On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

Abstract

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.