The proposed research is a study of dynamical properties of the Teichmueller geodesic flow on the moduli space of compact Riemann surfaces, with a special emphasis on applications in geometry. The main tool is a symbolic coding for the flow, given by the theory of renormalization for interval exchange maps. This technique is applied by the PI to the study of periodic Teichmueller geodesics and related counting problems. The second, more analytic, part of the project is a quantitative study of chaotic behavior of the Teichmueller geodesic flow,extending the Central Limit Theorem obtained by the PI. The broad aim is to show that Teichmueller geodesics behave as trajectories of Brownian motion. Using renormalization for interval exchange maps, the PI aims to carry over classical results on compact negatively curved manifolds to the context of the Teichmueller space, which is neither compact nor Gromov hyperbolic.
The theory of dynamical chaos plays a crucial role in modern science. Originating in 1890 in Poincare''s memoir on the stability of planetary motion, today the theory has applications in climate prediction and oceanography, statistical physics and complexity theory in computer science. The proposed research is a study of dynamical chaos in the moduli space of compact Riemann surfaces.
