Post date: Jun 29, 2011 5:59:9 AM
Mathematical study of closed billiard trajectories is due to George Birkhoff. A billiard ball is a point that moves inside a given domain along a straight line and reflects from the boundary making the angle of incidence equal the angle of reflection. Given a convex domain in R2 with a smooth boundary, how can one estimate from below the number of closed billiard trajectories of period p? In other words, if the point reflects p times and then starts moving along the same path, how many different trajectories it could be moving along?
George Birkoff proved that if p is prime, then the number of different p-periodic trajectories is at least p-1 (for non-prime p this estimate needs to be modified). A natural generalisation of this problem is the same question for a domain in Rn instead of R2. It was solved by M. Farber and S. Tabachnikov in 2002.
The topic of my Ph.D. thesis was to study the same problem in a more (maybe most?) general setting. Recall that the billiard table in classical Birkhoff′s question is a convex domain in R2, so the boundary of the domain is a topological circle. It is possible though to define a generalised billiard when the billiard ball reflects from an arbiatrary manifold M embedded in Rn. The manifold M can be even of codimension higher than 1, so it is probably more natural to think about a ray of light instead of a billiard ball then.
I managed to obtain a general estimate for the number of p-periodic billiard trajectories, but my theorem still can be improved within the same theory for p≥5. Also, the question of whether these estimates are sharp remains open.
TEXTBOOKS
[shop] George Birkhoff Dynamical Systems 1927
[shop] Serge Tabachnikov Geometry and Billiards (Student Mathematical Library)
REFERENCES
[pdf] M. Farber, S. Tabachnikov Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards, Topology, 41 (2002), 553-589
[pdf] M. Farber, S. Tabachnikov Periodic trajectories in 3-dimensional convex billiards, Manuscripta Math., 108, (2002), 431-437
DOWNLOADS
[pdf] F. Duzhin Lower bounds for the number of closed billiard trajectories of period 2 and 3 in manifolds embedded in Euclidean space., International Mathematics Research Notices, Vol. 2003, No. 8, pp. 425-449
[pdf] F. Duzhin Bounds for the number of periodic trajectories of generalized billiards, Journal of Mathematical Sciences, Vol. 138 (2006), No. 3, pp. 5691-5698
LINKS
[url] Serge Tabachnikov's homepage
[url] Michael Farber's homepage