Prerequisites: Standard measure theory, differentiable manifolds, basic Lie groups, basic functional analysis. Some familiarity with basics of ergodic theory and hyperbolic dynamics is desirable but not completely necessary.
1. Standard examples of smooth actions of higher-rank abelian groups: automorphisms of tori and nil-manifolds, Weyl chamber flows, twisted Weyl chamber flows, futher modifications.
2. Topological and differentiable conjugacy, orbit equivalence, structural stability, moduli and differentiable rigidity.
3. Overview of hyperbolic dynamics. Absence of differentiable rigidity for diffeomorphisms and flows. Anosov, normally hyperbolic, and partially hyperbolic actions of higher-rank abelian groups.
4. Cocycles and invariant distributions.
5. Various types of rigidity for smooth actions of higher-rank abelian groups: cocycle rigidity, local and global differentiable rigidity, rigidity of invariant measures.
6. Detailed discussion of the simplest nontrivial example: Cartan action of Z^2 on the three--dimensional torus.
7. Methods for proving cocycle rigidity and differentiable rigidity.
8. Beyoud uniformly hyperbolic actions. Mutliplicative ergodic theorem for actions of higher-rank abelian groups.
9. Elements of Pesin theory for nonuniformly hyperbolic actions of higher-rank abelian groups.
10. Invariant geometric structures and global rigidity.
Zeit: Freitag, 10 - 12 Uhr
Ort: HG G 43 (Hermann-Weyl-Zimmer)
Beginn: 12. April