Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well known; for an example of the latter, see e.g. section 1 of Charles Radin's article "Dynamics of Limit Models", available at http://www.ma.utexas.edu/users/radin/papers/limitmodels.pdf.)

Can single-particle billiards exhibit such catastrophes in a domain with smooth boundary?

(If the answer depends on the degree of smoothness, e.g. if the answer is different for $C^1$ vs $C^\infty$, then interpret my question as "For what forms of smoothness is it the case that ..., and for what forms of smoothness is it not the case that ...?")

Note that this is not the same as asking the corresponding question about the discrete billiards map. The discrete map can be iterated unboundedly many times, but if the return-time to the boundary shrinks quickly, "time infinity" under discrete (return-map) dynamics could correspond to a finite-time catastrophe in the flow.

Note also that I am not asking whether billiards flow is well-defined for a set of initial conditions of full measure; I am asking whether it is well-defined for ALL initial conditions. (So please don't add an ergodic-theory tag to my post!)

Note also that the catastrophe phenomenon is not the same as the ill-definedness of the velocity at the moment of rebound. This ill-definedness does indeed make it slightly tricky to define billiard flow, but this is merely a technical problem that is easily surmountable. Extending the dynamics beyond a catastrophe is more problematic; it entails breaking symmetry and/or introducing discontinuities.

Hopefully I have anticipated all misunderstandings of my question that are likely to arise.

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    $\begingroup$ By a "catastrophe" I meant a point in time beyond which the specified evolution rule cannot be applied. E.g., a billiards shot directly into a corner of a polygonal table is a catastrophe in my sense (even though in some cases, e.g. a rectangular table, there may be a natural way to evolve the system through the singularity). $\endgroup$ Jan 4 '14 at 5:59

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

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.

  • $\begingroup$ That's really lovely! Somehow I had always assumed that such things could not happen, but the construction is remarkably simple... $\endgroup$ Jan 4 '14 at 3:13
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    $\begingroup$ In addition to derivatives, Halpern assumes "nowhere vanishing curvature". So he does not appear to rule out the possibility that a catastrophe might exist if the curvature goes smoothly to zero. $\endgroup$
    – user25199
    Jan 4 '14 at 19:14
  • $\begingroup$ @Carl: Good point. I think I can construct an example of a catastrophe with a smooth boundary and curvature going to $0$. $\endgroup$ Jan 4 '14 at 19:36
  • $\begingroup$ Douglas, I just noticed this post, and I would be very much interested in your counterexample in the case of a smooth boundary but non-vanishing curvature. I asked a similar question today: mathoverflow.net/questions/203620/… $\endgroup$ Apr 22 '15 at 21:08

This is just a guess, I will admit (apologies if I'm among the misunderstandings you anticipated), but perhaps Peter Guber's paper,

Convex billiards. Geometriae Dedicata. February 1990, Volume 33, Issue 2, pp 205-226. (Springer link)

may be of some assistance? Here is a possibly relevant sentence from the Introduction:



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