Integrable Outer billiards and rigidity

TL;DR

This paper introduces a new generating function for outer billiards and proves that near a smooth convex curve, the only invariant curves are ellipses, establishing a rigidity result using integral geometry and the Blaschke-Santalo inequality.

Contribution

The paper presents a novel generating function for outer billiards and proves a rigidity theorem characterizing ellipses as the only invariant curves near smooth convex boundaries.

Findings

Invariant curves near smooth convex curves are ellipses.

A new generating function for outer billiards is introduced.

The proof overcomes noncompactness using weighted integral geometry.

Abstract

In the present paper we introduce a new generating function for outer billiards in the plane. Using this generating function, we prove the following rigidity result: if the vicinity of the smooth convex plane curve $\gamma$ of positive curvature is foliated by continuous curves which are invariant under outer billiard map, then the curve $\gamma$ must be an ellipse. In addition to the new generating function used in the proof, we also overcome the noncompactness of the phase space by finding suitable weights in the integral-geometric part of the proof. Thus, we reduce the result to the Blaschke-Santalo inequality.