This article is cited in 3 scientific papers (total in 3 papers)
The Fomenko–Zieschang invariants of nonconvex topological billiards
V. V. Vedyushkina
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Along with a classical planar billiard, one can consider a topological billiard for which the motion takes place on a locally planar surface obtained by an isometric gluing of several planar domains along boundaries that are arcs of confocal quadrics. Here, a point is moving inside every domain along segments of straight lines, passing from one domain into another when it hits the boundary of the gluing. The author has previously obtained the Liouville classification of all such topological billiards obtained by gluings along convex boundaries. In the present paper, we classify all topological integrable billiards obtained by gluing both along convex and along nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics. For all such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules $W^*$) of Liouville equivalence are calculated.
Bibliography: 25 titles.
integrable system, billiard, Liouville equivalence, Fomenko–Zieschang invariant.
|Russian Science Foundation
|This work was supported by the Russian Science Foundation (project no. 17-11-01303).
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Sbornik: Mathematics, 2019, 210:3, 310–363
MSC: Primary 37J35; Secondary 37G10, 70E40
V. V. Vedyushkina, “The Fomenko–Zieschang invariants of nonconvex topological billiards”, Mat. Sb., 210:3 (2019), 17–74; Sb. Math., 210:3 (2019), 310–363
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\paper The Fomenko--Zieschang invariants of nonconvex topological billiards
\jour Mat. Sb.
\jour Sb. Math.
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A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrability in geometry and physics. New scope and new potential”, Moscow University Mathematics Bulletin, 74:3 (2019), 98–107
V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173
V. V. Vedyushkina, “Integriruemye billiardy realizuyut toricheskie sloeniya na linzovykh prostranstvakh i 3-tore”, Matem. sb., 211:2 (2020), 46–73
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