The project involves measure rigidity of higher rank partially hyperbolic abelian actions and applications thereof. It is conjectured that under certain circumstances there are very few probability measures both invariant and ergodic under such actions (Furstenberg, Margulis). So far all of the current work relies on the additional assumption of positive entropy (Johnson, Rudolph), and sometimes on additional ergodicity along critical directions (Kalinin, Katok, Spatzier). Substantial progress has been made recently in avoiding unnecessary ergodicity-type assumptions as well as in applications of measure rigidity (Einsiedler, Katok, Lindenstrauss). Einsiedler will continue his study of invariant measures avoiding additional ergodicity-type assumptions, work on disjointness properties of various higher rank dynamical systems, further applications of rigidity, and related algebraic topics.
The theory of dynamical systems is a relatively new, but important mathematical theory with many connections to other parts of mathematics as well as other sciences such as physics, meteorology, or computer sciences. Historically it developed from the study of the evolution of a deterministic but complicated physical process (for instance in celestial mechanics) over time. Especially if this process is too complicated to predict long term outcomes precisely, the theory of dynamical systems is important because it can give qualitative predictions. More recently symbolic dynamics turned out to be crucial to find efficient and safe coding algorithms in computer sciences. There is also a long tradition of using dynamics to solve problems in other areas of mathematics. The study of higher rank dynamical systems (where time has more than one dimension) has received much attention during the last years, in part again because of its connections to physics (in particular statistical mechanics) and computer sciences (higher dimensional data storage methods). The proposed research links three separate areas of mathematics, namely ergodic theory, Lie theory, and algebra. The rich interplay among these fields helps to attack problems by using tools from different areas. The most prominent connection is between dynamics on homogeneous spaces and number theory, which holds the key to problems in the theory of Diophantine approximation. However, the theory of dynamical systems can benefit from this interaction too since dynamical systems of algebraic origin are often more amenable to a detailed study of their dynamical properties while the phenomena encountered are interesting in the larger context of general dynamical systems.
