Abstract:

Consider a strictly convex planar billiard bounded by a smooth closed curve. Its caustic is a curve $C$ whose tangent lines are reflected by the billiard to its tangent lines (e.g., a confocal ellipse in an elliptic billiard). The famous open Birkhoff Conjecture (late 1920-ths) deals with Birkhoff integrable billiards, for which there exists a foliation by closed caustics in a neighborhood of the boundary from the inner side, with boundary being a leaf. It states that the only Birkhoff integrable billiards are ellipses, and the caustics are confocal ellipses. It was studied by many mathematicians. Recent substantial progress was obtained by V. Kaloshin and A. Sorrentino [2], and by M. Bialy and A. Mironov [1].
For its survey see [2,1,3]. In [4, p. 103] Tabachnikov suggested a generalization of the Birkhoff Conjecture to planar projective billiards, which implies its versions for billiards on surfaces of constant curvature and for Euclidean outer billiards. Tabachnikov's Conjecture is stated in dual terms. Consider a \emph{planar strictly convex closed curve $C$ and a foliation by closed curves of its neighborhood on the concave side}, with $C$ being its leaf. For every projective line $L$ tangent to $C$ at a point $P$ consider the germ at $P$ of involution of the line $L$ fixing $P$ and permuting its intersection points with each individual leaf of the foliation. Let for every point $P\in C $ the latter involution be a  projective transformation $L\to L$. Tabachnikov's Conjecture affirms that under these assumptions the curve $C$ is an ellipse and the foliation is a pencil of conics.

In the talk we present a proof of the rational version of the Tabachnikov's Conjecture for $C^4$-smooth curves: the positive answer under the additional assumption that the foliation admits a rational first integral.

We also treat the local real and complex cases, when $C$ is a germ of curve and the germ of foliation admits a rational first integral. We prove that the curve $C$ is a conic, and we give a complete classification of the latter germs of foliations up to projective transformation. Their list includes:
- pencils of conics;
- two infinite series of exotic examples, with higher degree leaves;
- two real examples with leaves of degree four;
- two real examples with leaves of degrees six.

As an application, we describe projective billiards with $C^4$-smooth boundaries whose flows admit $0$-homogeneous rational first integrals in velocity.

The author is partially supported by Laboratory of Dynamical Systems and Applications, HSE University, of the Ministry of science and higher education of the RF grant ag.~No 075-15-2019-1931. Research was also partially supported by RFBR grants 16-01-00748, 16-01-00766 and 19-51-50005 JF a. This material is partly based upon work supported by the National Science Foundation under Grant No.~1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, USA, during the period August-September 2018.

References:
    
[1] M. Bialy, A. Mironov, The Birkhoff--Poritsky conjecture for centrally-symmetric billiard tables, preprint https://arxiv.org/abs/2008.03566
[2] V. Kaloshin, A. Sorrentino, On local Birkhoff Conjecture for convex billiards, Annals of Mathematics, 188:1 (2018), 315–380.
[3] V. Kaloshin, A. Sorrentino, On the integrability of Birkhoff billiards, Philosophical Transactions of Royal Society A, 376:2131 (2018), 20170419, 16 pp.
[4] S. Tabachnikov, On algebraically integrable outer billiards, Pacific Journal of Mathematics, 235:1 (2008), 101–104.