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Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with the walls, when it is subjected to an elastic response (i.e., its direction vector undergoes a reflection in the facet the point-mass has collided with).

The total trajectory covered by the point-mass is its orbit, and such an orbit is periodic if the point-mass eventually returns to its starting spot and its starting velocity. This billiards flow defines a fairly well-studied dynamical system.

I'm wondering if there are conditions on the symmetry group of $P$ which have consequences for the existence of periodic orbits inside $P$. More precisely, are there any conditions on a finite group $G$ which forbid the existence of a periodic orbit in any convex polytope having $G$ as its symmetry group?

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Dear Zach Conn,

I think your problem is very interesting but quite difficult to approach even in dimension 2: for instance, it is a well-known open problem to decide whether every irrational triangular billiard has a periodic orbit. As far as I know, the conjectural answer is yes, but, besides the case of acute triangles (where Fagnano's orbit can be constructed) and sufficiently small perturbations of isosceles triangles (after the work of R. Schwartz and W. P. Hooper), there is few progress in the general case.

Anyway, let me point out that this conjecture (if true) says that we can't hope (in general) to find conditions on the symmetry group of the billiard to avoid periodic orbits: in fact, the existence of periodic orbits in triangular billiards is expected in both irrational ("small symmetry group") and rational ("large symmetry group") cases.

Best,

Matheus

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Thanks! Actually I'm aware of the difficulty of the problem in 2 dimensions. One of my hopes with this approach was that it may be more effective in high dimensions. Perhaps a condition on the symmetry group could could be found which, for instance, only translates into the non-existence of periodic orbits for sufficiently high-dimensional polytopes having that group as their symmetry groups. Somehow this seems like a problem which may be easier in high dimensions and much, much harder in low dimensions. –  Zach Conn Jul 16 '10 at 17:51
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