Geodesic
Ergodicity of billiards in polygons
Sbornik. Mathematics, Tome 188 (1997) no. 3, pp. 389-434 
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/
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     title = {Ergodicity of billiards in polygons},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Zemlyakov A. N., Katok A. B., “Topologicheskaya tranzitivnost bilyardov v mnogougolnikakh”, Matem. zametki, 18:2 (1975), 291–300 | MR | Zbl

[2] Kerckhoff S., Masur H., Smillie J., “Ergodicity of billiard flows and quadratic differentials”, Ann. of Math., 124:2 (1986), 293–311 | DOI | MR | Zbl

[3] Pospiech F., Stepin A. M., “Über die vollständige Nichtintegrierbarkeit eines Systems zweier Teilchen auf einem Intervall”, Wiss. Z. Univ. Halle, 37:4 (1988), 89–98 | MR | Zbl

[4] Masur H., “The growth rate of trajectories of a quadratic differential”, Ergodic Theory Dynam. Systems, 10:1 (1990), 151–176 | DOI | MR | Zbl

[5] Boshernitzan M., “A condition for minimal interval exchange maps to be uniquely ergodic”, Duke Math. J., 52:3 (1985), 723–752 | DOI | MR | Zbl

[6] Vorobets Ya. B., “Ergodichnost bilyardov v mnogougolnikakh: yavnye primery”, UMN, 51:4 (1996), 151–152 | MR | Zbl