On algebraically integrable outer billiards

Abstract

We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse. In this note, an outer billiard table is a compact convex domain in the plane bounded by an oval (closed smooth strictly convex curve) C. Pick a point x outside of C. There are two tangent lines from x to C; choose one of them, say, the right one from the view-point of x, and reflect x in the tangency point. One obtains a new point, y, and the transformation T : x 7→ y is the outer (a.k.a. dual) billiard map. We refer to [3, 4, 5] for surveys of outer billiards. If C is an ellipse then the map T possesses a 1-parameter family of invariant curves, the homothetic ellipses; these invariant curves foliate the exterior of C. Conjecturally, if an outer neighborhood of an oval C is foliated by the invariant curves of the outer billiard map then C is an ellipse – this is an outer version of the famous Birkhoff conjecture concerning the conventional, inner billiards. In this note we show that ellipses are rigid in a much more restrictive sense of algebraically integrable outer billiards; see [2] for the case of inner billiards.