12 - Geodesic Flows

Summary

For a long time geodesic flows have played an important stimulating role in the development of hyperbolic theory. In the beginning of the twentieth century, Hadamard and Morse, while studying the statistics of geodesics on surfaces of negative curvature, pointed out that the local instability of trajectories gives rise to some global properties of dynamical systems such as ergodicity and topological transitivity. The further study of geodesic flows and some objects closely related to them (e.g., frame flows) later inspired the introduction of different classes of hyperbolic dynamical systems (e.g., Anosov systems, uniformly partially hyperbolic systems, and nonuniformly hyperbolic systems preserving volume). On the other hand, geodesic flows always were a touchstone for applying new advanced methods of the general theory of dynamical systems. This, in particular, has led to some new interesting results in differential and Riemannian geometry. In this chapter we will present some of these results. While describing ergodic properties of geodesic flows in the spirit of the book, we consider the Liouville measure that is invariant under the flow (we allow other invariant measures when discussing an upper bound for the measure-theoretic entropy), and we refer the reader to the excellent survey [115] where measures of maximal entropy are studied.