Exceptional and Generic Orbits in Homogeneous Dynamics and Number Theory

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This project deals with algebraic dynamical systems and their applications to number theory. It has been observed for a long time that many problems concerning diophantine approximation an be cast in terms of the behavior of orbits of a suitable homogeneous flow. In particular, various phenomena related to the theory of integer equations or inequalities can be in a useful way interpreted as certain trajectories being generic with respect to some natural measures on homogeneous spaces, while other call for constructing trajectories with exceptional behavior. The principal investigator plans to continue his study of those orbits, aiming at new applications to number theory. Dynamical tools to be used are: measure rigidity, ergodic theorems, mixing and equidistribution, reduction theory, and quantitative nondivergence.A dynamical system here stands for an abstract set of points together with an evolution law which governs the way points move over time. It turns out that many problems concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of certain orbits. Furthermore, systems that arise in this context are of algebraic nature, which makes it possible to use a wide variety f sophisticated tools for their investigation.