I think the paper you want is [B. Halpern, "Strange Billiard Tables." *Transactions of the AMS* Vol 232, 1977.][1]

Thanks to Carl for pointing out that Halpern considers tables with the additional condition of nonvanishing curvature. He constructs a $C^2$ catastrophe (with collision points on the unit circle, but an irregularly shaped table passing through those points) and rules out a $C^3$ catastrophe.

I believe that without the condition of nonvanishing curvature, Halpern's construction can be adapted to produce a smooth catastrophe with infinitely many collision points on $\exp(-1/x^2)$ approaching the origin. However, I haven't verified that the result can be made smooth.


  [1]: http://www.ams.org/journals/tran/1977-232-00/S0002-9947-1977-0451308-7/S0002-9947-1977-0451308-7.pdf