Rotation numbers and instability sets

Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows. Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G. D. Birkhoff was led to the concept of "regions of instability" for twist maps. We discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties.