Ergodic Theory of Group Actions
Seminar SS 2017
Organisers: Tanja Eisner, Tobias Finis, Artem Sapozhnikov
Time/Place: Tue, 09:15-11:45, SG 3-11
If you are interested to participate please write us an email.
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
|25.4. || cancelled
|2.5.|| An introduction to amenable groups, Vishal Gupta ([1-3])
9.5. ||Mean ergodic theorem for amenable groups; pointwise ergodic theorem along Følner sequences with Tempelman condition I, Tabea Bacher ()
16.5. ||Pointwise ergodic theorem along Følner sequences with Tempelman condition II, Mike Schnurr ()
30.5||Pointwise ergodic theorem for amenable groups: the general case I, Konrad Zimmermann ()
20.6 ||Pointwise ergodic theorem for amenable groups: the general case II, Tanja Eisner () Notes
27.6 ||Unitary representations of SL(n,R) and the mean ergodic theorem, Tobias Finis (, Chapter 5)
4.7 ||Unitary representations of SL(n,R) and decay of matrix coefficients, Tobias Finis (, Chapter 5)
- A. Paterson, Amenability. American Mathematical Society, Providence, RI, 1988.
- T. Tao, Some notes on amenability, blog.
- A. Garrido, An introduction to amenable groups, lecture notes. (link)
- D. Ornstein, B. Weiss, "The Shannon-McMillan-Breiman theorem for a class of amenable groups", Israel J. Math. 44 (1983), 53–60.
- E. Lindenstrauss, "Pointwise theorems for amenable groups", Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82-90. (link)
- A. Gorodnik, A. Nevo, The ergodic theory of lattice subgroups.
Princeton University Press, Princeton, NJ, 2010.
- B. Bekka, P. de la Harpe, A. Valette,
Kazhdan's property (T). Cambridge University Press, Cambridge, 2008. (link to pdf)
- G. A. Margulis, Discrete subgroups of semisimple Lie groups. Springer-Verlag, Berlin, 1991.
- 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.
- A. Nevo, E. M. Stein,
"A generalization of Birkhoff's pointwise ergodic theorem",
Acta Math. 173 (1994), 135–154.
- R. Howe, E.-Ch. Tan,
Nonabelian harmonic analysis.
Applications of SL(2,R). Springer-Verlag, New York, 1992.