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 Mosc. Math. J., 2014, Volume 14, Number 2, Pages 239–289 (Mi mmj522)

On quadrilateral orbits in complex algebraic planar billiards

Alexey Glutsyukabc

a CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon 07, France
b Laboratoire J.-V. Poncelet (UMI 2615 du CNRS and the Independent University of Moscow)
c National Research University Higher School of Economics, Russia

Abstract: The famous conjecture of V. Ya. Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of $4$-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we provide classification of the so-called real algebraic pseudo-billiards with open set of quadrilateral orbits: billiards formed by four real algebraic curves; the reflections allow to change the side with respect to the reflecting tangent line.

Key words and phrases: billiard, periodic orbit, complex algebraic curve, complex reflection law, complex Euclidean metric, isotropic line, complex confocal conics, birational transformation.

DOI: https://doi.org/10.17323/1609-4514-2014-14-2-239-289

Full text: http://www.mathjournals.org/.../2014-014-002-005.html
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Bibliographic databases:

MSC: 37C25, 37F05, 51N15, 14E15
Received: August 7, 2013; in revised form December 28, 2013
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Citation: Alexey Glutsyuk, “On quadrilateral orbits in complex algebraic planar billiards”, Mosc. Math. J., 14:2 (2014), 239–289

Citation in format AMSBIB
\Bibitem{Glu14} \by Alexey~Glutsyuk \paper On quadrilateral orbits in complex algebraic planar billiards \jour Mosc. Math.~J. \yr 2014 \vol 14 \issue 2 \pages 239--289 \mathnet{http://mi.mathnet.ru/mmj522} \crossref{https://doi.org/10.17323/1609-4514-2014-14-2-239-289} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3236494} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000342789300005} 

• http://mi.mathnet.ru/eng/mmj522
• http://mi.mathnet.ru/eng/mmj/v14/i2/p239

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Glutsyuk A., “on Odd-Periodic Orbits in Complex Planar Billiards”, J. Dyn. Control Syst., 20:3 (2014), 293–306
2. Jovanovic B., Jovanovic V., “Geodesic and Billiard Flows on Quadrics in Pseudo-Euclidean Spaces: l-a Pairs and Chasles Theorem”, Int. Math. Res. Notices, 2015, no. 15, 6618–6638
3. A. Glutsyuk, “On 4-reflective complex analytic planar billiards”, J. Geom. Anal., 27:1 (2017), 183–238
4. B. Jovanovic, V. Jovanovic, “Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5163–5190
5. B. Jovanović, V. Jovanovic, “Heisenberg Model in Pseudo-Euclidean Spaces II”, Regul. Chaotic Dyn., 23:4 (2018), 418–437
6. A. Glutsyuk, E. Shustin, “On polynomially integrable planar outer billiards and curves with symmetry property”, Math. Ann., 372:3-4 (2018), 1481–1501