Seminar on "Ergodic Theory of Group Actions"

In the last decade ergodic theorems for actions of non-amenable groups on probability spaces have been actively researched by A. Nevo and collaborators. An interesting feature is that for a broad class of groups, such as semisimple Lie groups with Kazhdan's property (T), the exponential convergence rate in the ergodic theorem arises naturally without any spectral assumptions, which is not the case, for instance, for actions of amenable groups.

The aim of this seminar is to understand in depth various techniques behind proofs of ergodic theorems for amenable and non-amenable groups. After reviewing ergodic theorems in the case of amenable groups, we proceed with that of non-amenable ones, following in parts the book "The ergodic theory of lattice subgroups" by A. Gorodnik and A. Nevo. Necessary tools will be introduced and discussed.

References

1. A. Paterson, Amenability.  American Mathematical Society, Providence, RI, 1988.
2. T. Tao, Some notes on amenability, blog.  
3. A. Garrido, An introduction to amenable groups, lecture notes.
4.  D. Ornstein, B. Weiss, "The Shannon-McMillan-Breiman theorem for a class of amenable groups", Israel J. Math. 44 (1983), 53–60.
5. E. Lindenstrauss, "Pointwise theorems for amenable groups", Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82-90. (link)
6. A. Gorodnik, A. Nevo, The ergodic theory of lattice subgroups.   Princeton University Press, Princeton, NJ, 2010.  
7. B. Bekka, P. de la Harpe, A. Valette,  Kazhdan's property (T). Cambridge University Press, Cambridge, 2008. (link to pdf)
8. G. A. Margulis, Discrete subgroups of semisimple Lie groups. Springer-Verlag, Berlin, 1991.
9. G. A. Margulis, A. Nevo, E. M. Stein, "Analogs of Wiener's ergodic theorems for semisimple Lie groups, II", Duke Math. J. 103 (2000), 233–259.
10. A. Nevo, E. M.  Stein, "A generalization of Birkhoff's pointwise ergodic theorem", Acta Math. 173 (1994), 135–154.
11. R. Howe, E.-Ch. Tan,  Nonabelian harmonic analysis. Applications of SL(2,R). Springer-Verlag, New York, 1992.

Date and time info
Tue, 09:15-11:45

Language
English