Dynamical systems and ergodic theory investigate the evolution of a physical or mathematical system over time, such as turbulence in a fluid flow or changing planetary systems. New ideas and concepts such as information, entropy, chaos and fractals have changed our understanding of the world. Dynamics and ergodic theory provide excellent mathematical tools, and have a strong impact on the sciences and engineering. Symbolic dynamics for example has been instrumental in developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology. Geometry is a highly developed and ancient field in mathematics of amazing vigor. It studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, starting with Gauss in the nineteenth century. It is closely linked with physics and other sciences and applied areas such as computer vision. Geometry and dynamics are closely related. Indeed, important dynamical systems come from geometry, and vice versa geometry provides tools to study dynamical systems. One main goal of this project studies when two dynamical systems commute, i.e. when one system is unaffected by the changes brought on by the other. Alternatively, these are systems with unexpected symmetries Important examples of such systems arise from geometry when the space contains many flat subspaces. Group theory finally enters both dynamics and geometry by studying the group of symmetries of a geometry or dynamical situation, or by investigating the dynamical and geometric behavior of the group of symmetries acting on a space.
This project centers on problems between dynamical systems, group theory and geometry. There are two main goals: First, establish exponential mixing properties for several different systems in dynamics, in particular frame flows from Riemannian geometry and solenoids coming from noninvertible systems. The principal investigator (PI) will draw tools from dynamics, geometry and number theory to accomplish these goals. Second, prove rigidity properties in geometry and in dynamical systems, in particular when the system and spaces in question are "higher rank", e.g. when spaces have flat subspaces or the dynamics has nontrivially commuting elements. Such systems appear naturally in seemingly quite separate areas, for example in number theory or in studying the spectrum of the Laplacian. The investigator will work on rigidity properties of actions of higher rank abelian and semi-simple Lie groups and their lattices striving to classify such systems under suitable geometric or dynamical hypotheses. The PI will employ tools from geometry, dynamics, Lie groups, and specifically exponential mixing properties. The PI will also investigate discrete faithful representations of hyperbolic groups in p-adic Lie groups, equilibrium states for partially hyperbolic dynamical systems and spherical higher rank in Riemannian geometry.
