Geodesic
Topological type of isoenergy surfaces of billiard books
Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1660-1674 Cet article a éte moissonné depuis la source Math-Net.Ru

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The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products $S^1\times S^2$. The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms – that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems – are calculated for several integrable billiards of this type. Bibliography: 14 titles.
Keywords: integrable system
Mots-clés : billiard book, Liouville equivalence, Fomenko-Zieschang invariant. 
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V. V. Vedyushkina. Topological type of isoenergy surfaces of billiard books. Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1660-1674. http://geodesic.mathdoc.fr/item/SM_2021_212_12_a0/

[1] A. T. Fomenko, H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1991), 567–596 | DOI | MR | Zbl

[2] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | DOI | MR | MR | Zbl | Zbl

[3] G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York, 1927, viii+295 pp. | MR | Zbl | Zbl

[4] V. V. Kozlov, D. V. Treshchev, Billiards. A genetic introduction to the dynamics of systems with impacts, Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991, viii+171 pp. | MR | MR | Zbl | Zbl

[5] E. Gutkin, “Billiard dynamics: a survey with the emphasis on open problems”, Regul. Chaotic Dyn., 8:1 (2003), 1–13 | DOI | MR | Zbl

[6] I. S. Kharcheva, “Isoenergetic manifolds of integrable billiard books”, Moscow Univ. Math. Bull., 75:4 (2020), 149–160 | DOI | MR | Zbl

[7] A. T. Fomenko, S. V. Matveev, Algorithmic and computer methods for three-manifolds, Math. Appl., 425, Kluwer Acad. Publ., Dordrecht, 1997, xii+334 pp. | DOI | MR | MR | Zbl | Zbl

[8] V. V. Fokicheva, “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics”, Sb. Math., 206:10 (2015), 1463–1507 | DOI | DOI | MR | Zbl

[9] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[10] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl

[11] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219 | DOI | MR | Zbl

[12] V. V. Vedyushkina, “Integrable billiard systems realize toric foliations on lens spaces and the 3-torus”, Sb. Math., 211:2 (2020), 201–225 | DOI | DOI | MR | Zbl

[13] K. I. Solodskikh, “Predstavlenie izoenergeticheskikh poverkhnostei gamiltonovykh sistem zatsepleniyami v $S^3$”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh. (to appear)

[14] A. Hatcher, Basic topology of 3-manifolds http://www.math.cornell.edu/~hatcher