Dynamical properties of algebraic systems : a study in closed geodesics
To a great extent, rigidity theory is the study of boundaries of semisimple groups. Here we investigate the action of a lattice on such a boundary. While we can construct topological factors for real rank 1 groups we show the nonexistence of such factors in higher rank for some cases. We also study the geodesic flow on a compact locally symmetric manifold of the noncompact type. He calculate metric and topological entropies and see that the Liouville measure is a measure or maximal entropy. This leads to a study of compact maximal flats. We give a new proof of their density in the space of all flats. We prove specification and expansiveness theorems for the geodesic flow and apply them to determine a growth rate for compact maximal flats. Finally, we give an example of a space with infinitely many closed singular geodesics.