On rationally integrable projective billiards

Consider a billiard in a strictly convex planar domain bounded by a smooth curve. An oriented straight line intersecting the billiard is reflected at its last intersection point with the boundary according to the classical reflection law: the angle of incidence is equal to the angle of reflection.
A caustic of a convex billiard 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 Birkhoff Conjecture stated in late 1920-ths deals with those bounded planar strictly convex billiards that are Birkhoff integrable: admit a foliation by closed caustics in a neighborhood of the boundary from the inner side, with boundary being a leaf. It affirms that the only Birkhoff integrable billiards are ellipses and the caustics are confocal ellipses. This open conjecture was studied by many mathematicians. Recent substantial progress was obtained in joint papers by V.Kaloshin and A.Sorrentino [KS] and by M.Bialy and A.Mironov [BM]. For its survey see [KS1, BM, KS2].
In [T1] Sergei Tabachnikov introduced projective billiards, which generalize the usual billiards on standard surfaces of constant curvature: the Euclidean plane, the hyperbolic plane and the round sphere. A planar projective billiard is a planar curve $\gamma$ equipped with a transversal line field $\mathcal N$. A line intersecting $\gamma$ is reflected at their intersection point $P$ via the linear involution $\sigma_P:T_P\mathbb R^2\to T_P\mathbb R^2$ with invariant subspaces $T_P\gamma$ and $\mathcal N(P)$, acting trivially on $T_P\gamma$ and as central symmetry $v\mapsto-v$ on $\mathcal N(P)$. In [T2, p.103] Tabachnikov suggested a generalization of the Birkhoff Conjecture to strictly convex planar projective billiards, which implies its versions for billiards on surfaces of constant curvature and for Euclidean outer billiards.
Projective duality sends $\gamma$ and $\sigma_P$ to a curve $C$ equipped with a family of projective involutions acting on its projective tangent lines and fixing tangency points. Tabachnikov's Conjecture is stated in dual terms. Namely, consider a 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. Suppose that for every point $P\in C$ the latter involution is a projective transformation of the tangent line $L$. Tabachnikov's Conjecture affirms that under these assumptions the curve $C$ is an ellipse and the foliation in question 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 prove a local version: in the case when $C$ is a germ of real $C^4$-smooth (or holomorphic) planar curve and the germ of foliation admits a rational first integral. We prove that in this general case the curve $C$ is also a conic. But the leaves of the foliation may be higher degree algebraic curves. We give a complete classification of germs of foliations satisfying the conditions of local Tabachnikov's Conjecture and admitting rational first integrals, 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 get the descriptions of germs of projective billiards with $C^4$-smooth boundaries whose billiard flows admit $0$-homogeneous rational first integral 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.
[BM] M. Bialy, A. Mironov, The Birkhoff–Poritsky conjecture for centrally-symmetric billiard tables, Preprint https://arxiv.org/abs/2008.03566
[KS1] V. Kaloshin, A. Sorrentino, On local Birkhoff Conjecture for convex billiards, Annals of Mathematics, 188:1 (2018), 315–380.
[KS2] V. Kaloshin, A. Sorrentino, On the integrability of Birkhoff billiards, Philosophical Transactions of Royal Society A 376:2131 (2018), 20170419, 16 pp.
[P] H. Poritsky, The billiard ball problem on a table with a convex boundary—an illustrative dynamical problem, Annals of Mathematics 51:2 (1950), 446–470.
[T1] S. Tabachnikov, Introducing projective billiards, Ergodic Theory and Dynamical Systems, 17 (1997), 957–976.
[T2] S. Tabachnikov, On algebraically integrable outer billiards, Pacific Journal of Mathematics, 235:1 (2008), 101–104.