Breadcrumb
Rigidity phenomena in dynamical systems
Supervisor: Alexander Gorodnik
Theme: Ergodic Theory and Dynamical Systems
The abstract theory of dynamical systems studies a space X, equipped with some some geometric structure, and an action of a group G on X that preserves this structure. In this context one is intersted, for instance, in
- morphisms between dynamical systems,
- perturbations of dynamical systems,
- various objects on X preserved by the action.
It turns out that under fairly weak conditions, dynamical properties of the action and/or structure of the group G force surprisingly strong conclusions such as
- morphisms preserving a weak structure preserve a strong structure,
- perturbations are isomorphic to the orginal system,
- the invariant objects are scarse and can be completely classified.
This program has been actively persued for some classes of actions, and there are still many new instances of these phenomena too be discovered.
1. M. Gromov and P. Pansu, Rigidity of lattices: an introduction. Geometric topology: recent developments (Montecatini Terme, 1990), 39-137, Lecture Notes in Math., 1504, Springer, Berlin, 1991.
2. G. Margulis, Problems and conjectures in rigidity theory. Mathematics: frontiers and perspectives, 161-174, Amer. Math. Soc., Providence, RI, 2000.
3. R. Spatzier, An invitation to rigidity theory. Modern dynamical systems and applications, 211-231, Cambridge University Press 2004.
4. D. Witte Morris, Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005.