Necessary and sufficient condition for M-2-convergence to a Levy process for billiards with cusps at flat points

Abstract

We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding Lévy process in the M1 and M2 Skorohod topologies has been studied in recent work. Here, we show that certain sufficient conditions for M2-convergence are also necessary.