Counting Periodic Trajectories of Finsler Billiards

Pavle V. M. Blagojević ; Michael Harrison ; S. Tabachnikov ; Günter M. Ziegler

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group $\mathbb{Z}_r$ acts on these extremal polygons, and one counts the $\mathbb{Z}_r$-orbits. Using Morse and Lusternik–Schnirelmann theories, we prove that if $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.

Détail

Keywords: mathematical billiards, Finsler manifolds, magnetic billiards, Morse and Lusternik–Schnirelmann theories, unlabeled cyclic configuration spaces.

Bibliographie

[1] Adem A., Milgram R. J., Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, 309, 2nd ed., Springer-Verlag, Berlin, 2004 | DOI | MR | Zbl

[2] Akopyan A. V., Zaslavsky A. A., Geometry of conics, Mathematical World, 26, Amer. Math. Soc., Providence, RI, 2007 | DOI | MR | Zbl

[3] Álvarez J. C., Durán C., An introduction to Finsler geometry, Notas de la Escuela Venezolana de Matématicas, 1998

[4] Artstein-Avidan S., Karasev R., Ostrover Y., “From symplectic measurements to the Mahler conjecture”, Duke Math. J., 163 (2014), 2003–2022, arXiv: 1303.4197 | DOI | MR | Zbl

[5] Artstein-Avidan S., Ostrover Y., “Bounds for Minkowski billiard trajectories in convex bodies”, Int. Math. Res. Not., 2014 (2014), 165–193, arXiv: 1111.2353 | DOI | MR | Zbl

[6] Babenko I. K., “Periodic trajectories of three-dimensional Birkhoff billiards”, Math. USSR Sb., 71 (1992), 1–13 | DOI | MR | Zbl

[7] Bangert V., “On the existence of closed geodesics on two-spheres”, Internat. J. Math., 4 (1993), 1–10 | DOI | MR | Zbl

[8] Bao D., Chern S.-S., Shen Z., An introduction to Riemann–Finsler geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000 | DOI | MR | Zbl

[9] Berglund N., Kunz H., “Integrability and ergodicity of classical billiards in a magnetic field”, J. Statist. Phys., 83 (1996), 81–126, arXiv: chao-dyn/9501009 | DOI | MR | Zbl

[10] Bialy M., “On totally integrable magnetic billiards on constant curvature surface”, Electron. Res. Announc. Math. Sci., 19 (2012), 112–119, arXiv: 1208.2455 | DOI | MR | Zbl

[11] Birkhoff G. D., “On the periodic motions of dynamical systems”, Acta Math., 50 (1927), 359–379 | DOI | MR | Zbl

[12] Blagojević P.V.M., Lück W., Ziegler G. M., “Equivariant topology of configuration spaces”, J. Topol., 8 (2015), 414–456, arXiv: 1207.2852 | DOI | MR | Zbl

[13] Busemann H., “Problem IV: Desarguesian spaces”, Mathematical Developments Arising from Hilbert Problems (Northern Illinois Univ., De Kalb, Ill., 1974), Proc. Sympos. Pure Math., 1976, 131–141 | DOI | MR | Zbl

[14] Chern S.-S., “Finsler geometry is just Riemannian geometry without the quadratic restriction”, Notices Amer. Math. Soc., 43 (1996), 959–963 | MR | Zbl

[15] Chernov N., Markarian R., Chaotic billiards, Mathematical Surveys and Monographs, 127, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR | Zbl

[16] Cohen F. R., Lada T. J., May J. P., The homology of iterated loop spaces, Lecture Notes in Math., 533, Springer-Verlag, Berlin–New York, 1976 | DOI | MR | Zbl

[17] Cornea O., Lupton G., Oprea J., Tanré D., Lusternik–Schnirelmann category, Mathematical Surveys and Monographs, 103, Amer. Math. Soc., Providence, RI, 2003 | DOI | MR | Zbl

[18] Fadell E., Husseini S., “Category weight and Steenrod operations”, Bol. Soc. Mat. Mexicana, 37 (1992), 151–161 | MR | Zbl

[19] Fadell E., Husseini S., Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl

[20] Farber M., “Topology of billiard problems. II”, Duke Math. J., 115 (2002), 587–621, arXiv: math.AT/0006085 | DOI | MR | Zbl

[21] Farber M., Tabachnikov S., “Periodic trajectories in 3-dimensional convex billiards”, Manuscripta Math., 108 (2002), 431–437, arXiv: math.DG/0106049 | DOI | MR | Zbl

[22] Farber M., Tabachnikov S., “Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards”, Topology, 41 (2002), 553–589, arXiv: math.DG/9911226 | DOI | MR | Zbl

[23] Fenchel W., “On the differential geometry of closed space curves”, Bull. Amer. Math. Soc., 57 (1951), 44–54 | DOI | MR | Zbl

[24] Fomenko A., Fuchs D., Homotopical topology, Graduate Texts in Mathematics, 273, 2nd ed., Springer, Cham, 2016 | DOI | MR | Zbl

[25] Franks J., “Geodesics on $S^2$ and periodic points of annulus homeomorphisms”, Invent. Math., 108 (1992), 403–418 | DOI | MR | Zbl

[26] Gutkin B., “Hyperbolic magnetic billiards on surfaces of constant curvature”, Comm. Math. Phys., 217 (2001), 33–53, arXiv: nlin.CD/0003033 | DOI | MR | Zbl

[27] Gutkin E., Tabachnikov S., “Billiards in Finsler and Minkowski geometries”, J. Geom. Phys., 40 (2002), 277–301 | DOI | MR | Zbl

[28] Hatcher A., Algebraic topology, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[29] Karasev R. N., “Periodic billiard trajectories in smooth convex bodies”, Geom. Funct. Anal., 19 (2009), 423–428, arXiv: 0905.1761 | DOI | MR | Zbl

[30] Katok A. B., “Ergodic perturbations of degenerate integrable Hamiltonian systems”, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539–576 | MR

[31] Kozlov V. V., Treshchev D. V., Billiards. A genetic introduction to the dynamics of systems with impacts, Translations of Mathematical Monographs, 89, Amer. Math. Soc., Providence, RI, 1991 | DOI | MR | Zbl

[32] Laudenbach F., “A Morse complex on manifolds with boundary”, Geom. Dedicata, 153 (2011), 47–57, arXiv: 1003.5077 | DOI | MR | Zbl

[33] Mazzucchelli M., “On the multiplicity of non-iterated periodic billiard trajectories”, Pacific J. Math., 252 (2011), 181–205, arXiv: 1012.5593 | DOI | MR | Zbl

[34] McCleary J., A user's guide to spectral sequences, Cambridge Studies in Advanced Mathematics, 58, 2nd ed., Cambridge University Press, Cambridge, 2001 | MR | Zbl

[35] Robnik M., Berry M. V., “Classical billiards in magnetic fields”, J. Phys. A: Math. Gen., 18 (1985), 1361–1378 | DOI | MR | Zbl

[36] Roth F., “On the category of Euclidean configuration spaces and associated fibrations”, Groups, Homotopy and Configuration Spaces, Geom. Topol. Monogr., 13, Geom. Topol. Publ., Coventry, 2008, 447–461, arXiv: 0904.1013 | DOI | MR | Zbl

[37] Rudyak Yu.B., “On category weight and its applications”, Topology, 38 (1999), 37–55 | DOI | MR | Zbl

[38] Shen Y.-B., Shen Z., Introduction to modern Finsler geometry, World Scientific Publishing Co., Singapore, 2016 | DOI | MR | Zbl

[39] Strom J. A., Category weight and essential category weight, Ph.D. Thesis, The University of Wisconsin, USA, 1997 | MR | Zbl

[40] Tabachnikov S., Billiards, Panor. Synth., 1, 1995, vi+142 pp. | MR | Zbl

[41] Tabachnikov S., “Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem”, Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, 2004, 233–250, arXiv: math.DG/0302288 | MR | Zbl

[42] Tabachnikov S., Geometry and billiards, Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI, 2005 | DOI | MR | Zbl

[43] Tasnádi T., “The behavior of nearby trajectories in magnetic billiards”, J. Math. Phys., 37 (1996), 5577–5598 | DOI | MR | Zbl

[44] Zharnitsky V., “Invariant tori in Hamiltonian systems with impacts”, Comm. Math. Phys., 211 (2000), 289–302 | DOI | MR | Zbl