Abstract
This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For anyn≧3 there exists a corresponding convexn-agon (forn=3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, then-agon serving as the base.
The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons.
The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.
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Communicated by Ya. G. Sinai
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Galperin, G.A. Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. Commun.Math. Phys. 91, 187–211 (1983). https://doi.org/10.1007/BF01211158
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DOI: https://doi.org/10.1007/BF01211158