Geodesic
Ergodicity of billiards in polygons
Sbornik. Mathematics, Tome 188 (1997) no. 3, pp. 389-434

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In the space of all polygons, a topologically massive subset consisting of polygons with ergodic billiard flows is explicitly described. The elements of this set have a specified order of approximation by rational polygons. As intermediate results, constructive versions of the ergodic theorem for the billiard in a rational polygon and for the geodesic flow on a surface with flat structure, and also a constructive quadratic estimate for the growth of the number of saddle connections (singular trajectories) in a flat structure, are proved. 
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     author = {Ya. B. Vorobets},
     title = {Ergodicity of billiards in polygons},
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     number = {3},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_3_a3/}
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Ya. B. Vorobets. Ergodicity of billiards in polygons. Sbornik. Mathematics, Tome 188 (1997) no. 3, pp. 389-434. http://geodesic.mathdoc.fr/item/SM_1997_188_3_a3/