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On 4-Reflective Complex Analytic Planar Billiards

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Abstract

The famous conjecture of Ivrii (Funct Anal Appl 14(2):98–106, 1980) 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 its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the author’s previous result Glutsyuk (Moscow Math J 14(2):239–289, 2014) classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar \(C^4\)-smooth pseudo-billiards; solutions of Tabachnikov’s Commuting Billiard Conjecture and the 4-reflective case of Plakhov’s Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise \(C^4\)-smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

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Notes

  1. Everywhere in the paper by cusp we mean the singularity of an arbitrary irreducible singular germ of analytic curve, not necessarily the one given by equation \(x^2=y^3+\dots \) in appropriate coordinates.

  2. Real triangular spirals were introduced in [10, p. 320], where a real version of Proposition 2.19 was proved. Our proof of Proposition 2.19 is analogous to arguments from loc. cit.

  3. Recall that a mapping \(V\rightarrow W\) of complex manifolds (or analytic sets in complex manifolds) is meromorphic, if it is well-defined and holomorphic on an open and dense subset in V, and the closure of its graph is an analytic subset in \(V\times W\), see Convention 2.31. It is well-known that if W is compact and V is irreducible, then the set of indeterminacies of every meromorphic mapping \(V\rightarrow W\) is contained in the union of the singular set of V (which has codimension at least two in V, if V is normal) and an analytic subset in V of codimension at least two. A mapping is bimeromorphic, if it is meromorphic together with its inverse.

  4. Everywhere below for an analytic set M by \(M_{reg}\) (\(M_{sing}\)) we denote the set of its smooth (respectively, singular) points.

  5. Everywhere below by analytic variety we mean an analytic subset in a complex manifold.

  6. The edge lengths denoted by \(L_j\) in [10] are denoted here by \(l_j\).

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Acknowledgments

I am grateful to Yu.S. Ilyashenko, Yu.G. Kudryashov, A.Yu. Plakhov and S.L. Tabachnikov for attracting my attention to Ivrii’s Conjecture, invisibility and Commuting Billiard Conjecture. I wish to thank Yu.G.Kudryashov, to whom this paper owes much: some of his arguments from our joint paper [10] were used here in a crucial way. I am grateful to them and to E.M. Chirka, E. Ghys, A.L. Gorodentsev, Z. Hajto, B. Sevennec, V.V. Shevchishin, J.-C. Sikorav, M.S. Verbitsky, O.Ya. Viro, V. Zharnitsky for helpful discussions.

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Authors and Affiliations

  1. CNRS (UMR 5669 (UMPA, ENS de Lyon), UMI 2615 (Lab.J.-V.Poncelet)). Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364, Lyon 07, France

    Alexey Glutsyuk

  2. National Research University Higher School of Economics (HSE), Moscow, Russia

    Alexey Glutsyuk

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  1. Alexey Glutsyuk
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Correspondence to Alexey Glutsyuk.

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Research supported by part by RFBR Grants 10-01-00739-a, 13-01-00969-a, 16-01-00748, 16-01-00766 and NTsNIL_a 10-01-93115 (RFBR-CNRS grant), by ANR grant ANR-13-JS01-0010.

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Glutsyuk, A. On 4-Reflective Complex Analytic Planar Billiards. J Geom Anal 27, 183–238 (2017). https://doi.org/10.1007/s12220-016-9679-x

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