Convergence to a Lévy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps

Ian Melbourne, Paulo Varandas

Published 2018-09-18Version 1

We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. As far as we know, this is the first instance where the M_2 topology arises naturally. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.