Starting the study of outer length billiards

Abstract

We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal Taylor expansion of the corresponding Mather's ${\hat{I}}^2$-function. Finally, we provide explicit Lazutkin coordinates up to order 4.