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Abstract:
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Let P be a simple, closed polygon in the plane, all interior angles of which are rational multiples of $\pi$. We consider the possible paths of a point, rebounding in the interior of P with constant speed and elastic reflections. Such a dynamical system is known as "billiards in P". By means of a well-known construction, "billiard" trajectories in such a polygon P are identified with geodesic paths on a closed Riemann surface $X\sp{P}$, where the Riemannian metric is one of zero curvature with isolated singularities, and is given by a holomorphic one-form $\omega$ on the surface.
To this holomorphic one-form one can canonically associate a discrete subgroup $\Gamma$ of $PSL(2,\IR$). If $\Gamma$ happens to be a lattice (has cofinite volume), then it is known that all geodesic paths in the zero-curvature metric given by $\omega$ must either be closed or uniformly distributed in the surface $X\sp{P}$. As a corollary, all billiard paths in the original polygon P must either be finite or uniformly distributed in P.
A new class of examples of polygons P, whose associated group $\Gamma$ is, in fact, a lattice have been discovered. At the same time, we have discovered the first examples of triangles P, as above, for which the associated groups $\Gamma$ are not lattices (i.e. have infinite covolume). Finally, it is shown how to derive, in an explicit way, algebraic equations which specify the Riemann surface $X\sp{P}$ and one-form $\omega$, which before were only described geometrically. |
