M 393C (57695) Topics in Ergodic Theory
Time: TTH 3:30-5:00 pm
Room: RLM 10.176
Instructor: L. Bowen
Course Description:
Rigidity theory seeks to show that two mathematical objects that are equivalent in some apriori weak sense are actually equivalent in a much stronger sense. Mostow rigidity, for example, shows that lattices in a semi simple Lie group that are abstractly isomorphic are actually conjugate. Zimmer's cocycle super-rigidity shows that cocycles from actions of lattices in higher rank simple Lie groups are cohomologous to homomorphisms (under appropriate extra hypotheses). This generalizes Margulis' normal subgroup theorem that such lattices have no infinite infinite-index normal subgroups. These results form the basis and inspiration for much current research in rigidity theory.
This course begins with an introduction to ergodic theory and dynamics with an emphasis on actions of semi-simple Lie groups. We will develop enough tools to prove Zimmer cocycle super-rigidity theorem. Along the way we will study topics of independent interest such as: Poisson boundaries of random walks, stationary measures, orbit-equivalence relations, cohomology of ergodic actions, property (T), amenability and Mostow rigidity.
Prerequisites:
Will be kept to a minimum.
References:
1. Zimmer, Robert J. Strong rigidity for ergodic actions of semisimple Lie groups. Ann. of Math. (2) 112 (1980), no. 3, 511–529.
2. Zimmer, Robert J. Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984. x+209
3. Feres, Renato Dynamical systems and semisimple groups: an introduction. Cambridge Tracts in Mathematics, 126. Cambridge University Press, Cambridge, 1998. xvi+245 pp.
4. Furstenberg, Harry Boundary theory and stochastic processes on homogeneous spaces. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), 193–229. Amer. Math. Soc., Providence, R.I., 1973.
5. Furstenberg, Harry Rigidity and cocycles for ergodic actions of semisimple Lie groups (after G. A. Margulis and R. Zimmer). Bourbaki Seminar, Vol. 1979/80, pp. 273–292, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981.
6. Margulis, G. A. Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17. Springer-Verlag, Berlin, 1991. x+388 pp.