Abstract:
A local version of A.T. Fomenko's conjecture on modeling of integrable systems by billiards is formulated. It is proved that billiard systems realize arbitrary numerical marks of Fomenko–Zieschang invariants. Thus, numerical marks are not a priori a topological obstacle to the realization of the Liouville foliation of integrable systems by billiards.
Citation:
V. V. Vedyushkina, V. A. Kibkalo, A. T. Fomenko, “Topological modeling of integrable systems by billiards: realization of numerical invariants”, Dokl. RAN. Math. Inf. Proc. Upr., 493 (2020), 9–12; Dokl. Math., 102:1 (2020), 269–271
\Bibitem{VedKibFom20}
\by V.~V.~Vedyushkina, V.~A.~Kibkalo, A.~T.~Fomenko
\paper Topological modeling of integrable systems by billiards: realization of numerical invariants
\jour Dokl. RAN. Math. Inf. Proc. Upr.
\yr 2020
\vol 493
\pages 9--12
\mathnet{http://mi.mathnet.ru/danma86}
\crossref{https://doi.org/10.31857/S2686954320040207}
\zmath{https://zbmath.org/?q=an:7424607}
\elib{https://elibrary.ru/item.asp?id=43795337}
\transl
\jour Dokl. Math.
\yr 2020
\vol 102
\issue 1
\pages 269--271
\crossref{https://doi.org/10.1134/S1064562420040201}
Linking options:
https://www.mathnet.ru/eng/danma86
https://www.mathnet.ru/eng/danma/v493/p9
This publication is cited in the following 12 articles:
V. V. Vedyushkina, S. E. Pustovoitov, “Classification of Liouville foliations of integrable topological billiards in magnetic fields”, Sb. Math., 214:2 (2023), 166–196
V. N. Zav'yalov, “Billiard with slipping by an arbitrary rational angle”, Sb. Math., 214:9 (2023), 1191–1211
A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrable systems”, Russian Math. Surveys, 78:5 (2023), 881–954
A. A. Kuznetsova, “Modeling of degenerate peculiarities of integrable billiard systems by billiard books”, Moscow University Mathematics Bulletin, 78:5 (2023), 207–215
A. T. Fomenko, V. V. Vedyushkina, “Evolutionary force billiards”, Izv. Math., 86:5 (2022), 943–979
V. V. Vedyushkina, V. N. Zav'yalov, “Realization of geodesic flows with a linear first integral by billiards with slipping”, Sb. Math., 213:12 (2022), 1645–1664
G. V. Belozerov, “Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space”, Sb. Math., 213:2 (2022), 129–160
A. T. Fomenko, V. A. Kibkalo, “Topology of Liouville foliations of integrable billiards on table-complexes”, European Journal of Mathematics, 8:4 (2022), 1392
V. V. Vedyushkina, V. A. Kibkalo, “Billiardnye knizhki maloi slozhnosti i realizatsiya sloenii Liuvillya integriruemykh sistem”, Chebyshevskii sb., 23:1 (2022), 53–82
A. T. Fomenko, V. V. Vedyushkina, “Billiards with Changing Geometry and Their Connection with the Implementation of the Zhukovsky and Kovalevskaya Cases”, Russ. J. Math. Phys., 28:3 (2021), 317
V. V. Vedyushkina, “Local modeling of Liouville foliations by billiards: implementation of edge invariants”, Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 76:2 (2021), 60–64
A. T. Fomenko, V. V. Vedyushkina, V. N. Zav'yalov, “Liouville Foliations of Topological Billiards with Slipping”, Russ. J. Math. Phys., 28:1 (2021), 37
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