## Institute for Mathematical Sciences

## Preprint ims94-13

** P. Boyland**
* Dual Billiards, Twist Maps, and Impact Oscillators.*

Abstract: In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex
closed curve G in the plane. It is shown that there do not
exist invariant circles near G when there is a point on G where
the radius of curvature vanishes or is discontinuous. In
addition, when the radius of curvature is not $C^1$ there are
examples with orbits that converge to a point of G. If the
derivative of the radius of curvature is bounded, such orbits
cannot exist. The final section of the paper concerns an impact
oscillator whose dynamics are the same as a dual billiards map.
The appendix is a remark on the connection of the inverse
problems for invariant circles in billiards and dual billiards.

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