Cohomology of Dynamical Systems

The theme of this project is the cohomology of dynamical systems and applications of cohomological methods in ergodic theory, differentiable dynamics and related subjects. Although cohomological methods have been among the standard tools in dynamics almost since the beginning of the subject, recent years have seen some interesting new developments in this area due to the discovery of a number of unexpected rigidity properties of certain classes of dynamical systems (especially of Z-d-actions with d>1). Roughly speaking, these rigidity properties mean that certain usually abundant objects are very rare for these actions: they have few invariant measures, are difficult to `deform`, have few isomorphisms and automorphisms, or they have very few cocycles with values in certain classes of groups. The extent of - and the connections between - these different rigidity properties are quite subtle and not very well understood. In addition to the study of some of the more classical aspects and applications of dynamical cohomology (especially of connections between cohomology and probability theory, nonabelian cohomology and Livsic theory) this project is hoping to contribute to the understanding of cohomological rigidity properties of algebraic and symbolic Z-d-actions, and to the connection between cohomological rigidity, isomorphism rigidity and other properties of actions of higher-rank abelian groups currently under investigation.

The aim of this project was to study the cohomology of dynamical systems and applications of cohomological methods in dynamics and related subjects. The mathematical theory of dynamical systems is concerned with group actions on measure spaces, topological spaces or smooth manifolds. If the acting group is the real line one usually interprets this action as a time evolution of the system. Many questions about dynamical systems of this kind (especially about perturbations, extens- ions or certain isomorphisms of the systems) lead to so-called cohomological equations: these are the analogues of group homomorphisms and isomorphisms for actions of groups rather than for the groups themselves. Recent years have seen renewed interest in cohomology of dynamical. One of the reasons for this was the discovery of some unexpected and surprising geometric, algebraic and cohomological rigidity properties of dynamical systems (especially of actions of multi- parameter groups). Another active area of study is the investigation of nonabelian cohomology and skew-product extensions of dynamical systems. The results which arose from this research project contributed to both of these aspects of dynamical cohomology. Here is a short summary of some of the contributions. Nonabelian cocycles in topological dynamics: Gernot Greschonig and Ulrich Haböck proved very interesting regularity properties of continuous cocycles over isometries with values in nilpotent groups. Cohomological rigidity: Danijela Damjanovic proved in joint work with Anatole Katok strong differentiable rigidity results for smooth actions of partially hyperbolic higher rank abelian groups. Invariant sets and measures of partially hyperbolic toral automorphisms: in joint work with Elon Lindenstrauss the investigator showed that such sets and measures have some quite remarkable properties, and that they are closely related to cocycles on shift-spaces satisfying certain rigidity properties (or, equivalently, to certain highly constrained random walks). These results made it possible to characterize and construct such sets and measures. All these results have since been published in leading international mathematical journals.