For more information about this meeting, contact Yakov Pesin, Anatole Katok, Svetlana Katok, Dmitri Burago.
|Title:||Absolute Focusing, Elliptic Periodic Orbits, and C^2-Stadia|
|Seminar:||Center for Dynamics and Geometry Seminars|
|Speaker:||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)|
Room Reservation Information
|Date:||03 / 23 / 2009|
|Time:||03:30pm - 05:30pm|