# Meeting Details

Title: Absolute Focusing, Elliptic Periodic Orbits, and C^2-Stadia Center for Dynamics and Geometry Seminars A. Grigo, Georgia Tech As is well known since the 1970's, dispersing and defocusing are the two mechanisms leading to hyperbolic behavior in billiards. In this talk we show that whenever a non-absolutely focusing boundary component is used, the general strategy to construct hyperbolic billiard tables might fail. In fact, we show how the presence of a non-absolutely focusing boundary component can lead to stable periodic orbits of arbitrary long free paths. As a result of this, existing method of constructing hyperbolic billiards with focusing components must be restricted to absolutely focusing ones. As an example we conside smoothing out the ends of the circular arcs of the usual stadium billiard such that the curvature of the resulting curved segment vanishes at its endpoint. We show that for an arbitrarily short smoothed out part of the originally circular arcs the resulting $C^2$-smooth stadium possess elliptic periodic orbits for arbitrary short and also for arbitrary large separation distances of the two curved segments. (This is joint work with L.Bunimovich)