Billiards by Tabachnikov S.

By Tabachnikov S.

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Thus the symplectic invariant of a periodic orbit specializes to the length of a closed billiard trajectory. S. Marvizi and R. Melrose ([M-Me]) studied the asymptotics of the lengths of n-periodic billiard trajectories in a smooth strictly convex plane domain as n → ∞. Let L n be the supremum and ln – the infimum of the perimeters of simple billiard n-gons. Theorem 1. lim nk (Ln − ln ) = 0 n→∞ for any positive k. Moreover, Ln has an asymptotic expansion as n → ∞: Ln ∼ l + ∞ i=1 ci , n2i where l is the length of the billiard table and ci are constants, depending on the curvature of the table.

We will outline some results on the set of lengths of the closed trajectories in a convex billiard, the set that is called the length spectrum of a billiard. To start with, let us explain the symplectic meaning of the length spectrum, following V. Guillemin and R. Melrose ([G-M]). Let T be an exact symplectomorphism of an exact symplectic manifold (M, ω). , equal to dF for a generating function F . Let x be an n-periodic point: T n x = x. Assign to this point the number n F (T i x). L(x) = i=1 This number depends on the choice of the generating function, determined up to an additive constant.

The system of three equations is easily solved: multiply all three, take the square root, and divide by one of them. One gets: H1 + H2 − H3 = H1 H2 /C3 = 2H1 H2 K3 / sin α3 . Now apply the Cosine Theorem to the triangle X1 X2 X3 . It boils down to H1 + H 2 − H 3 = 4H1 H2 sin2 α3 . H1 + H 2 + H 3 Notice that a 3-periodic trajectory is an extremum of the perimeter length functional on inscribed triangles. This functional is constant on its critical set: H1 + H2 + H3 = L for some constant L and all nearby 3-periodic trajectories.

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