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Topology of Liouville foliations of integrable billiards on table-complexes

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Abstract

We review recent results on the class of integrable billiards and the class of integrable Hamiltonian systems. Recently introduced by Vedyushkina, classes of integrable billiards on piecewise flat manifolds or CW-complexes glued from several flat domains bounded by confocal quadrics and equipped with commuting permutations (called topological billiards or billiard books respectively), made it possible to obtain significant progress in the proof of Fomenko billiard conjecture. According to this conjecture, each topological class of Liouville foliations and non-degenerate singularities of integrable Hamiltonian systems appears in the corresponding class of integrable billiards. In other words, each non-degenerate integrable system can be realized by integrable billiards from the topological point of view. New results on the classification problem for billiard books are obtained and the general problem is reduced to classification of five finite systems of permutations with commutation conditions. Also we formulate several open problems.

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Notes

  1. i.e. for various definitions of “integrability” and for various classes of billiards

  2. because of the reflection from boundary which can identify phase points which were “far” from each other

References

  1. Bialy, M., Mironov, A.E.: Polynomial nonintegrability of magnetic billiards on the sphere and the hyperbolic plane. Russian Math. Surveys 74(2), 187–209 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bolsinov, A., Matveev, V.S., Miranda, E., Tabachnikov, S.: Open problems, questions and challenges in finite-dimensional integrable systems. Philos. Trans. Roy. Soc. A 376(2131), Art. No. 20170430 (2018)

  3. Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification. Chapman and Hall/CRC, Boca Raton (2004)

  4. Bolsinov, A.V., Matveev, S.V., Fomenko, A.T.: Topological classification of integrable Hamiltonian systems with two degrees of freedom. A list of systems of small complexity. Russian Math. Surveys 45(2), 59–94 (1990)

  5. Brailov, A.V., Fomenko, A.T.: The topology of integral submanifolds of completely integrable Hamiltonian systems. Sb. Math. 62(2), 373–383 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dragović, V., Gasiorek, S., Radnović, M.: Billiard ordered games and books. Regul. Chaotic Dyn. 27(2), 132–150 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dragović, V., Gasiorek, S., Radnović, M.: Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology. Mat. Sb. 213(9), 34–69 (2022)

    Google Scholar 

  8. Dragović, V., Radnović, M.: Bifurcations of Liouville tori in elliptical billiards. Regul. Chaotic Dyn. 14(4–5), 479–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dragović, V., Radnović, M.: Poncelet Porisms and Beyond: Integrable Billiards. Hyperelliptic Jacobians and Pencils of Quadrics. Frontiers in Mathematics. Birkhäuser, Basel (2011)

    Book  MATH  Google Scholar 

  10. Dragović, V., Radnović, M.: Pseudo-integrable billiards and arithmetic dynamics. J. Mod. Dyn. 8(10, 109–132 (2014)

  11. Dragović, V.I., Radnović, M.: Pseudo-integrable billiards and double-reflection nets. Russ. Math. Surv. 70(1), 1–31 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fedoseev, D.A., Fomenko, A.T.: Noncompact bifurcations of integrable dynamic systems. J. Math. Sc. 248, 810–827 (2020)

    Article  MATH  Google Scholar 

  13. Fokicheva, V.V.: Description of singularities for system billiard in an ellipse system. Moscow Univ. Math. Bull. 67(5–6), 217–220 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fokicheva, V.V.: Classification of billiard motions in domains bounded by confocal parabolas. Sb. Math. 205(8), 1201–1221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fokicheva, V.V.: Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas. Moscow Univ. Math. Bull. 69(4), 148–158 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fokicheva, V.V.: A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics. Sb. Math. 206(10), 1463–1507 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fokicheva, V.V.: Topological Classification of Integrable Billiards. Ph.D. Thesis. Lomonosov Moscow State University, Moscow (2016)

  18. Fomenko, A.T.: A Morse theory for integrable Hamiltonian systems. Dokl. Akad. Nauk SSSR 287(5), 1071–1075 (1986) (in Russian)

  19. Fomenko, A.T.: The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability. Math. USSR-Izv. 29(3), 629–658 (1987)

    Article  MATH  Google Scholar 

  20. Fomenko, A.T.: Local modeling of Liouville foliations by billiards: implementation of edge invariants. Moscow Univ. Math. Bull. 76(2), 60–64 (2021)

    Article  MathSciNet  Google Scholar 

  21. Fomenko, A.T., Kibkalo, V.A.: Saddle singularities in integrable Hamiltonian systems: Examples and algorithms. In: Sadovnichiy, V.A., Zgurovsky, M.Z. (eds.): Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics. Understanding Complex Systems, pp. 3–26. Springer, Cham (2021)

  22. Fomenko, A.T., Matveev, S.V.: Algorithmic and Computer Methods for the Three-Manifolds. Mathematics and its Applications, vol. 425. Kluwer, Dordrecht (1997)

  23. Fomenko, A.T., Vedyushkina, V.V.: Billiards and integrability in geometry and physics. New scope and new potential. Moscow Univ. Math. Bull. 74(3), 98–107 (2019)

  24. Fomenko, A.T., Vedyushkina, V.V.: Implementation of integrable systems by topological, geodesic billiards with potential and magnetic field. Russ. J. Math. Phys. 26(3), 320–333 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fomenko, A.T., Vedyushkina, V.V.: Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards. Izv. Math. 83(6), 1137–1173 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fomenko, A.T., Zieschang, H.: On the topology of the three-dimensional manifolds arising in Hamiltonian mechanics. Soviet Math. Dokl. 35(2), 520–534 (1987)

    MathSciNet  MATH  Google Scholar 

  27. Fomenko, A.T., Zieschang, H.: On typical topological properties of integrable Hamiltonian systems. Math. USSR-Izv. 32(2), 385–412 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Glutsyuk, A.: On polynomially integrable Birkhoff billiards on surfaces of constant curvature. J. Eur. Math. Soc. (JEMS) 23(3), 995–1049 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Glutsyuk, A.A.: On two-dimensional polynomially integrable billiards on surfaces of constant curvature. Dokl. Math. 98(1), 382–385 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gutkin, E.: Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8(1), 1–13 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kaloshin, V., Sorrentino, A.: On the local Birkhoff conjecture for convex billiards. Ann. Math. 188(1), 315–380 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kapovich, M., Millson, J.: On the moduli space of polygons in the Euclidean plane. J. Diff. Geom. 42(2), 430–464 (1995)

    MathSciNet  MATH  Google Scholar 

  33. Kapovich, M., Millson, J.J.: The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 4(3), 479–513 (1996)

    MathSciNet  MATH  Google Scholar 

  34. Kapovich, M., Millson, J.J.: Universality theorems for configuration spaces of planar linkages. Topology 41(6), 1051–1107 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Katok, S.: Bifurcation sets and integral manifolds in the heavy rigid body problem. Uspekhi Mat. Nauk 27(2), 124–133 (1972)

    Google Scholar 

  36. Kharcheva, I.S.: Isoenergetic manifolds of integrable billiard books. Moscow Univ. Math. Bull. 75(4), 149–160 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kharlamov, M.P.: Topological analysis of integrable problems of rigid body dynamics. LSU Publications, Leningrad (1998) (in Russian)

  38. Kharlamov, M.P., Ryabov, P.E., Savushkin, AYu.: Topological atlas of the Kowalevski-Sokolov top. Regul. Chaotic Dyn. 21(1), 24–65 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kibkalo, V.A., Fomenko, A.T., Kharcheva, I.S.: Realizing integrable Hamiltonian systems by means of billiard books. Trans. Moscow Math. Soc. 75(4), 37–64 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Kozlov, V.V., Treshchev, D.V.: Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts. American Mathematical Society, Providence (1991)

    Book  MATH  Google Scholar 

  41. Kudryavtseva, E.A.: Liouville integrable generalized billiard flows and Poncelet type theorems. J. Math. Sci. (N.Y.) 225(4), 611–638 (2017)

  42. Lazutkin, V.F.: KAM Theory and Semiclassical Approximations to Eigenfunctions. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 24. Springer, Berlin (1993)

  43. Moskvin, V.A.: Topology of Liouville bundles of integrable billiard in non-convex domains. Moscow Univ. Math. Bull. 73(3), 103–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Moskvin, V.A.: Algorithmic construction of two-dimensional singular layers of billiard atoms in non-convex domains. Moscow Univ. Math. Bull. 75(3), 91–101 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nikolaenko, S.S.: Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds. Sb. Math. 211(8), 1127–1158 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Oshemkov, A.A.: Fomenko invariants for the main integrable cases of the rigid body motion equations. In: Fomenko, A.T. (ed.) Topological Classification of Integrable Systems. Advances in Soviet Mathematics, vol. 6, pp. 67–146. American Mathematical Society, Providence (1991)

    Chapter  MATH  Google Scholar 

  47. Oshemkov, A.A.: Morse functions on two-dimensional surfaces. Encoding of singularities. Proc. Steklov Inst. Math. 205(4), 119–127 (1995)

  48. Oshemkov, A.A.: Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems. Sb. Math. 201(7–8), 1153–1191 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Pnueli, M., Rom-Kedar, V.: On the structure of Hamiltonian impact systems. Nonlinearity 34(4), 2611–2658 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ratiu, T.S., Zung, N.T.: Integrable systems in planar robotics. Chebyshevskiĭ Sb. 21(2), 320–340 (2020)

    MathSciNet  MATH  Google Scholar 

  51. Smale, S.: Topology and mechanics. I. Invent. Math. 10, 305–331 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  52. Solodskikh, K.I.: Graph-manifolds and integrable Hamiltonian systems. Sb. Math. 209(5), 739–758 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  53. Tabachnikov, S.L.: Geometry and Billiards. American Mathematical Society, Providence (2005)

    Book  MATH  Google Scholar 

  54. Tatarinov, Ya.V.: On the investigation of the phase topology of compact configurations with symmetry. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 28(5), 70–77 (1973) (in Russian)

  55. Tatarinov, Ya.V.: Portraits of classical integrals of the problem of the rotation of a rigid body around a fixed point. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 29(6), 99–105 (1974) (in Russian)

  56. Topalov, P.I.: Homological properties of labels of the Fomenko–Zieschang invariant. Proc. Steklov Inst. Math. 205, 151–156 (1995)

    Google Scholar 

  57. Vedyushkina, V.V.: Fomenko–Zieschang invariants of topological billiards bounded by confocal parabolas. Moscow Univ. Math. Bull. 73(4), 150–155 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  58. Vedyushkina, V.V.: The Fomenko–Zieschang invariants of nonconvex topological billiards. Sb. Math. 210(3), 310–363 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  59. Vedyushkina, V.V.: Integrable billiard systems realize toric foliations on lens spaces and the 3-torus. Sb. Math. 211(2), 201–225 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  60. Vedyushkina, V.V.: Integrable billiards on CW-complexes and integrable Hamiltonian flows. Sc.D. (Habilitation) Thesis, Lomonosov MSU, Moscow (2020)

  61. Vedyushkina, V.V.: Topological type of isoenergy surfaces of billiard books. Sb. Math. 212(12), 1660–1674 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  62. Vedyushkina, V.V., Fomenko, A.T., Kharcheva, I.S.: Modeling nondegenerate bifurcations of closures of solutions for integrable systems with two degrees of freedom by integrable topological billiards. Dokl. Math. 97(2), 174–176 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  63. Vedyushkina, V.V., Kharcheva, I.S.: Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems. Sb. Math. 209(12), 1690–1727 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  64. Vedyushkina, V.V., Kharcheva, I.S.: Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems. Sb. Math. 212(8), 1122–1179 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  65. Vedyushkina, V.V., Kibkalo, V.A.: Realization of numerical invariant of the Seifert bundle of integrable systems by billiards. Moscow Univ. Math. Bull. 75(4), 161–168 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  66. Vedyushkina, V.V., Kibkalo, V.A.: Billiard books of low complexity and realization of Liouville foliations of integrable systems. Chebyshevskiĭ sbornik 23(1), 53–82 (2022)

    Article  MathSciNet  Google Scholar 

  67. Vedyushkina, V.V., Kibkalo, V.A., Fomenko, A.T.: Topological modeling of integrable systems by billiards: realization of numerical invariants. Dokl. Math. 102(1), 269–271 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  68. Vedyushkina, V.V., Kibkalo, V.A., Pustovoitov, S.E.: Realization of focal singularities of integrable systems using billiard books with a Hooke potential field. Chebyshevskiĭ sbornik 22, 44–57 (2021)

    Article  MathSciNet  Google Scholar 

  69. Vedyushkina (Fokicheva), V.V., Fomenko, A.T.: Integrable topological billiards and equivalent dynamical systems. Izv. Math. 81(4), 688–733 (2017)

  70. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. I. Invent. Math. 3(4), 308–333 (1967)

    Article  MATH  Google Scholar 

  71. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. II. Invent. Math. 4(2), 88–117 (1967)

    Google Scholar 

  72. Zhila, A.I.: Topological types of isoenergy surfaces in the system of the Chaplygin ball with a rotor. Moscow Univ. Math. Bull. 75(3), 134–138 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, Russia, 119991

    Anatoly T. Fomenko & Vladislav A. Kibkalo

  2. Moscow Center For Fundamental and Applied Mathematics, Moscow, Russia

    Anatoly T. Fomenko & Vladislav A. Kibkalo

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Fomenko, A.T., Kibkalo, V.A. Topology of Liouville foliations of integrable billiards on table-complexes. European Journal of Mathematics 8, 1392–1423 (2022). https://doi.org/10.1007/s40879-022-00589-7

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