T 11:30 a.m. - 12:30 p.m. TH 10 a.m. -11 a.m.

### References

**Basic References in ergodic theory:**
K. E. Peterson, Ergodic Theory (Cambridge Studies in Advanced Mathematics)
P. Walters, An Introduction to Ergodic Theory (GTM, Springer)
M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory
B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications)
I. Cornfeld, S. Fomin, Y. Sinai, Ergodic Theory

** Ergodic theory towards Number Theory (additive combinatorics)**
M. Einsiedler, T. Ward, Ergodic Theory: with a view towards Number Theory (pdf)
B. Green's lecture note of Lent 2008 Part III course on Ergodic Theory (pdf)
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (Porter Lectures)

**More Advanced References (on ergodic theory for group actions on homogeneous spaces):**
There is an excellent paper by M. Einsiedler on Ratner's theorem on SL_{2}(R) invariant measures. (Section 2.1. will be helpful to see SL_{2}(R) action on the hyperbolic plane.) You can find the paper on the webpage of Einsiedler.
D. Morris, Ratner's Theorems on Unipotent Flows (pdf)
P. Nicholls, The Ergodic Theory on Discrete Groups
M. Bekka, M. Mayer, Ergodic Theory and Topological dnamics of Group Actions on Homogeneous Spaces

### Assignments

home